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Faculty of MathematicsUniversity “Al. I. Cuza” of Iasi
Institute “Octav Mayer” of theRomanian Academy, Iasi, Romania
International Conference on Applied Analysis
and Differential Equations
Iasi, Romania
September 4–9, 2006
ABSTRACTS
Invited Speakers
Variational Methods in Metric Regularity and ImplicitMultifunction Theorems
D. AZE
UMR CNRS MIP, Universite Paul Sabatier118 Route de Narbonne, 31062 Toulouse cedex 4, France
Relying on the notion of strong slope introduced by De Giorgi, Marino and Tosquesin the eighties, we develop a general framework leading to several characterizations ofmetric regularity of multifunctions in the complete metric space setting. We focus ourattention on parametric metric regularity, leading to implicit multifunction theorems.We give several results on existence and regularity of the implicit multifunctions, and,surveying some recent results on this topic, we show how they can be derived fromour general method. We also give an exact formula for the derivative of the implicitmultifunction. At last, some applications are given to differential inclusions and tostability in nonsmooth analysis.
Stochastic Taylor expansion and stochastic viscosity solutionsfor nonlinear SPDEs
Rainer BUCKDAHN
Universite de Bretagne Occidentale, UFR Sciences et TechniquesLaboratoire de Mathematiques, CNRS - UMR 6204
In an earlier work R.Buckdahn and J.Ma (2001) introduced a notion of stochasticviscosity solution, inspired by earlier results of P.L.Lions and P.E.Souganidis (1998). Byusing a Doss-Sussmann-type transformation and the so-called backward doubly stochas-tic differential equations (BDSDEs) introduced by E.Pardoux and S.Peng (1994), theyestablished the existence and uniqueness of stochastic viscosity solution to the stochasticpartial differential equation (SPDE)
u(t, x) = u(0, x) +∫ t
0(Au + f(x, u, σ∗∇u) (t, x)dt +
∫ t
0g(t, x, u(t, x))dBt,
where B is a Brownian motion and A is the generator of a diffusion process. Ourcontribution extends the study of stochastic viscosity solutions to the class of SPDEs
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for which the dependence of the diffusion coefficient g on the solution u is replaced bythat on its the spatial derivative ∇u .
Our main tool is a new type of stochastic time-space Taylor expansion for Ito randomfields which holds outside some universal null set N for every random expansion point. Itgeneralizes the work on stochastic Taylor development by R. Buckdahn and J. Ma (2002)The second principal tool is the recently developed theory on BDSDEs (E. Pardoux,S. Peng, 1994). It is mainly used to prove existence of the stochastic viscosity solution.
The talk is based on a common with with I. Bulla (Universite de Brest, France) andJ. Ma (Purdue University, U.S.A).
Radius Theorems and Conditioning
A. L. DONTCHEV
Mathematical Reviews, Ann Arbor, Michigan 48107-8604, [email protected]
As known from linear algebra, the distance from a nonsingular matrix A to the set ofsingular matrices is equal to the reciprocal of the norm of the inverse of A. The standardcondition number of a matrix is then just the normalized reciprocal to the distance tosingularity. This result is sometimes called the Eckart–Young theorem but the actualpaper by Eckart and Young from 1936 is only indirectly related to the issue. In this talkwe will take a quick tour through basic ideas that lead to expressions for the distancesto irregularity of mappings. Then we outline generalizations in various directions andpose some open problems.
On weak solutions of BSDEs
Hans-Jurgen ENGELBERT
Institute for Stochastics, Friedrich Schiller-UniversityJena, GERMANY
This talk is based on joint work with R. Buckdahn (Brest, France) and A. Rascanu(Iasi, Romania) (cf. [1]–[4]). The main objective consists in introducing and discussingthe concept of weak solutions of certain backward stochastic differential equations (BS-DEs):
Yt = E[H(X) +
∫ T
t
f(s,X, Y )ds
∣∣∣∣Ft
], t ∈ [0, T ]. (1)
condition H depends in functional form on a driving cadlag process X, and the coefficientf (called the generator or driver of the BSDE) depends on time t and, in functionalform, on X and the solution process Y . The functional f(t, x, y), (t, x, y) ∈ [0, T ] ×D
([0, T ]; Rd+p
), is assumed to be bounded and continuous in (x, y) on the Skorohod
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space D([0, T ]; Rd+p
)in the Meyer–Zheng topology. Using weak convergence in the
Meyer–Zheng topology, we shall give a general result on the existence of a weak solutionY , with driving process X admitting a given distribution, defined on some filteredprobability space (Ω,F , P ;F). By examples, we can show that there are, indeed, weaksolutions which are not strong, i.e., are not solutions in the usual sense adapted to thefiltration FX generated by X. We will also discuss pathwise uniqueness and uniquenessin law of the solution and conclude, similar to the Yamada–Watanabe theorem, thatpathwise uniqueness and weak existence ensure the existence of a (uniquely determined)strong solution. Applying these concepts, finding a unique strong solution is divided intotwo subtasks: To prove pathwise uniqueness and to prove weak existence for BSDE (1).It turns out that pathwise uniqueness holds whenever every weak solution of BSDE (1)has a.s. continuous paths, and this condition is even necessary if the driving process Xis a continuous local martingale satisfying the previsible representation property.
References:
[1] Buckdahn, R.; Engelbert, H.-J.; Rascanu, A., On weak solutions of backwardstochastic differential equations, Theory Probab. and its Appl. Vol. 49, No.1, 70–108(2004).
[2] Buckdahn, R.; Engelbert, H.-J., A backward stochastic differential equationswithout strong solution, Theory Probab. and its Appl. Vol. 50, No.2 (2005).
[3] Buckdahn, R.; Engelbert, H.-J., On the notion of weak solutions of backwardstochastic differential equations, pp. 21, to appear in: Proceedings of the Fourth Collo-quium on Backward Stochastic Differential Equations and Their Applications , Shanghai,P.R. China, May 29 – June 1, 2005.
[4] Buckdahn, R.; Engelbert, H.-J., On the continuity of weak solutions of backwardstochastic differential equations, manuscript, pp. 13, submitted.
Regular and singular optimal controls
H. O. FATTORINI
University of California, Department of MathematicsLos Angeles, California 90095-1555
Let E be a Banach space, S(t) a strongly continuous semigroup in E. We dealmostly with the time optimal control problem for
y′(t) = Ay(t) + u(t), y(0) = ζ (1)
with point target condition y(T ) = y and control constraint ‖u(t)‖ = 1 a. e. Amultiplier space Z is a space Z ⊃ E∗ such that S(t)∗Z ⊆ E∗(t > 0). Pontryagin’smaximum principle for a control u(t) in 0 ≤ t ≤ T is
〈S(T − t)∗z, u(t)) = max‖u‖≤1
〈S(T − t)∗z, u〉 a.e. in 0 = t = T (2)
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with z in some multiplier space. Time optimal controls for (1) are divided into fiveclasses determined by their adherence to the maximum principle (or lack thereof). Acontrol u(t) satisfying (2) with z ∈ E∗ is strongly regular. If z belongs to the space ofmultipliers Z(T ) ⊃ E∗ characterized by
∫ T
0
‖S(t)∗z‖dt < 8
the control is regular. If z belongs to an arbitrary multiplier space, the control is weaklysingular. If u(t) does not satisfy (2) for z in any multiplier space then u(t) is singular.Finally, a control not satisfying (2) in any subinterval [a, b] ⊆ [0, T ] is hypersingular.Enough examples are known to show that each class is nonempty and there are alsoresults relating time optimality and membership in various classes. For instance: (a)if u(t) is time optimal and y ∈ D(A) then u(t) is regular, (b) regularity implies timeoptimality, (c) every time optimal control is weakly singular if E is a Hilbert space andS(t) is analytic, (d) weak singularity does not imply time optimality, although there aretime optimal controls that are only weakly singular (not regular). This talk will be onwhat is known (and also what is not known) about each class of controls and on theirrelations. For instance, it is known that the condition y ∈ D(A) on the target does notguarantee strong regularity, but the only available counterexample lives in a nonsmoothspace E (the sequence space `1) while it may be true that y ∈ D(A) implies strongregularity for a self adjoint infinitesimal generator A in a Hilbert space.
Analyticity of stable invariant manifold and stabilizationproblem for semilinear parabolic equation
A. V. FURSIKOV
Department of Mechanics and MathematicsMoscow State University, 119992 Moscow, RUSSIA
In a bounded domain Ω ⊂ Rd, d ≤ 3 we consider semilinear parabolic equation
∂ty(t, x)−∆y + f(y) = h(x), y|∂Ω = 0, y|t=0 = y0(x) (1)
with analytic function f(y) =∑∞
k=1 fkyk where |fk| ≤ γ0ρ
k0 and h(x) ∈ L2(Ω). We
consider (1) in phase space V = H2(Ω) ∩H10 (Ω). Let y(x) ∈ V be an unstable steady-
state solution of (1). Then orthogonal decomposition V = V+ ⊕ V− holds where finite-dimensional subspace V+ is generated by eigenvectors of the operator Az = −∆z(x) +f ′(y(x))z(x), z∂Ω = 0 with non negative eigenvalues and V− is generated by eigenvectorswith negative eigenvalues.
Denote resolving operator of (1) by St(y0) : St(y0) ≡ y(t). Recall that stableinvariant manifold M−(y) of (1) is the manifold defined in an neigborhood of y such thatfor y0 ∈ M−(y) inclusion St(y0) ∈ M−(y) is true for all t > 0, and ‖St(y0)− y‖V ≤ ce−αt
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as t → ∞ with some positive c, α. It is known that M−(y) = y + y− + F (y−), y− ∈O(V−) whereO(V−) is a neigborhood of origin in V− and F : O(V−) → V+ is a nonlinearoperator.
Theorem 1 The operator F (y−) is analytic: F (y−) =∑∞
r=2 Fr(y−, . . . , y−), whereFr(y1, . . . , yr) : V− × · · · × V− → V+ are r-linear operators, and ‖Fr‖ ≤ γρr withγ > 0, ρ > 0 depending on γ0, ρ0.
Theorem 1 has been applied for numerical stabilization of the semilinear equationfrom (1) with help of approach proposed in [1]. This stabilization is local since it wasmade under addition assumption that ‖y − y0‖ is small enough. In the case of one-dimensional Ω Theorem 1 has been obtained in [2].
The following result can be applied in the case of unlocal stabilization problem forequation from (1).
Theorem 2 For each R > 0 there exist K > 0 and subspace VK ⊂ V− of codimensionK such that restriction of F (y−) on VK can be extended analytically on the ball y− ∈VK : ‖y−‖V− ≤ RReferences:
[1] A.V. Fursikov, Stabilizability of quasilinear parabolic equation by feedback bound-ary control, Matematicheskii Sbornik 192:4 (2001), 115-160 (in Russian); Sbornik:Mathematics, 192:4 (2001),593-639.
[2] A.V. Fursikov, Analyticity of stable invariant manifold of 1D-semilinear parabolicequation. Proceedings of Joint Summer Research Conference on Control Methods inPDE-Dynamical Systems. AMS (to appear)
The Complex Dynamics of Age-Structured Populations
Mimmo IANNELLI
Mathematics Department, University of [email protected]
Modeling of age-structured populations is governed by the Gurtin–MacCamy sys-tem that takes into account age-structure and nonlinear effects on the vital rates. Theframework of this problem allows to treat ecological mechanisms for the intra-specificinteraction such as juvenile-adult competition, Allee effect, cannibalism. It is knownthat the models that take into account age-structure are related to delay equations(distributed delay, but also concentrated delay) and that stability of steady states isgoverned by transcendental characteristic equations that are not easy to analyze ana-lytically. Thus numerical methods for the analysis of such characteristic equations area powerful tool for exploring the behavior of the models, tracing asymptotical stability,Hopf bifurcations and possible chaos. Recent numerical approaches for characteristicroots of delay differential are based on the discretization of either the associated solu-tion operator semigroup or its infinitesimal generator whose spectra are related to thecharacteristic roots. The idea is to turn the characteristic roots approximation prob-lem into a corresponding eigenvalue problem for a suitable matrix. This approach has
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been applied to the specific case of the Gurtin–MacCamy model in order to exploreits behaviour versus some significant parameters. In this talk we present the complexdynamics of the Gurtin–MacCamy model, the numerical method set up to locate thecharacteristic roots, the results we can draw on the behaviour of specific models.
Variational analysis of evolution and partial differentialinclusions
Boris S. MORDUKHOVICH
Department of Mathematics, Wayne State UniversityDetroit, Michigan 48202, USA
This talk is devoted to optimal control problems governed by evolution/differentialinclusions in infinite-dimensional spaces and also by semilinear partial differential in-clusions. We pursue a twofold goal: to develop the method of discrete approximationsfor such problems and to derive necessary optimality conditions of the Euler-Lagrangetype under natural assumptions. First we consider a generalized Bolza problems forinfinite-dimensional differential inclusions with endpoint constraints. One of the princi-pal differences between finite-dimensional and infinite-dimensional dynamic systems isthe lack of compactness in infinite dimensions. Constructing well-posed discrete approx-imations and using advanced tools of variational analysis and generalized differentiation,we derive necessary conditions for discrete-time problems and then, by passing to thelimit, for continuous-time evolution inclusions. A similar procedure is developed for op-timal control problems of the Mayer type governed by constrained semilinear inclusionswith unbounded operators generating compact semigroups. This particularly coversparabolic partial differential inclusions whose solutions are understood in the conven-tional mild sense.
Slant Differentiability and Semismooth Methods for OperatorEquations
Zuhair NASHED
Department of Mathematics, University of Central FloridaOrlando, FL 32816, [email protected]
Let X and Y be Banach spaces, and F : D ⊂ X → Y be a continuous mapping on anopen domain D. The following concepts of slant differentiability and slanting functionwere introduced in [1]. A function F : D ⊂ X → Y is said to be slantly differentiableat x ∈ D if there exist a mapping f : D → L(X, Y ) such that the family f (x + h)
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of bounded linear operators is uniformly bounded in the operator norm for h sufficientlysmall and
limh→0
F (x + h)− F (x)− f 0(x + h)h
‖ h ‖ = 0.
The function f is called a slanting function for F at x. A function F : D ⊂X → Y is said to be slantly differentiable in an open domain D0 ⊂ D if there existsa mapping f 0 : D → L(X, Y ) such that f is a slanting function for F at every pointx ∈ D0. In this case, f is called a slanting function for F in D0.
In this talk I will discuss operator-theoretic aspects of slant differentiability and re-lated variants. Application to Newton-like methods, optimal control theory, and nonlin-ear ill-posed problems will be indicated. A unifying framework for semismooth analysiswill be sketched and compared with the setting in [2] for smooth analysis.
References:
[1] X. Chen, Z. Nashed, and L.Qi, Smoothing methods and semismooth methodsfor nondifferentiable operator equations, SIAM J. Numer. Anal. 38 (2000), no. 4,1200-1216.
[2] M. Z. Nashed, Differentiability and related properties of nonlinear operators:Some aspects of the role of differentials in nonlinear functional analysis, in L.B. Rall,ed., ”Nonlinear Functional Analysis and Applications”, Academic Press, New York,1971, pp. 103-309.
Homogenization of periodic linear degenerate PDEs
Etienne PARDOUX
Universite de Provence, [email protected]
Our goal is to study, by a probabilistic method, the limit as ε → 0 of the solutionuε(t, x) of an elliptic PDE in the regular bounded domain D ⊂ Rd
Lεu
ε(x) + f(x,
x
ε
)uε(x) = 0, x ∈ D,
uε(x) = g(x), x ∈ ∂D,
where f is bounded from above, and g is continuous, as well as the limit of uε(t, x), thesolution of a parabolic PDE of the form
∂uε(t, x)
∂t= Lεu
ε(t, x) +
(1
εe(
x
ε) + f(x,
x
ε)
)uε(t, x)
uε(0, x) = g(x), x ∈ Rd,
where
Lε =1
2
d∑i,j=1
aij(x
ε)
∂2
∂xi∂xj
+d∑
i=1
(1
εbi(
x
ε) + ci(
x
ε)
)∂
∂xi
.
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Homogenization, in particular in the periodic case, is an old problem. We refer to [3]and [8] for many results in this field, to [7] for the first presentation of the probabilisticapproach to this problem (which will be our point of view), and to [9] for the modernPDE approach to homogenization.
The novelty of our result lies in the fact that we allow the matrix a to degenerate(and even possibly to vanish) in some open subset D of Rd. There is by now quite a vastliterature concerning the homogenization of second order elliptic and parabolic PDEswith a possibly degenerating matrix of second order coefficients a, see among others [1],[2], [4], [6], [13]. But, as far as we know, in all of these works, either the coefficient a isallowed to degenerate in certain directions only, or else it may vanish on sets of measurezero only. It seems that our paper presents the first result where the matrix a is allowedto vanish on an open set.
References:
[1] R. De Arcangelis, F. Serra Cassano, On the homogenization of degenerate ellipticequations in divergence form, J. Math. Pures Appl. 71, 119–138, 1992.
[2] M. Bellieud, G. Bouchitte, Homogeneisation de problemes elliptiques degeneres,CRAS Ser. I Math. 327, 787–792, 1998.
[3] A. Bensoussan, J.L. Lions, G. Papanicolaou, Asymptotic analysis for periodicstructures, Studies in Mathematics and its Applications, 5, North-Holland PublishingCo., Amsterdam-New York, 1978.
[4] M. Biroli, U. Mosco, N. Tchou, Homogenization for degenerate operators withperiodical coefficients with respect to the Heisenberg group, CRAS Ser. I Math. 322,439–44, 1996.
[5] A. Diedhiou, E. Pardoux, Homogenization of semilinear hypoelliptic PDEs, Preprint.[6] J. Engstrom, L.–E. Persson, A. Piatnitski, P. Wall, Homogenization of random
degenerated nonlinear monotone operators, Preprint.[7] M.I. Freidlin, The Dirichlet problem for an equation with periodic coefficients
depending on a small parameter. (Russian) Teor. Verojatnost. i Primenen. 9, 133–139,1964.
[8] V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of differential operatorsand integral functionals, Springer Verlag, 1994.
[9] F. Murat, L. Tartar, Calculus of variations and homogenization, in Topics in themathematical modelling of composite materials, 139–173, Progr. Nonlinear DifferentialEquations Appl., 31, Birkhauser Boston, Boston, MA, 1997.
[10] E. Pardoux, Homogenization of Linear and Semilinear Second Order ParabolicPDEs with Periodic Coefficients: A Probabilistic Approach, Journal of Functional Anal-ysis 167, 498-520, 1999.
[11] E. Pardoux, A. Yu. Veretennikov, On Poisson equation and diffusion approxi-mation 1, Ann. Probab. 29, 1061-1085, 2001.
[12] E. Pardoux, A. Yu. Veretennikov, On Poisson equation and diffusion approxi-mation 3, Ann. Probab., to appear.
[13] F. Paronetto, Homogenization of degenerate elliptic–parabolic equations, Asymp-tot. Anal. 37, 21–56, 2004.
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Viability with probabilistic knowledge of initial condition,application to differential games
Marc QUINCAMPOIX
Departement de Mathematiques, Universite de Bretagne Occidentale6 Avenue Victor Le Gorgeu F-29200 Brest, France
We consider a deterministic control system with state constraints. The main speci-ficity of the question we address now concerns with the case where the state-space isonly unperfectly known by the controller: Instead of knowing the initial condition , heknow only the probability measure of the initial condition. The problem is then reducedto a new problem where the state variable appears to be be a measure. We provide aresult characterizing the compatibility of the constraints and the control systems withprobabilistic knowledge of the state space.
Then we use our approach to characterize the value function of an differential gamewith probabilistic knowledge of initial condition in term of the unique solution of a suit-ably defined Hamilton Jacobi Isaacs equation (written on the set of measures). Thiscreates some difficulties mainly because the set of measure on RN is not finite dimen-sional and it is not a normed space: we will introduce and use the so-called Wassersteindistance between probability measures.Nevertheless, we give a new result of existenceof a value for a differential game with unperfect information.
Elliptic and parabolic PDE for measures
Michael G. ROECKNER
Fakultat fur Mathematik, Universitat BielefeldPostfach 100131, D-33501 Bielefeld, Germany
Invariant measures and transition probabilities of continuous stochastic processessatisfy second order PDE of elliptic and parabolic type respectively, however, with co-efficients of possibly low regularity. This motivates the study of such equations formeasures from a purely analytic point of view. In the first part of the talk we shallstart with reviewing existence, uniqueness and local regularity results in the ellipticcase. In particular, the measures solving the PDE (essentially) always have densitieswith respect to Lebesgue measure. Subsequently, we shall present some recent globalregularity results for these densities as well as conditions implying that they decay poly-nomially or exponentially at infinity. In the second part of the talk we shall pass to theparabolic case. Here the solutions will be measures on space time. Results on existenceand local regularity are quite similar to the elliptic case. Results on uniqueness andglobal regularity have been established only very recently and are quite different fromthose in the elliptic case. Finally, it should be mentioned that though in this talk only
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the finite dimensional case is discussed, the same circle of problems is being analyzed ininfinite dimensions and some of the above results have also been proved there.
A continuation method for for a class of periodic evolutionvariational inequalities
Michel THERA
XLIM (UMR-CNRS 6172), Universite de Limoges, [email protected]
In this presentation we will give new existence results for finite variational inequalitiesand we will derive the existence of solutions of periodic solutions for a class of evolutionvariational inequalities.
Optimization of fundamental mechanical structures
Dan TIBA
Institute of Mathematics, [email protected]
We study structures like beams, plates, arches, curved rods and shells. For shellsand curved rods, we introduce new models of generalized Naghdi type and of asymptotictype.
The optimization problems concern the thickness for beams and plates or the geom-etry for arches, curved rods and shells. They enter the class of control by coefficientsproblems, known for their high nonconvexity and their stiff character.
Our approach allows the relaxation of the regularity hypotheses on the geometry andgives a partial answer for the “locking problem” in the case of arches and curved rods.We also present numerical examples with have a clear physical interpretation, whichvalidates our methods and results.
Theoretical approaches and numerical implementations ofquantum control
Gabriel TURINICI
CEREMADE, Universite Paris DauphinePlace du Marechal De Lattre De Tassigny, 75775 PARIS CEDEX 16, FRANCE
The control of quantum phenomena is being increasingly explored, both theoreticallyand in the laboratory and has reached a development allowing to consider practically
10
relevant circumstances : selective dissociation, creation of particular molecular states,remote detection, high frequency laser generation, etc.
These models are expressed in terms of equations that are bi-linear (the control mul-tiplies the state). After an initial presentation of the applications and mathematicalimplications of the experimental settings, we will discuss in this talk two types of re-sults: ensemble controllability and numerical algorithms. While in the first, geometricalLie group controllability theory is used, in the second we will deal with numerical al-gorithms and their relationship with the designing of optimal control cost functionals.Relationships with both theoretical works on PDE control and experimental algorithmswill also be presented.
Local uniform linear openness of multifunctions and calculus ofBouligand–Severi tangent sets
Corneliu URSESCU
“Octav Mayer” Institute of Mathematics, Romanian Academy, Iasi Branch8 Carol I blvd., 700506 Iasi, Romania
Let X and Y be linear spaces, and let F : X → Y be a multifunction. For everytangency concept τ and for every point (a, b) ∈ X × Y the equality
graph(τF (a, b)) = τgraph(F )(a, b)
defines a multifunction τF (a, b) : X → Y , and moreover, the equality
τF−1(b)(a) = (τF (a, b))−1(0)
is expected. Here, F−1 : Y → X stands for the inverse of the multifunction F . Inthe following we consider the specific tangency concept K which originates from thetangent half-lines considered by Bouligand (1931) and Severi (1931). The multifunctionKF (a, b) was introduced by Aubin (1981). Because of the elementary inclusion
KF−1(b)(a) ⊆ (KF (a, b))−1(0),
the equalityKF−1(b)(a) = (KF (a, b))−1(0)
is strongly expected. This equality is established at all points (a, b) ∈ graph(F ) assuminglocal uniform linear openness of F . Local uniform linear openness of F is investigatedthrough a metric result concerning local uniform ω−openness of F . It is also establishedthe equivalence between local uniform ω−openness of F and local metric regularity ofF .
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Short Communications
Internal stabilization of nonnegative solutions of parabolicsystems
Sebastian ANITA
Faculty of Mathematics, “Al. I. Cuza” University of Iasi“O. Mayer” Mathematics Institute of the Romanian Academy, Iasi, Romania
We investigate the internal stabilization of nonnegative solutions of a linear parabolicequation. We find a necessary condition and a sufficient condition for the zero-stabilizabi-lity of the nonnegative solution with respect to the value of the principal eigenvalue ofa certain elliptic operator. The value of this principal eigenvalue is strongly related tothe convergence rate of the nonnegative solution. We estimate this principal eigenvalueas a function of the shape of the domain and the support of the control. The approachfrom the case of the parabolic equation is used also in the case of a system of reaction-diffusion type with two components. This models the dynamic of prey-predator typesystem. Several stabilization strategies are compared.
Influence of variable permeability on vortex instability of ahorizontal non-darcy free or mixed convection flow in a
saturated porous medium
A.A. BAKER, A.M. ELAIW and F.S. IBRAHIM
Department of Mathematics, Faculty of ScienceAl-Azhar University, Assiut, EGYPT
Department of Mathematics, Faculty of ScienceAssiut University, Assiut, EGYPT
A linear stability theory is used to analyze the vortex instability of free or mixedconvection boundary layer flow in a saturated porous medium. The non-Darcian effects,which include the inertia force and surface mass flux are examined. The variation ofpermeability in the vicinity of the solid boundary is approximated by an exponentialfunction. The variation rate itself depends slowly on the streamwise coordinate, such asto allow the problem to possess a set of solutions, invariant under a group of transfor-mations. Velocity and temperature profiles as well as local Nusselt number for the baseflow are presented for the uniform permeability UP and variable permeability VP cases.An implicit finite difference method is used to solve the base flow and the resulting
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eigenvalue problems are solved numerically. The critical Peclet and Rayleigh numbersand the associated wave numbers for both UP and VP cases are obtained. The re-sults indicate that the inertial coefficients reduces the heat transfer rate and destabilizesthe flow to the vortex mode of disturbance. Moreover, the variable permeability effecttends to increase the heat transfer rate and destabilize the flow to the vortex mode ofdisturbance.
Endpoint Strichartz estimates in 3d for nonselfadjointSchrodinger operators
Marius BECEANU
Department of Mathematics, University of Chicago5734 S. University Ave., Chicago, IL 60637
Endpoint Strichartz estimates in R3 are obtained for nonselfadjoint Schrodinger op-erators.
Consider operators in R3 of the form H = H0 + V , where
H0 =
( −∆ + µ 00 ∆− µ
), V =
( −U −WW U
).
We assume that U and W are real-valued and have a suitable amount of polynomialdecay: |U |+ |W | ≤ C〈x〉−7/2−, where 〈x〉 = 1 + |x|.
The operator H has σ(H) ⊂ R∪ iR and σess(H) = (−∞,−µ]∪ [µ,∞). We make thespectral assumption that H has no eigenvalues in the set (−∞,−µ) ∪ (µ,∞) and thatthe thresholds ±µ are also regular, meaning that I +(H0−µ± i0)−1V : L2,−1− → L2,−1−
is invertible. Then σc(H) = σess(H).Let Pc = 1−Ppp be the Riesz projection on the continuous spectrum of the operator
H. Under these conditions, the following estimates hold for all admissible pairs (q, r)and (q′, r′), where 2 ≤ q, r ≤ ∞, 1 ≤ q′, r′ ≤ 2, 1
q+ 3
2r= 3
4, and 1
q′ + 32r′ = 7
4:
‖eitHPcf‖Lqt Lr
x≤ ‖f‖2,
∥∥∥∥∫
e−isHPcF (s)ds
∥∥∥∥2
≤ ‖F‖Lq′
t Lr′x,
∥∥∥∥∫
s<t
eitHe−isH∗PcFds
∥∥∥∥Lq
t Lrx
≤ ‖F‖Lq′
t Lr′x,
∥∥∥∥∫
s<t
ei(t−s)HPcFds
∥∥∥∥Lq
t Lrx
≤ ‖F‖Lq′
t Lr′x.
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Boundary value problems with non-surjective φ-Laplacian andone-sided bounded nonlinearity
Cristian BEREANU
Departement de Mathematique, Universite Catholique de LouvainB-1348 Louvain-la-Neuve, Belgium
Using Leray-Schauder degree theory we obtain various existence results for nonlinearboundary value problems
(φ(u′))′ = f(t, u, u′), l(u, u′) = 0
where l(u, u′) = 0 denotes the periodic, Neumann or Dirichlet boundary conditions on[0, T ], φ : R→ ]− a, a[ is a homeomorphism, φ(0) = 0.This is a joint work with Prof. Dr. Jean Mawhin.
Markov processes associated with Lp-resolvents andapplications to stochastic differential equations on Hilbert
space
Lucian BEZNEA
“Simion Stoilow” Institute of Mathematics of the Romanian AcademyP.O. Box 1-764, RO-014700, Bucharest, Romania
The talk is based on joint works with Nicu Boboc and Michael Rockner.We give general conditions on a generator of a C0-semigroup (resp. of a C0-resolvent)
on Lp(E, µ), p ≥ 1, where E is an arbitrary (Lusin) topological space and µ a σ-finitemeasure on its Borel σ-algebra, so that it generates a sufficiently regular Markov processon E. We present a general method how these conditions can be checked in manysituations. Applications to solve stochastic differential equations on Hilbert space inthe sense of a martingale problem are given.
On the use of Korn’s type inequalities in the existence theoryfor Cosserat elastic surfaces with voids
Mircea BIRSAN
University “A.I. Cuza” Iasi, Faculty of MathematicsBd. Carol I, no. 11, 700506, Iasi, Romania
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The theory of Cosserat surfaces is a direct approach to the mechanics of thin elasticshells in which the shell-like body is modeled as a two-dimensional continuum endowedwith a deformable vector assigned to every point of the surface. A detailed analysisof the theory of Cosserat shells is included in the classical monograph by Naghdi [1]and in the more recent book of Rubin [2]. The differential equations governing thedeformation of porous Cosserat shells have been presented by Birsan [3]. The porosityof the material is described using the Nunziato-Cowin theory for elastic media withvoids. In the present paper, we investigate the existence of solutions to the boundaryvalue problems associated to the deformation of Cosserat shells with voids. We establishfirst the inequalities of Korn’s type which are valid for Cosserat surfaces and employthem to show the existence of solution to the variational equations in elastostatics.The proof of the Korn’s type inequality rely in a crucial manner on a lemma of J.L.Lions [4]. We employ a method similar to that used in the classical linear shell theory byCiarlet [5]. For the dynamic problems, we show the existence, uniqueness and continuousdependence of solution to the boundary-initial-value problem using the results of thesemigroup of linear operators theory.
References:
1. Naghdi, P. M.: The Theory of Shells and Plates. In: Handbuch der Physik, Vol.VI a/2, pp. 425-640, Springer-Verlag, Berlin Heidelberg New York (1972).
2. Rubin, M. B.: Cosserat Theories: Shells, Rods, and Points. Kluwer AcademicPublishers, Dordrecht (2000).
3. Birsan, M.: On the theory of elastic shells made from a material with voids.International Journal of Solids and Structures 43, 3106-3123 (2006).
4. Duvaut, G., Lions, J. L.: Inequalities in Mechanics and Physics. Springer-Verlag,Berlin (1976).
5. Ciarlet, P. G.: Mathematical Elasticity, Vol. III: Theory of Shells. North-Holland,Amsterdam (2000).
On the dynamical reconstruction of control in systems ofordinary differential equations
M. BLIZORUKOVA, V. MAKSIMOV and N. FEDINA
Institute of Mathematics and MechanicsUral Branch of Russian Academy of Sciences
S. Kovalevskaya Str. 16, Ekaterinburg, 620219, [email protected]
Problems of dynamical identification of unknown control acting upon a nonlinearsystem described by ordinary differential equations are discussed. Under the measure-ments (with errors) of phase state of the system, a regularizing algorithm that allows toidentify the control is indicated. The algorithm is stable with respect to informationalnoises and computational errors. The procedure suggested is based on the combina-tion of the methods of the theory of guaranteed control and of the theory of ill-posedproblems.
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Time periodic viscosity solutions of Hamilton–Jacobi equations
Mihai BOSTAN
Universite de Franche-Comte, [email protected]
We study the existence of time periodic viscosity solution of first order Hamilton–Jacobi equations. Existence results are presented under usual hypotheses. The mainidea is to reduce the analysis of time periodic problems to the study of stationaryproblems obtained by averaging the source term over a period. We investigate also thelong time behavior and high frequency asymptotic behavior (leading to homogenizationproblems). Most of these results can be generalized in the framework of almost-periodicviscosity solutions.
Brezis–Haraux - type approximation of the range of amonotone operator composed with a linear mapping
Radu Ioan BOT, Sorin Mihai GRAD, Gert WANKA
Chemnitz University of Technology, [email protected]
Given two monotone operators, the sum of their ranges is usually larger than therange of their sum, but there are some situations where these sets are almost equal, i.e.their interiors and closures coincide. The problem of finding conditions under which thesum of the ranges of two monotone operators is almost equal to the range of their sumis known as the Brezis–Haraux approximation problem ([1]). We give a Brezis–Haraux- type approximation of the range of the monotone operator TA = A∗ T A whereA is a linear continuous mapping between two non-reflexive Banach spaces and T is amaximal monotone operator. Then we specialize the result for a Brezis–Haraux - typeapproximation of the range of the subdifferential of the precomposition to A of a properconvex lower semicontinuous function defined on a non-reflexive Banach space, which isproven to hold under a weak sufficient condition. This extends and corrects some olderresults due to Riahi ([2]) that consist in the approximation of the range of the sum ofthe subdifferentials of two proper convex lower semicontinuous functions. Finally wegive two applications, one in optimization and the other to a complementarity problem.
References:
[1] Brezis, H., Haraux, A. Image d’une somme d’operateurs monotones et applica-tions, Israel Journal of Mathematics No. 23 (1976), 165-186.
[2] Riahi, H. On the range of the sum of monotone operators in general Banachspaces, Proceedings of the American Mathematical Society No. 124 (1996), 3333–3338.
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On a wrong solution to a trivial optimal control problem inmathematical economics
Stefan MIRICA and Touffik BOUREMANI
Faculty of Mathematics, University of BucharestAcademiei 14, 010014 Bucharest, ROMANIA
[email protected], [email protected]
This work is intended, in the first place, as a warning to the increasing number ofauthors that try to solve concrete optimal control problems without enough knowledgeand even basic mathematical abilities; secondly, our aim is to show that the DynamicProgramming approach in [2] is much more efficient than the PMP approach, in thestudy of this type of problems.
A tipical example is the one in the recent paper, [1], in which the authors believethat they solved the problem of minimizing the cost functional
J(P (.), I(.)) :=
∫ T
0
e−ρt[h(I(t)) + K(P (t))]dt
subject to:I ′(t) = P (t)−D(t, I(t)), P (t) ≥ D(t, I(t)) > 0,
a.e. ([0, T ], I(0) = I0, I(T ) = IT that arise from some (not very convincing) “model inmathematical economics”.
Applying a non-existent (in [3]) variant of Pontryagin’s Maximum Principle (PMP)as a “necessary optimality condition”, the authors of [1] conclude in their Th.1 thatin the particular case in which: K(p) := 1
2kp2, h(I) := 1
2hI2, D(t, I) := d1(t) +
d2I, k, h, d2, d1(t) > 0, the “optimal trajectory” of the problem above is of the form:I∗(t) := a1e
m1t + a2em2t + Q(t).
On the other hand, using the Dynamic Programming approach in [2] (adapted tonon-autonomous problems) we prove that the only optimal trajectories are the constant
functions, I(t) ≡ I0, hence the solution in [1] is wrong and the problem above is rather“trivial”. Moreover, we also identify the main scientific errors made by the authorsof [1], among which, the first one is the fact that they apply the “classical form” ofPMP to a problem defined by a differential inclusion since in this case the set of controlparameters is variable (depending on the time and the state).
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Derived cones to reachable sets of discrete hyperbolicinclusions
Aurelian CERNEA
Faculty of Mathematics and Informatics, University of BucharestAcademiei 14, 010014 Bucharest, Romania
We consider a multiparameter discrete inclusion that describes Roesser’s model andwe prove that the reachable set of a certain variational multiparameter inclusion is aderived cone in the sense of Hestenes to the reachable set of the discrete inclusion. Thisresult allows to obtain sufficient conditions for local controllability along a referencetrajectory and a new proof of the minimum principle for an optimization problem givenby a hyperbolic discrete inclusion with end point constraints.
Numerical solution of variational problems using Walshwavelet packets
Heena D. CHALISHAJAR and Pragna KANTAWALA
Department of Applied Mathematics, Babaria Institute of Technology (BIT)Varnama-391240, Vadodara, Gujarat, India
Department of Applied Mathematics, Faculty of Technology and EngineeringM.S. University of Baroda, Vadodara- 390001, Gujarat, India
In 2004, Glabisz [2] solved linear boundary value problems using the operationalmatrices obtained from Walsh wavelet packets i.e. using Walsh Bases and Haar bases.
In this paper the authors are solving the variational problems using the operationalmatrices for Walsh wavelet packets i.e. using Walsh bases and Haar bases. An il-lustrative example has been included. Also a comparative study has been made forthe solutions obtained using the Haar bases, Walsh bases, the Exact solution and thesolution obtained by using Walsh functions.
References:
[1] Chalishajar H.D and Kantawala P.S – A Direct Method for solving VariationalProblems using Walsh Wavelet packets and Error Estimates, Modern MathematicalModels, Methods and Algorithms for Real World Systems, Anamaya Publishers, Indiapp. 235-245 (2006)
[2] Glabisz Wojciech – The use of Walsh wavelet packets in linear boundary valueproblems, Computers and Structures, Vol.82, 131-141 (2004).
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Controllability of Damped Second Order Initial Value Problemfor a class of Differential Inclusions with Nonlocal Conditions
on Noncompact Intervals
Dimplekumar N. CHALISHAJAR
Department of Applied Mathematics, Sardar Vallabhbhai Patel Institute ofTechnology (SVIT), Gujarat University
Vasad-388 306, DIST: Anand, Gujarat State, [email protected]
In this article, we investigate sufficient conditions for controllability of second-ordersemi-linear initial value problem with nonlocal conditions for the class of differentialinclusions in Banach spaces using the theory of strongly continuous cosine families. Weshall rely on a fixed point theorem due to Ma for multi-valued maps. An example isprovided to illustrate the result. This work is motivated by the paper of Benchohra andNtouyas [1] and Benchohra, Gatsori and Ntouyas [2].
We have considered the following second order inclusion system with non local con-ditions
y′′(t)− Ay(t) ∈ Gy′(t) + Bu(t) + F (t, yt, y′(t)), t ∈ J
y(0) + g(y) = φ, y′(0) = y0.
(1)
Here the state y(t) takes values in Banach space E and the control u ∈ L2(J, U), thespace of admissible controls, where J = (0,∞).
Our aim is to study the exact controllability of the above abstract system which willhave applications to many interesting systems including PDE systems. We reduce thecontrollability problem (1) to the search for fixed points of a suitable multi-valued mapon a subspace of the Frechet space C(J,E).
References:
[1] Benchohra, M. and Ntouyas, S. K., Controllability for an infinite time horizoneof second order differential inclusions in Banach spaces with Nonlocal conditions, J.Optim. Theory Appl. 109 (2001), 85–98.
[2] Benchohra, M., Gatsori, E. P. and Ntouyas, S. K., Nonlocal quasilinear dampeddifferential inclusions, Electronic journal of Differential Equations, Vol.2002 (2002),No.7, 1–14.
A (p− q) coupled system in elliptic nonlinear boundary valueproblems
L. CONSIGLIERI
Department of Mathematics and CMAFSciences Faculty of University of Lisbon, 1749-016 Lisboa, Portugal
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In the present work we deal with a problem motivated by the solid and/or fluidthermomechanics, and we establish an existence result of a weak solution. For Ω anopen bounded set of Rn (n > 1) with a sufficiently smooth boundary ∂Ω constitutedby two disjoint complementary open subsets Γ0 and Γ, we study the elliptic boundaryvalue problem: find u, e : Ω → R and τ : Ω → Rn such that
−∇ · τ = f(e, u,∇u) in Ω;
τ ∈ ∂F(e,∇u) in Ω;
−τ · n ∈ ∂G(e, u) on Γ;
−∇ · A(e,∇e) = g(u, e,∇e) + τ · ∇u in Ω;
A(e,∇e) · n + γ(e) = −(τ · n)u on Γ;
u = e = 0 on Γ0 := ∂Ω \ Γ.
Here n denotes the unit outward normal vector to Γ. The (p− q) structure is related tothe functionals F and A. The proof is based on variational and compactness methods.
Some remarks on functional equations and their applications
C. CORDUNEANU
Department of Mathematics, University of Texas, Arlington, [email protected]
We shall consider functional differential equations with causal operators,and dealwith the following type of problems:
1) Some existence theorem,with special concern on global existence and uniqueness;2) Neutral functional/functional differential equations with causal operators and
their reduction to equations of normal form;3) Control problems for some classes of functional equations with causal operators;4) A duality principle for dynamical systems described by functional differential
equations.The following papers of the author are taken as departing point for conducting the
research:
[1] Functional Equations with Causal operators. Taylor and Francis, London, 2002;[2] A duality principle for dynamical systems described by functional equations.
Nonlinear Dynamics and Systems Theory, vol. 5 (2005).
On nonlinear mixed Volterra–Fredholm integrodifferentialequations in Banach spaces
M.B. DHAKNE and S.D. KENDRE
Department of Mathematics, Dr.Babasaheb Ambedkar Marathwada University,Aurangabad-431004, India (M.S.), [email protected]
Department of Mathematics, University of Pune, Pune-411007, India (M.S.)
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Let X be a Banach space with norm ‖ . ‖. Let B = C([0, α], X) be the Banach spaceof all continuous functions from [0, α] into X endowed with supremum norm ‖ x ‖B =sup ‖ x(t) ‖ : t ∈ [0, α]. In the present paper, we investigate the global existence ofmild solutions of nonlinear mixed Volterra-Fredholm integrodifferential equation of thetype
x′(t) + Ax(t) = f
(t, x(t),
∫ t
0
k(t, s, x(s))ds,
∫ α
0
h(t, s, x(s))ds
),
x(0) = x0 ∈ X; t ∈ [0, α],
where −A is the infinitesimal generator of a strongly continuous semigroup of boundedlinear operators T (t) in X, f : [0, α] ×X ×X ×X → X, k, h : [0, α] × [0, α] ×X → Xare continuous functions and x0 is a given element of X. The main tool employed inour analysis is based on an application of the Leray-Schauder alternative and rely on apriori bounds of solutions.
References:
[1] M. B. Dhakne; Global existence of solutions of nonlinear functional integral equa-tions, Indian J. Pure. Appl. Math., 30(7) (1999), 735-744.
[2] J. Dugundji and A. Granas; Fixed point theory, Vol.1 Monografie Matematyczne,PWN, Warrsw (1982).
[3] J. W. Lee and D. O’Regan; Existence results for differential delay equations, I.J. Differential equations, 102 (1993), 342-359.
Singularly perturbed evolution inclusions
Tzanko DONCHEV
Department of Mathematics, University of Architecture and Civil Engineering1 “Hr. Smirnenski” str., 1046 Sofia, Bulgaria
In this paper we study a control evolution system described by two evolution inclu-sions: a “slow” and a “fast” one:
x(t) + A1x ∈ F (x, y, u(t)), x(0) = x0, u(t) ∈ U
εy(t) + A2y ∈ G(x, y, u(t)), y(0) = y0, t ∈ I = [0, 1].
We describe the limit of the solution set as the small parameter ε tends to 0+. Wepresent an example of concrete system, where our results are applicable.
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Robust `-step receding horizon control of sampled-datanonlinear systems with bounded additive disturbances with
application to a HIV/AIDS model
A. M. ELAIW
Al-Azhar University (Assiut), Faculty of ScienceDepartment of Mathematics, Assiut, EGYPT
In this paper, a robust receding horizon control for multirate sampled-data non-linear systems with bounded additive disturbances is presented. Sufficient conditionsare established which guarantee that the `-step receding horizon controller that stabi-lizes the nominal approximate discrete-time model also practically stabilizes the exactdiscrete-time system with small disturbances. This version of the method is motivatedby recently developed models of the interaction of the HIV virus and the immune sys-tem of the human body. In this model the drug dose is considered as control input, andthe uninfected steady state is to be stabilized. Reverse transcriptase inhibitors is used.Simulation results are discussed.
Approximate solutions for non-linear autonomous ODEs onthe basis of PWL approximation theory
A. GARCIAa, S. BIAGIOLAb, J. FIGUEROAc, L. CASTROd
O. AGAMENNONIe
a,b,c,e: Departamento de Ingenierıa Electrica y de Computadoras, UniversidadNacional del Sur, Alem 1253, 8000 Bahıa Blanca, Argentinaagarcia, biagiola, figueroa, [email protected]
d: Departamento de Matematica, Universidad Nacional del Sur, Alem 1253, 8000Bahıa Blanca, Argentina, [email protected]
The present work introduces a new approach to the problem of describing the dy-namic behavior of nonlinear autonomous systems. A general method to approximatethe solutions of a set of ODEs is presented. The technique makes use of a PWL approx-imation of the ODEs vector field which describes the dynamics of a system. A measureof the dynamics approximation error is used to estimate upper and lower bounds for theerror between the real and the approximate trajectories (i.e. the real and approximatesolutions to the ODE). The proposed solution approach can be used in many applica-tions. For instance, in the field of physical and (bio) chemical processes that exhibitnonlinear behavior. These ones, are good examples since many of them are modelled bysystems of ODEs in continuous-time domain. Moreover, it could be used in the solutionof dynamic optimization problems with significant advantages as regards computationaltime consumption. It is important to remark the possibility of the implementation ofan integrated circuit capable to emulate PWL functions, thus providing a very efficientreal time application.
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A fourth order problem in a thin multidomain
Antonio GAUDIELLO
DAEIMI, Universita degli Studi di Cassinovia G. Di Biasio 43, 03043 Cassino (FR), ITALIA
This is joint work with E. Zappale.We consider a thin multidomain of RN , N ≥ 2, consisting (e.g. in a 3D setting) of a
vertical rod upon a horizontal disk. In this thin multidomain we introduce a bulk energydensity of the kind W (D2U), where W is a convex function with growth p ∈]1, +∞[,and D2U denotes the Hessian tensor of a scalar (or vector valued) function U . Byassuming that the two volumes tend to zero with same rate, under suitable boundaryconditions, we prove that the limit model is well posed in the union of the limit domains,with dimensions, respectively, 1 and N − 1. Moreover, we show that the limit problemis uncoupled if 1 < p ≤ N−1
2, “partially” coupled if N−1
2< p ≤ N − 1, and coupled if
N − 1 < p. The main result is applied in order to derive the equilibrium configurationof two joint beams, T-shaped, clamped at the three endpoints and subject to transverseloads. The main result is also applied in order to describe the equilibrium configurationof a wire upon a thin film with contact at the origin, when the thin structure is filledwith a martensitic material.
Conjugacy and Fenchel duality for almost convex and nearlyconvex functions
Radu Ioan BOT, Sorin Mihai GRAD, Gert WANKA
Chemnitz University of Technology, [email protected]
A function f : Rn → R is called (cf. [2]) almost convex if f is convex and ri(epi(f)) ⊆epi(f), nearly convex if epi(f) is nearly convex and closely convex if epi(f) is convex(i.e. f is convex), where f denotes the lower-semicontinuous hull of f .We show that the classical formulae of the conjugates of the precomposition with alinear operator, of the sum of finitely many functions and of the sum between a functionand the precomposition of another one with a linear operator hold even when the usualconvexity assumptions (cf. [3]) are replaced by almost convexity or near convexity.We also prove that the hypotheses of duality statements due to Fenchel can be weakenedwhen the functions involved are almost (nearly) convex.
References:
[1] R. I. Bot, S. M. Grad, and G. Wanka – Almost convex functions: conjugacy andduality, to appear in Proceedings of the 8th International Symposium on GeneralizedConvexity and Generalized Monotonicity, Varese, Italy, 4-8 July, 2005.
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[2] W. W. Breckner and G. Kassay – A systematization of convexity concepts forsets and functions, Journal of Convex Analysis No. 4 Vol. 1 (1997), 109–127.
[3] R. T. Rockafellar – Convex analysis, Princeton University Press, 1970.
Semimonotone stochastic integral equations and explosion time
Hamideh D. HAMEDANI
Department of Statistics, Faculty of Mathematical SciencesShahid Beheshti University Tehran, IRAN
In this talk, we first give an existence and uniqueness result for the following Hilbertspace valued semimonotone nonlinear integral equation
Xt = U(t, 0)X0 +
∫ t
0
U(t, s)F (s,Xs)ds+
+
∫
(o,t]
U(t, s)G(s−, Xs−)dMs + Vt; t ≤ τ,(1)
where U(t, s) is a contraction-type evolution operator, Ft(.) = F (t, ω, .) is a semimono-tone function, G is a good operator-valued Lipschitz integrand, M is a cadlag martingale,V is a cadlag adapted process, X0 is a random variable and τ is a stopping time.
Then we define the explosion time of (1) and in the lack of linear growth condition,we show that the solution will be exploded.
On the integral and asymptotical representation of singularsolutions of elliptic equations near boundary
Nicolae JITARASU
USM, Chisnau, Republic of [email protected]
Let G0 ∈ Rn be a bounded domain with (n−1)-dimensional boundary Γ0 ∈ C∞ andthe nk-dimensional manifolds Γk without boundary lying inside of Γ0, 0 ≤ nk ≤ n − 1.Assume that Γk ∈ C∞ and Γk∩Γj = ∅ for k 6= j. Denote by G the domain G0 \∪χ
k=1Γk,Γ = ∪χ
k=0Γj the boundary of domain G. In the domain G we consider the ellipticboundary value problem (BVP)
L(x, ∂)u(x) = f(x), ordL = 2m, (1)
Bj(x, ∂)u(x)|Γ0 = ϕj(x), j = 1, 2, . . . , m, (2)
in the scale of Sobolev spaces Hs(G), s ∈ R1. Using the Green function G(x) of BVP(1), (2) in G0 [1], we obtain the integral representations of singular solutions u(x) of
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BVP (1), (2) in G near variety Γk, u(x) ∈ H−s(G), s > 0. In the model case ofthe operators with constant coefficients we obtain the asymptotical representations ofsingular solutions u(x) ∈ H−s(G) near Γk. These asymptotical representations are usedfor correct formulation of BVP with singular boundary conditions on Γk [2].
References:
[1] Solonnikov V.A., On Green’s matrices for elliptic boundary value problems, TrudyMat. Inst. Acad. Nauk SSSR, v. 110, 1970, p. 107 - 145.
[2] Jitarasu N., On the Sobolev boundary value problem with singular and regularizedboundary conditions for elliptic equations, Anal. and Optimiz. of Differential systems,Kluwer Acad. Publ., 2003, p.219 - 226.
Duality in vector optimization
Frank HEYDE, Andreas LOHNE and Christiane TAMMER
Universitat Halle-WittenbergFB Mathematik und Informatik, 06099 [email protected]
A new approach to duality theory for convex vector optimization problems is devel-oped. We modify a given (set-valued) vector optimization problem such that the imagespace becomes a complete lattice (a sublattice of the power set of the original imagespace), where the corresponding infimum and supremum are sets that are related tothe usual solution concepts of vector optimization. In doing so we can carry over thestructures of the duality theory of scalar convex programming. It follows a discussionof the special case of multi-objective linear programming, in particular, the possibilityto develop a dual simplex algorithm based on this duality theory.
An existence result for a class of nonlinear differential systems
Rodica LUCA-TUDORACHE
Department of Mathematics, “Gh. Asachi” Technical University11 Bd. Carol I, Iasi 700506, ROMANIA
We investigate the existence, uniqueness and asymptotic behaviour of the strong andweak solutions to the nonlinear discrete hyperbolic system
(S)
dun
dt(t) +
vn(t)− vn−1(t)
h+ cnA(un(t)) 3 fn(t),
dvn
dt(t) +
un+1(t)− un(t)
h+ dnB(vn(t)) 3 gn(t),
n = 1, 2, . . . , 0 < t < T, in H,
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with the extreme condition
(EC) (v0(t), s1w′1(t), . . . , smw′
m(t))T ∈ −Λ((u1(t), w1(t), . . . , wm(t))T ), 0 < t < T
and the initial data
(ID)
un(0) = un0, vn(0) = vn0, n = 1, 2, . . . ,wi(0) = wi0, i = 1,m,
where H is a real Hilbert space, T > 0, m ∈ IN , h > 0, cn > 0, dn > 0, ∀n ∈ IN ,si > 0, ∀ i = 1,m, and A, B are multivalued operators in H and Λ is a multivaluedoperator in Hm+1 which satisfy some assumptions.
This problem is a discrete version with respect to x (with H = IR) of some problemswhich have applications in the theory of integrated circuits. For the proofs of ourtheorems we use some results from the theory of monotone operators and nonlinearevolution equations of monotone type in Hilbert spaces.
Analytical solutions for integral operator’s nonlinearoptimization in case of the airfoils curvilinear of maximal drag
in aero hydrodynamics
Mircea LUPU and Ernest SCHEIBER
Faculty of Mathematics-InformaticsTransilvania University of Brasov
[email protected], [email protected]
In the paper there are solved direct and inverse Dirichlet or Riemann - Hilbertproblems and analytical solution are obtained for optimization problems in the caseof some nonlinear integral operators. It is modeled the plane potential flow of an in-viscid, incompressible jet, is eliminated from the channel and encounters a curvilinearsymmetrical obstacle (airfoils/hydrofoils) which must modeled geometrically obtainedthe deflector of maximal drag. There are delivered integral singular equations, for di-rect and inverse problems and the movement in the auxiliary canonical half-plane isobtained. Next, the optimization problem is solved in analytical manner. The mathe-matical model used is “Valcovich–Birkhoff–Popp” in the case of the curvilinear profile;it is used integral Jensen’s inequality for increase the integral nonlinear operator - whorepresents the resistance to advancement - determined the global maximum accepted.The author established one of the most general velocity distributions on the airfoil, withdistributed parameters, for getting the optimal forms of the maximal drag airfoil in theBrillouin–Villat’s condition. The design of the optimal airfoils is performed, numericalcomputation concerning the drag coefficient.
In particularly case is obtained the limited jets which encountered the curvilinearairfoils (the Cisotti–Iacob’s model) and the case of the obstacle situated and the freeunlimited flow (the Helmholtz’s model — impermeable parachute). The constructive
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techniques have a theoretical and practical importance. The problems of maximal dragare very important, in relation with applications to the most reversal devices, or thedirection control of the reactive vehicles. We notice also other applications to the solvingby means of fluid jets or to the jet flaps system from the airplane wings or turbine bucket.
On the method of Lyapunov functionals in inverse problems ofdistributed systems
Vyacheslav MAKSIMOV
Institute of Mathematics and MechanicsUral Branch of the Russian Academy of Sciences
S. Kovalevskoi Str. 16, Ekaterinburg, 620219, [email protected]
Inverse problems of dynamical reconstruction of unknown characteristics for dis-tributed systems are considered. The role of these characteristics may be played bydistributed or boundary disturbances, by varying coefficient at higher derivative of el-liptic operator. Solving algorithms, which are stable with respect to informational noisesand computational errors and operate in real time mode, are designed. These algorithmsare based on the method of Lyapunov functions, on the theory of positional differentialgames principle of control with a model. Inaccurate measurements of current phasestates of systems represent input data for the algorithm, which provide some valuesapproximating real (but unknown) characteristics as outputs. The basic elements of thealgorithms are stabilization procedures (functioning by feedback principle) for appro-priate Lyapunov functionals.
Nonlinear Multiobjective Transportation Problem: A FuzzyGoal Programming Approach
H.R. MALEKI and H. MISHMASTE NEHI
Department of Basic Sciences, Shiraz University of Technology, Shiraz, [email protected]
Department of Mathematics, Sistan and Baluchestan University, Zahedan, [email protected]
The nonlinear multiobjective transportation problem refers to a special class of non-linear multiobjective problem. In this paper we present a fuzzy goal programmingapproach to solve the nonlinear multiobjective transportation problem. To this end weuse a special type of nonlinear (hyperbolic) membership functions. A numerical exampleis given to illustrate the efficiency of the proposed approach.
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References:
[1] A.K. Bit, M.P. Biswal and S.S. Alam, Fuzzy programming approach to multiob-jective solid transportation problem, Fuzzy Sets and Systems 57 (1993), pp. 183-194.
[2] R.H. Mohamed, The relationship between goal programming and fuzzy program-ming, Fuzzy Sets and Systems 89 (1997), pp. 215–222.
[3] B.B. Pal, B.N. Moitra, U. Maulik, A goal programming procedure for fuzzymultiobjective linear programming problem, Fuzzy Sets and Systems 139 (2003), pp.395–405.
[4] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965), pp. 338–353.
On Schrodinger operators with multipolar inverse-squarepotentials
Elsa MARCHINI
Dipartimento di Matematica e ApplicazioniUniversita di Milano-Bicocca Via R. Cozzi 53 - Edificio U5 - 20125 Milano
Positivity, essential self-adjointness, and spectral properties of a class of Schrodingeroperators with multipolar inverse-square potentials are discussed. In particular a nec-essary and sufficient condition on the masses of singularities for the existence of atleast a configuration of poles ensuring the positivity of the associated quadratic form isestablished.
A mathematical model for the flow in a porous medium with aspace and time-dependent porosity
Gabriela MARINOSCHI
Institute of Mathematical Statistics and Applied MathematicsBucharest, [email protected]
The paper deals with the study of the well-posedness of a model describing thewater flow in a porous medium characterized by a time and space variability of thepore structure. The model consists in a parabolic partial differential equation with ablowing-up diffusivity and with initial and boundary data. The model leads to a Cauchyproblem involving a time-dependent multivalued operator, for which an existence anduniqueness result is proved.
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Petri net toolbox for MATLAB in modeling and simulation ofdiscrete-event and hybrid systems
Mihaela MATCOVSCHI and Octavian PASTRAVANU
Department of Automatic Control and Applied InformaticsTechnical University “Gh. Asachi” of Iasi
Blvd. Mangeron 53A, Iasi 700050, Romaniaopastrav, [email protected]
The Petri Net Toolbox (PN Toolbox) for MATLAB is a software package that pro-vides instruments for the simulation, analysis and design of the discrete-event systemsmodeled via the Petri net formalism. The toolbox is equipped with a user-friendlygraphical interface and can handle five types of Petri nets (untimed, transition-timed,place-timed, stochastic and generalized stochastic) with finite or infinite capacity. Threesimulation modes, accompanied or not by animation, are available. Dedicated proce-dures cover the key topics of analysis such as behavioral properties, structural properties,time-dependent performance indices, max-plus state-space representations. A designprocedure is also available, based on parameterized models. The Petri Net SimulinkBlock (PNSB) allows the modeling and analysis of hybrid systems whose event-drivenpart(s) is (are) modeled based on the PN formalism. A synchronized Petri net containtransitions that are triggered by external events (corresponding to the evolution of thecontinuous part(s) of the hybrid system) and exports internal events, generated by tran-sition firings, and its current marking as input signals for the continuous part(s). ThePN Toolbox is included in the Connections Program of The MathWorks Inc., as a thirdparty product.
Stochastic approach for multivalued Dirichlet–Neumannproblems
Lucian MATICIUC and Aurel RASCANU
Department of Mathematics, “Gheorghe Asachi” Technical University of IasiBd. Carol I, no. 11, Iasi - 700506, Romania
Faculty of Mathematics, “Alexandru Ioan Cuza” University of Iasi and“Octav Mayer” Mathematics Institute of the Romanian Academy
Bd. Carol I, no. 11, Iasi - 700506, [email protected]
We prove the existence and uniqueness of a viscosity solution of the parabolic vari-ational inequality with a mixed nonlinear multivalued Neumann-Dirichlet boundary
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condition:
∂u(t, x)
∂t+ Ltu (t, x) + f(t, x, u(t, x), (∇uσ)(t, x)) ∈ ∂ϕ(u(t, x)),
t ∈ [0, T ], x ∈ D,
∂u(t, x)
∂n+ ∂ψ(u(t, x) 3 g(t, x, u(t, x)), t ∈ [0, T ], x ∈ Bd (D) ,
u(T, x) = h(x), x ∈ D,
(1)
where ∂ϕ and ∂ψ are subdifferential operators and Lt is a second differential operatorgiven by
Ltϕ (x) =1
2
d∑i,j=1
(σσ∗)ij(t, x)∂2ϕ (x)
∂xi∂xj
+d∑
i=1
bi(t, x)∂ϕ (x)
∂xi
.
The result is obtained by a stochastic approach: we study a backward stochasticgeneralized variational inequality
dYt + F (t, Yt, Zt) dt + G (t, Yt) dAt ∈ ∂ϕ (Yt) dt + ∂ψ (Yt) dAt + ZtdWt ,
0 ≤ t ≤ T ,
YT = ξ
and we obtain a Feynman-Kac representation formula for the viscosity solution u of theproblem (1).
Reducing a differential game to a pair of nonsmooth optimalcontrol problems
Stefan MIRICA
Faculty of Mathematics, University of BucharestAcademiei 14, 010014 Bucharest, Romania
We are extending to the case of non-autonomous differential games author’s recentconcepts and results introduced and developed for autonomous (i.e. time-invariant)differential games.
The main idea is to introduce first the concept of an “admissible pair of (multivalued)feedback strategies” to which one may associate a value function and which reduces thedifferential game to a pair of symmetric nonsmooth optimal control problems for dif-ferential inclusions; secondly, one introduces the concepts of bilaterally-optimal pairs offeedback strategies and one proves an abstract verification theorem containing necessaryand sufficient optimality conditions; next, this approach is made more realistic by theproof of several “practical verification theorems” containing corresponding differentialinequalities and regularity hypotheses on the value function that imply the optimality.
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Moreover, one may prove that under certain conditions, the value function associatedto a pair of bilaterally-optimal feedback strategies, is a generalized solution and, inparticular, a viscosity solution, of Isaacs’ main equation and suggest the possibility touse suitable extensions of Cauchy’s Method of Characteristics to construct, both, thevalue function and the pair of optimal feedback strategies. Already tested on severalnon-trivial examples, this work may be considered as a ”rehabilitation” on a rigorousbasis of Isaacs’ (1965) original approach.
A polytope approach for quadratic assignment problem
H. MISHMASTE NEHI and H.R. MALEKI
Department of Mathematics, Sistan and Baluchestan UniversityZahedan, IRAN
Department of Basic Sciences, Shiraz University of Technology, Shiraz, [email protected]
The Quadratic Assignment Problem (QAP) is one of the classical optimization prob-lem and is widely regarded as one of the most difficult problem in this class. Manyresearchers proposed different formulation for this problem based on a special approach.Given a set of N = 1, 2, ..., n, and n×n matrices F = fij, called flow matrix, D = dij,as distance matrix, and C = cij, as setup cost. The QAP is to find a permutation φof the set N which minimizes:
z =n∑
i=1
n∑j=1
fijdφ(i)φ(j) +n∑
i=1
ciφ(i).
The quadratic assignment polytope will be defined in section 2 as the convex hull ofthe feasible solutions to a suitable linearization of QAP. In order to profit from theconvenient notions of graph theory we formulate the quadratic assignment problem asa graph problem before starting its polyhedral investigations in Section 3. In Section4 connections between the quadratic assignment polytope and polytopes like the linearordering polytope and the traveling salesman polytope via certain projections are con-sidered. The quadratic assignment polytope will be proved to be a face of the booleanquadric as well as of the cut polytope. Finally we present an integer linear programmingformulation in Section 5.
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A delayed prey-predator system with parasitic infection
Debasis MUKHERJEE
Department of Mathematics, Vivekananda CollegeThakurpukur, Kolkata-700 063, INDIAdebasis [email protected]
This paper analyzes a prey-predator system in which some members of the preypopulation and all predators are subjected to infection by a parasite. The predatorfunctional response is a function of weighted sum of prey abundances. Persistenceand extinction criteria are derived. The stability of the interior equilibrium point isdiscussed. The role of delay is also addressed. Lastly the results are verified throughcomputer simulation. Numerical simulation suggests that the delay has a destabilizingeffect.
Numerical aspects on computational electrocardiology
Marilena MUNTEANU
University of Milanvia C. Saldini, 50 CAP 20133, Milan, Italy
We present numerical results regarding the stability and parallel scalability of semi-implicit and implicit discretizations of Fitz Hugh-Nagumo, monodomain and bidomainsystems.
On the convergent solutions of a class of nonlinear ordinarydifferential equations
Octavian G. MUSTAFA
Department of Mathematics, University of CraiovaAl. I. Cuza 13, Craiova, [email protected]
Via a special integral transformation, asymptotic integration results for ordinarydifferential equations are used to establish accurate asymptotic developments for radialsolutions of the elliptic equation ∆u + K(|x|)eu = 0, |x| > x0 > 0, in the bidimensionalcase.
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Time-dependent invariant sets in system dynamics
Octavian PASTRAVANU, Mihaela-Hanako MATCOVSCHI andMihail VOICU
Department of Automatic Control and Applied InformaticsTechnical University “Gh. Asachi” of Iasi
Blvd. Mangeron 53A, Iasi 700050, Romaniamvoicu, opastrav, [email protected]
A general framework has been developed to explore the positively (flow) invariantsets, for a large class of time-variant, nonlinear dynamical systems. We introduce theconcept of “diagonal invariance” defined by time-dependent diagonal matrices and forarbitrary Holder norms. The flow invariance results are formulated as necessary andsufficient conditions for linear systems. The approach to nonlinear systems relies on suf-ficient conditions that allow formulating a comparison theorem where the comparisonsystem has linear dynamics. The time-dependence of the invariant sets is consideredeither arbitrary, or constrained by requirements such as: boundedness, approaching theequilibrium point in accordance with a certain law, in particular decreasing exponen-tially. The special forms of the time-dependence exhibited by the invariant sets inducestability properties stronger than the standard ones known from the qualitative analysis.These properties are called “diagonally invariant stability”, “diagonally invariant asymp-totic stability” and “diagonally invariant exponential stability”, and for their study wepropose a methodology based on comparison principles. Thus, we also point out therole of essentially nonnegative matrices and of M matrices in defining the dynamics ofthe comparison system. We illustrate the applicability of our results for the nonlinearsystems defined by Hopfield neural networks.
Range condition in optimization
N. H. PAVEL
Department of Mathematics, Ohio UniversityAthens OH 45701, USA
In the early 1996 I have started the investigation of the problem:
Minimize the cost functional L(y, u)subject to the constraints Ay = Bu + f .
I have observed that if the linear operators A (unbounded) and B are densely definedon a Hilbert space, with closed range and if the range condition:(RC) R(A) ⊆ R(B), or vice versa: R(B) ⊆ R(A),holds, then one can obtain maximum principles of the form:If (yo, uo) is an optimal pair, then there is p such that:A∗p ∈ −∂yL(yo, uo), B∗p ∈ ∂uL(yo, uo).
33
Here ∂L could mean Frechet derivative, or subdifferential or Clarke’s generalized gra-dients, depending on the conditions on L. Such maximum principles have applicationsto optimal control of some differential systems including some PDE. Moreover, this ab-stract scheme includes the essentials of many existing results on optimal control, i.e.it has a unifying effect, too. This idea was extensively extended to more general casesby Pavel and in some joint papers by Pavel, Aizicovici, Motreanu, and by Voisei in hisPh.D thesis.We are currently investigating some cases in which the above (RC) is not necessary.
Limits of solutions to the initial boundary Dirichlet problemfor semilinear hyperbolic equation with small parameter
Andrei PERJAN
USM, Chisnau, Republic of [email protected]
Let Ω ∈ Rn be an open and bounded set with the smooth boundary ∂Ω. Considerthe following problems which will called (Pε) and (P0) respectively:
εutt(x, t) + ut(x, t)−∆u(x, t) + |u(x, t)|pu(x, t) = f(x, t), x ∈ Ω, t > 0,
u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω, u(x, t)∣∣∣x∈∂Ω
= 0, t ≥ 0,
vt(x, t)−∆v(x, t) + |v(x, t)|pv(x, t) = f(x, t), x ∈ Ω, t > 0,
v(x, 0) = u0(x), x ∈ Ω, v(x, t)∣∣∣x∈∂Ω
= 0, t ≥ 0.
Theorem. Suppose that p ∈ [0, 2/(n − 2)] if n ≥ 3 and p ∈ [0,∞) if n = 1, 2.If f ∈ W 2,1(0, T ; L2(Ω)), u0, u1, α ∈ H1
0 (Ω) ∩ H2(Ω), then for any strong solution ofthe problem (Pε) the following relationships: u → v in C([0, T ]; L2(Ω)), u → v inL∞(0, T ; H1
0 (Ω)) and u′−v′−αe−t/ε → 0 in L∞(0, T ; L2(Ω)), as ε → 0, are valid, whereα = f(0)− u1 + ∆u0 − |u0|pu0.
Well-posedness and fixed point problems
Adrian PETRUSEL and Ioan A. RUS
Department of Applied Mathematics, Babes-Bolyai University Cluj-Napoca400084, Cluj-Napoca, ROMANIA
petrusel, [email protected]
Definition 1. Let (X.d) be a metric space, Y ∈ P (X) and T : Y → Pcl(X) be amultivalued operator. The fixed point problem is well-posed for T with respect to Dd
iff:(a1) FT = x∗
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(b1) If xn ∈ Y , n ∈ N and Dd(xn, T (xn)) → 0, as n → +∞ then xn → x∗, asn → +∞.
Definition 2. Let (X.d) be a metric space, Y ∈ P (X) and T : Y → Pcl(X) be amultivalued operator. The fixed point problem is well-posed for T with respect to Hd
iff:(a2) (SF )T = x∗(b2) If xn ∈ Y , n ∈ N and Hd(xn, T (xn)) → 0, as n → +∞ then xn → x∗, as
n → +∞.
The purpose of this paper is to define the concept of well-posed fixed point probleman to study it for the case of multi-valued operators. Several examples of well-posedfixed point problem are given.
Numerical solutions of two-point boundary value problems forordinary differential equations using particular Newton
interpolating series
Ghiocel GROZA and Nicolae POP
Department of Mathematics, Technical University of Civil EngineeringLacul Tei 124, Sect. 2, 020396-Bucharest, Romania
North University, Baia Mare, Department of Mathematics and Computer ScienceVictoriei 76, 43012-Baia Mare, Romania
Let xkk≥1 be a sequence of real numbers. We construct the polynomials
u0(x) = 1, ui(x) =i∏
k=1
(x− xk), i = 1, 2, ...,
where x is a real variable. We call an infinite series of the form
∞∑i=0
aiui(x), (1)
where ai ∈ R, a Newton interpolating series with real coefficients ai at xkk≥1. Theseseries are useful generalization of power series which, in particular forms, were used innumber theory to prove the transcendence of some values of exponential series. Takinginto account the importance of power series in the theory of initial value problems fordifferential equations, it seems to be very useful to study Newton interpolating seriesin order to find the solution of the multipoint boundary value problems for differentialequations. If we consider a function f : [0, 1] → R and the points xk ∈ [0, 1], thensay that the function f is representable into a Newton interpolating series at xkk≥1 ifthere exists a series of the form (1), which converges uniformly to f on [0, 1].
35
We obtain results concerning Newton interpolating series and their derivatives andsufficient conditions for a function to be representable into a Newton interpolating se-ries. The representation of solutions of particular differential equations through Newtoninterpolating series are also considered.
Pareto reducibility and contractibility in vector optimization
Nicolae POPOVICI
Babes-Bolyai University of Cluj, [email protected]
A multicriteria optimization problem is said to be Pareto reducible if the set ofweakly efficient solutions can be represented as the union of the sets of (Pareto) efficientsolutions of its subproblems (i.e. optimization problems obtained from the originalone by selecting certain criteria). The principal aim of this presentation is to providesufficient conditions for Pareto reducibility, under appropriate generalized convexityassumptions. We will also show that Pareto reducibility is intimately related to thecontractibility of efficient sets.
References:
[1] J. Benoist, Contractibility of efficient frontier of simply shaded sets, Journal of GlobalOptimization 25, (2003), 321–335.
[2] N. Popovici, Pareto reducible multicriteria optimization problems, Optimization 54,(2005), 253–263.
[3] N. Popovici, Structure of efficient sets in lexicographic quasiconvex multicriteriaoptimization, Operations Research Letters 34, (2006), 142–148.
Singular phenomena in nonlinear elliptic problems
Vicentiu RADULESCU
Department of Mathematics, University of Craiova200585 Craiova, Romania
We consider two classes of nonlinear elliptic equations and we are concerned withthe existence and the uniqueness of solutions, as well as with the study of the growth ofsolutions near the boundary. We first consider singular solutions of the logistic equationin anisotropic media and we discuss the blow-up rate of solutions in terms of Karamata’sregular variation theory. Next, we establish several bifurcation results for the Lane-Emden-Fowler equation with singular nonlinearity and convection term.
36
Functional monotone VP and normed coercivity
Mihai Turinici
“A. Myller” Mathematical Seminar; “Al. I. Cuza” University11, Copou Boulevard; 700506 Iasi, Romania
A functional extension is given for the monotone variational principle in Turinici [An.St. UAIC Iasi, 36 (1990), 329–352]. The obtained facts are then applied to establish(via conical Palais–Smale techniques) a monotone functional version of the coercivityresult in Zhong [Nonlinear Analysis, 29 (1997), 1421–1431].
Viability for Nonlinear Reaction-Diffusion Systems
Mihai NECULA and Ioan I. VRABIE
Faculty of Mathematics, “Al. I. Cuza” University of Iasi, [email protected]
Faculty of Mathematics, “Al. I. Cuza” University of Iasi“O. Mayer” Mathematics Institute of the Romanian Academy, Iasi, Romania
We prove several necessary and/or sufficient conditions for viability for certain classesof nonlinear reaction-diffusion systems governed by continuous perturbations of m-dissipative operators.
Integro-differential hamilton-jacobi-bellman equationsassociated to SDES driven by stable processes
A. ZALINESCU
Laboratoire de Mathematiques et ApplicationsUniversite de La Rochelle, Avenue Michel Crepeau
17042 La Rochelle, [email protected]
We are interested in the way in which (nonlinear) Hamilton–Jacobi–Bellman equa-tions (or variational inequalities) involving an integro-differential operator relate to jumpdiffusion processes via optimal stochastic control (and optimal stopping) problems.
Our main domain of interest lies in the case where the dynamics has infinite variance,especially in the case where the jump diffusion process is a solution of a SDE driven bystable processes.
We prove that the value function of the optimal control problem is a viscosity solutionof the integro-differential variational inequality arising from the associated dynamic
37
programming. We also establish comparison principles in the class of semi-continuousfunctions with polynomial growth of a given order.
Regularity and qualitative properties for models of complexnon-newtonian fluids
Arghir ZARNESCU
University of Chicago, Chicago, USA5480 S. Cornell Ave. Apt. 517, Chicago, IL 60615, USA
I will discuss models describing nematic liquid crystalline polymers. The modelsconsist of a coupling between a nonlinear Fokker-Planck equation and a Navier-Stokessystem. In the first part of the talk I will present regularity results for the systems, whoseproof involve new estimates for the 2D Navier-Stokes equations with nearly singularforcings. In the second part of the talk I will discuss qualitative properties of solutionsfor simplified models.
The first part is joint work with P. Constantin, C. Fefferman and E. S. Titi.
38
Posters
On the convergence of solutions to a certain fifth ordernonlinear differential equation
Olufemi Adeyinka ADESINA
Department of Mathematics, Obafemi awolowo University, Ile-Ife, [email protected]
In this paper, sufficient conditions for convergence of solutions to the fifth ordernonlinear differential equation:
x(v) + ax(iv) + bx′′′ + cx′′ + g(x′) + h(x) = p(t, x, x′, x′′, x′′′, x(iv)),
in which a, b and c are positive constants, functions h(x) and p(t, x, x′,x′′, x′′′, x(iv)) are real valued and continuous in their respective arguments are obtained.The function h(x) is not necessarily differentiable but satisfies an incrementally ratio
h(ζ + η)− h(ζ)
η∈ I0, η 6= 0,
where I0 is a closed Routh–Hurwitz interval.
Competition in patchy space with cross diffusion and toxicsubstances
Shaban ALY
Department of Mathematics, Faculty of Science, Al-Azhar UniversityAssiut 71511, [email protected]
In this paper we formulate a Lotka-Volterra competitive system affected by toxicsubstances in two patches in which the per capita migration rate of each species isin.uenced not only by its own but also by the other one.s density, i.e. there is crossdiffusion present. Numerical studies show that at a critical value of the bifurcationparameter the system undergoes a Turing bifurcation and the cross migration responseis an important factor that should not be ignored when pattern emerges.
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A Stability Theorem for Functional Differential Equations withImpulse Effect
J. O. ALZABUT
Department of Mathematics and Computer ScienceCankaya University, 06530 Ankara, Turkey
In this paper, we are concerned with functional differential equation with impulseeffect of the form
x′(t) = A(t)x(t) + B(t)x(t− τ), t 6= θi,∆x(θi) = Cix(θi) + Dix(θi−j), i ∈ N,
where A, B are n × n continuous bounded matrices, τ > 0 is a positive real number,Ci, Di are n × n bounded matrices and j ∈ N is fixed. It is shown that if a functionaldifferential equation with impulse effect of the above form verifies a Perron conditionthen its trivial solution is uniformly asymptotically stable.
On nonlinear diffusion problems with strong degeneracy
Kaouther AMMAR
TU Berlin, Institut fur MathematikStrasse des 17 juni 135 MA 6-4 10623 Berlin, Germany
In this paper we prove existence of a unique weak entropy solution for a degenerateproblem of the type
(Pb,g)(v0, a, f)
b(v)t −∆g(v) + div Φ(v) = f on Q :=]0, T [×Ω
g(v) = g(a) on Σ :=]0, T [×∂Ω
b(v)(0, ·) = b(v0) on Ω
where b, g : R → R are Lipschitz continuous, non-decreasing such that b(0) = g(0) = 0and R(g + b) = R.
Deterministic multivariate model for simulation of downstreamBIVAL automatic controller in irrigation systems
L. ROSU, A. BARBULESCU, S. HANCU and A. DUMITRU
Department of Mathematics, Ovidius University, Constanta, [email protected]
40
Providing irrigation canals with automatic controllers leads to increase the exploita-tion performance.
The major part of the mathematical simulation models are based on the numerical oranalytical solution of the nonlinear equations with partial derivatives of hyperbolic typethat govern the unsteady flow in the canals equipped with the automatic regulators.
In this paper we offer an answer on how to conceive and use an analytical modelfor the design of the automatic irrigation canals for practical important situations. Themodel was obtained by analytical integration of the linearized equations based on thehypothesis of small oscillation theory and the properties of the Fourier transforms. Itcan be used to predict the behavior of the system under the perturbation factors.
Semi-cardinal models for multivariable interpolation
Aurelian BEJANCU
Kuwait [email protected]
Schoenberg’s ‘semi-cardinal interpolation’ (SCI) model in univariate spline theoryconstructs a polynomial spline function that interpolates values given on the grid Z+ ofnon-negative integers. We present an overview of recent multivariable extensions of theSCI model, focusing on the complete results obtained for interpolation on the semi-planegrid Z+×Z from a space of triangular box-splines. The box-spline SCI schemes employboundary conditions that extend the ‘natural’ and ‘not-a-knot’ end-point conditions ofcubic spline interpolation. The analysis of the localization and polynomial reproductionproperties of the bivariate SCI schemes proves that the ‘natural’-type boundary condi-tions induce a halving effect in accuracy, while the ‘not-a-knot’-type conditions achievemaximal accuracy (Bejancu A., J. Comput. Appl. Math., to appear; Bejancu A., SabinM.A., Adv. Comput. Math. 22 (2005), 275-298).
A Viability Result for Semilinear Reaction-Diffusion Sytems
Monica BURLICA and Daniela ROSU
Chair of Mathematics, “Gheorghe Asachi” Technical University of Iasi, [email protected], [email protected]
Using some some topological assumptions, expressed by the Kuratowski measure ofnoncompactness, we establish several necessary and sufficient conditions for viability forvarious classes of semilinear reaction-diffusion systems.
41
Some new regularity conditions for Fenchel duality in reallinear spaces
Radu Ioan BOT, Erno Robert CSETNEK, Gert WANKA
Chemnitz University of Technology, [email protected]
We consider a convex optimization problem in a real linear space. Using the theoryof conjugate functions we attach to it the Fenchel dual problem. Then we give a newregularity condition which guarantees strong duality between these two problems. Tothis end, we employ some abstract convexity notions. In contrast to other conditionsgiven in the literature by Elster and Nehse ([1]) and Lassonde ([2]), written in termsof some algebraic interior point conditions of the effective domains of the functionsinvolved, the one given by us is formulated by using the epigraphs of their conjugates.We prove that our condition is weaker than the aforementioned regularity conditions.We treat some particular cases of these convex optimization problems and also we givea sufficient condition for the subdifferential sum formula of a convex function with theprecomposition of another convex function with a linear mapping.
References:
[1] Elster, K.H., Nehse, R. Zum Dualitatssatz von Fenchel, Mathematische Opera-tionsforschung und Statistik 5, vol. 4/5, (1974), 269–280.
[2] Lassonde, M. Hahn-Banach theorems for convex functions, Minimax theory andapplications, 135–145, Nonconvex Optimization and its Applications 26, Kluwer Aca-demic Publishers, Dordrecht, (1998).
Study of a carrier dependent infectious disease-cholera
Prasenjit DAS
Research Scholar, Department of Mathematics, Jadavpur UniversityKolkata-700 032, Indiajit [email protected]
This paper analyzes an epidemic model for carrier dependent infectious disease -cholera. Existence criteria of carrier-free equilibrium point and endemic equilibriumpoint (unique or multiple) are discussed. Some threshold conditions are derived forwhich disease-free, carrier-free as well as endemic equilibrium become locally stable.Further global stability criteria of the carrier-free equilibrium and endemic equilibriumare achieved. Conditions for survival of all populations are also determined. Lastlynumerical simulations are performed to validate the results obtained.
42
Output feedback stabilization of sampled-data nonlinearsystems by receding horizon control via discrete-time
approximations
A. M. ELAIW
Al-Azhar University (Assiut), Faculty of ScienceDepartment of Mathematics, Assiut, EGYPT
This paper is devoted to the stabilization problem of sampled-data nonlinear systemsby output feedback receding horizon control. The observer is designed via an approxi-mate discrete-time model of the plant. We investigate under what conditions this designachieve practical stability for the exact discrete-time model.
On the Crossing-Time for Two-dimensional Piecewise LinearSystems
A. GARCIAa, S. BIAGIOLAb, J. FIGUEROAc, L. CASTROd
O. AGAMENNONIe
a,b,c,e: Departamento de Ingenierıa Electrica y de Computadoras, UniversidadNacional del Sur, Alem 1253, 8000 Bahıa Blanca, Argentinaagarcia, biagiola, figueroa, [email protected]
d: Departamento de Matematica, Universidad Nacional del Sur, Alem 1253, 8000Bahıa Blanca, Argentina, [email protected]
Piecewise Linear (PWL) models have proved to be a useful tool for the approxi-mation of nonlinear dynamical systems. The dynamics of the real physical system isapproximated in a region of interest. For this goal, a finite number of simplices is usedwhen a PWL model is chosen. The PWL systems is given by a linear time invariantsystem, which varies from one simplex to the other. A relevant topic when using PWLmodels is the determination of the state of the system when the trajectory crosses fromone simplex to another one. An equivalent dilemma is the calculus of the time valueat which this crossing takes place. In this paper, we provide a formula for determiningthe crossing time (CT) when a given linear time invariant system reaches a frontierbetween simplices. Explicit expressions for the R2 case are derived and some examplesare presented.
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Existence and uniqueness results in the micropolar mixturetheory of porous media
Ionel-Dumitrel GHIBA
“Octav Mayer” Mathematics Institute, Romanian Academy of ScienceIasi Branch, 8 Bd. Carol I, 700506-Iasi, Romania
ghiba [email protected]
In this paper we study the existence and uniqueness of solutions for the initial–boundary value problem associates with a mixture of a micropolar elastic solid and anincompressible micropolar viscous fluid. We use some results of the semigroup opera-tors to obtain an existence and uniqueness theorem for the initial value problem withhomogeneous Dirichlet boundary conditions. Continuous dependence of the solutionsupon the initial data and supply terms is also established.
Compactness for Linear Evolution Equations with Measures
Gabriela GROSU
Chair of Mathematics, “Gheorghe Asachi” Technical University of Iasi, [email protected]
We prove a compactness result for the solution operator g 7→ u attached to a linearevolution equation with measures, du = Adt + dg, u(0) = ξ, where A generates a C0-semigroup in a real Banach space X and g belongs to a certain family of functions, withbounded variation, from [ 0, T ] to X.
A sufficient condition for null controllability of nonlinearcontrol systems
A. HEYDARI and A.V. KAMYAD
Payame Noor Univ., Fariman, [email protected]
Ferdowsi Univ., Mashhad, IRAN
Classic control methods such as Minimum principle of Pontyagin, Bang-Bang princi-ple and other methods aren’t useful for solving optimal control systems (OCS) speciallyoptimal control of nonlinear systems (OCNS).
In this paper we introduce a new approach for ONCS, by using a combination ofatomic measures. We define a criterion for controllability of lumped nonlinear controlsystems and when the system is nearly null controllable, we determine controls andstates.
At last we use this criterion to solve some numerical examples.
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Farkas-type results for fractional programming problems
Radu Ioan BOT, Ioan Bogdan HODREA, Gert WANKA
Chemnitz University of Technology, [email protected]
Considering a constrained fractional programming problem, we present some nec-essary and sufficient conditions which ensure that the optimal objective value of theconsidered problem is greater than or equal to a given real constant. More precisely,we give necessary and sufficient conditions which ensure that x ∈ X, h(x) ≤ 0 ⇒f(x)/g(x) ≥ λ, where the nonempty convex set X ⊂ Rn, the functions f : Rn → R,g : Rn → R and h : Rn → Rm and the real constant λ are given. As usual for a frac-tional programming problem, we assume g(x) > 0 for all x feasible. The desired resultsare obtained using the Fenchel-Lagrange duality approach applied to some optimizationproblems with convex, respectively, difference of convex (DC) objective functions andfinitely many convex inequality constraints. Recently, Bot and Wanka ([1]) have pre-sented some Farkas-type results for inequality systems involving finitely many convexfunctions using an approach based on the theory of conjugate duality for convex opti-mization problems. Their results are naturally extended to the problem we treat and,moreover, it is shown that some other recent statements can be derived as special casesof our general result.
References:
[1] R. I. Bot and G. Wanka Farkas-type results with conjugate functions, SIAMJournal on Optimization No. 15 Vol. 2 (2005), 540–554.
[2] W. Dinkelbach On nonlinear fractional programming, Management Science No.13 Vol. 7 (1967), 492–498.
[3] J. E. Martinez-Legaz and M. Volle Duality in D. C. programming: The case ofseveral D. C. constraints, Journal of Mathematical Analysis and Applications No. 237Vol 2 (1999), 657–671.
The passage of long waves above a vertical barrier: method ofcomplex variable for calculation of the local disturbances
A. LAOUAR and A. GUERZIZ
Department of Mathematics, Faculty of SciencesUniversity of Annaba, P. O. Box 12, 23000 Annaba, ALGERIA
Department of Physics, Faculty of SciencesUniversity of Annaba, P. O. Box 12, 23000 Annaba, ALGERIA
The generalized theory of the shallow-water [1, 2] is applied to the first order ap-proximation for the calculation of the local disturbances caused by the presence of animmersed vertical barrier. By using the appropriate complex variable theory [3], the
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flow is entirely given. This method gives a new physical interpretation of calculations.For the certain forms of obstacles, the results can be obtained directly by the knowledgeof the flows at adapted classical potential.
References:
[1] Barthelemy E., Kabbaj A. & Germain J.P, Long surface wave scattered by a stepin a two-layer fluid. Fluid Dyn. Res., 26, pp 235-255, 2000.
[2] Dean W.R., On the reflexion of surface waves by a submerged plane barrier. Proc.Cam. Phil. Soc. vol 41 pp 231-238, 1945.
[3] Guerziz A., Etude theorique et experimentale de la cinematique fine des ondeslongues de gravite au voisinage des obstacles. These de doctorat U.J.F Grenoble, 1992
[4] Laouar A., The method of lines and invariant imbedding for free boundary prob-lems of Hel-Shaw (submetted 2006), Inter. J. An. Num. and Modeling.
Parallel backward shooting method for solving stiff IVPs
Dr. Abdulhabib A. A. MURSHED
Faculty of Applied Science (Vice Dean) Thamar University Yemen,[email protected]
This paper investigate the effectiveness of parallel shooting technique for solving stiffinitial value problems in such a way it can be computed in parallel. These techniqueshave the important capacity for controlling the numerical stability of the numericalintegration methods. The overall performance of these techniques can be improvedby increasing the number of the processors and corresponding matching points. Theefficiency of this algorithm is illustrated by means of some numerical examples of stiffsystems, and a comparison with Gear’s method is made.
Integral inclusions in Banach spaces using Henstock-typeintegrals
Bianca SATCO
Facultatea de Inginerie Electrica, Univ. “Stefan cel Mare”Universitatii 13, 720229 Suceava
We consider an integral inclusion, where the set-valued integral involved is of Henstock-type. An existence result is obtained via Monch’s fixed point theorem. A condition usinga measure of noncompactness, as well as uniform integrability conditions appropriate toHenstock integral are required. Since the vector Henstock integral is more general thanBochner and Pettis integrals, our result extends many of previously obtained existenceresults, in single- or set-valued setting.
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Regularization of a solution to the cauchy problem forgeneralized system of Cauchy–Riemann in infinite domains
Ermamat SATTOROV
Norkulovich State University, Samarkand, [email protected]
We consider the problem of analytic continuation of the solution of the generalizedsystem of Cauchy–Riemann in infinite domains through known values of the solution ona part of the boundary, i.e., the Cauchy problem. The generalized system of Cauchy-Riman is elliptic. The Cauchy problem for elliptic equations is well known to be ill-posed; a solution is unique but unstable (Hadamard’s example). To make the statementwell-posed, we have to restrict the class of solutions.
During the last decades the classical ill-posed problems of mathematical physics havebeen of constant interest. This direction in studying the properties of solutions to theCauchy problem for the Laplace equation was originated in the fifties in the articles byM. M. Lavrent’ev (1956 y.) and S. N. Mergelyan (1956 y.) and was developed by V. K.Ivanov (1965 y.), Sh. Ya. Yarmukhamedov (1977 y.), et al.
Good intermediate-rank lattice rules based on the L∞ weightedstar discrepancy
Vasile SINESCU
Department of Mathematics, University of WaikatoPrivate Bag 3105, Hamilton, New Zealand
Good lattice rules for numerical multiple integration may be constructed by usinga ‘component-by-component’ technique, which is a ‘greedy-type’ algorithm based onsuccessive 1-dimensional searches.
Here, we assume that variables are arranged in the decreasing order of their impor-tance. Since the first variables are in some sense more important than the rest, there isan interest in considering ‘intermediate-rank’ lattice rules. The ‘goodness’ of a latticerule is assessed here by a ‘weighted star discrepancy’, which is based on a L∞ maximumerror.
The talk intends to present a survey of results regarding the construction of intermediate-rank lattice rules under a ‘product-weighted’ setting. The existence and the constructionof ‘good’ intermediate-rank lattice rules is proved by using an averaging argument (ifthe average is good, there must be a good one). The weighted star discrepancy for thelattice rules constructed here has order of magnitude of O(n−1(ln n)d). Under appro-priate conditions over the weights, the weighted star discrepancy will have the optimalbound of O(n−1+δ) for any δ > 0, where the involved constant is independent of d andn. Subsequent results for the Lp star discrepancy can be also deduced from the resultson the L∞ weighted star discrepancy.
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Joint work with Stephen Joe (University of Waikato).
On the minimum energy problem for linear systems
Alina VIERU
Department of Mathematics, “Gh. Asachi” Technical University, Iasi, ROMANIAvieru [email protected]
This paper is concerned with some problems of controllability with minimum energyassociated with linear systems in Banach spaces. First, we give conditions to have thecontrollability of a given pair of elements of the Banach space. Also, we find alternateways of obtaining the minimum norm control by which a given initial state of thesystem can be steered to a given final state, within a given time interval. We treat theseproblems as a special case of an abstract minimum norm problem described by a linearmapping. We give examples in order to illustrate the theory.
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Contributing participants
Adesina, O. A., 39 [email protected], S., 39 [email protected], J. O., 40 [email protected], K., 40 [email protected], S., 12 [email protected], D., 1 [email protected]ırsan, M., 14 [email protected], A. A., 12 a a [email protected], A., 40 [email protected], M., 13 [email protected], A., 41 [email protected], C., 14 [email protected], L., 14 [email protected], M., 15 [email protected], R. I., 16, 23, 42, 45 [email protected], M., 16 [email protected], T., 17 [email protected], R., 1 [email protected], M., 41 [email protected], A., 18 [email protected], D. N., 19 [email protected], H. D., 18 [email protected], L., 19 [email protected], C., 20 [email protected], E. R., 42 [email protected], P., 42 jit [email protected], M. B., 20 [email protected], T., 21 [email protected], A. L., 2 [email protected], A. M., 12, 22, 43 a m [email protected], H.-J., 2 [email protected], H. O., 3 [email protected], A.V., 4 [email protected]ıa, A., 22, 43 [email protected], A., 23 [email protected], I.-D., 44 ghiba [email protected], S. M., 23 [email protected], G., 44 [email protected], H. D., 24 [email protected], A., 44 [email protected], I. B., 45 [email protected], M., 5 [email protected], N., 24 [email protected], A., 25 [email protected], A., 45 [email protected]
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Luca-Tudorache, R., 25 [email protected], M., 26 [email protected], V., 27 [email protected], H. R., 27 [email protected], E., 28 [email protected], G., 28 [email protected], M., 29, 33 [email protected], L., 29 [email protected], S., 30 [email protected] Nehi, H., 31 [email protected], B. S., 6 [email protected], D., 32 debasis [email protected], M., 32 [email protected], A. A., 46 [email protected], O. G., 32 [email protected], Z., 6 [email protected], M., 37 [email protected], O., 29, 33 [email protected], E., 7 [email protected], N. H., 33 [email protected], A., 34 [email protected], A., 34 [email protected], N., 35 nic [email protected], N., 36 [email protected], M., 9 [email protected], A., 29 [email protected], V., 36 [email protected], D., 41 [email protected], L., 40 [email protected], M. G., 9 [email protected], B., 46 [email protected], E., 47 [email protected], E., 26 [email protected], V., 47 [email protected], M., 10 [email protected], D., 10 [email protected], G., 10 [email protected], M., 37 [email protected], C., 11 [email protected], A., 48 vieru [email protected], M., 33 [email protected], I. I., 37 [email protected], A., 37 [email protected], A., 38 [email protected]
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