Book of Abstracts
CSMT 2017
PLENARY SESSIONS
High-order Bahri-Lions and Pohozaev Liouville-type theorems
Abdellaziz Harrabi
University of Kairouan , Tunisia
[email protected]
Abstract
We consider the following higher-order elliptic equations:
(−∆)r u = f (u) in IRn ,
(1)
r+1
where r ∈ IN∗ , n ≥ 2r + 1, f ∈ C(IR) and u ∈ Hloc
(IRn ) is a weak solution with f (u) ∈
2n
n+2r
Lloc
(IRn ). We first establish a variant of Pohozaev identity to derive some nonexistence
results when f has a global subcritical or supercritical growth and u verifies respectively
Z
Z
r 2
|∇ u| +
F (u) = o(1)
AR
or
Z
AR
r
2
|∇ u| + R
−2r
AR
Z
as R → ∞. Here F (s) =
Z
u2 = o(1)
AR
s
f (t)dt and AR = {β1 R < |x| < β2 R} for some 0 < β1 < β2 .
0
The now aspect here consists in proving similar nonexistence results on a half-space IRn+ =
{x ∈ IRn , xn > 0}, under the Dirichlet Boundary conditions, which in general is more difficult
to deal with, due to the boundary integrals on {xn = 0} raised in the Pohozaev identity, also
for r ≥ 2, the extension by reflection fails to exhibit an entire solution of (1).
Consequently, we extend the well known Bahri-Lions Liouville-type theorems [1] to (1) for a
large class of superlinear nonlinearities f having quasi-critical growth. In particular, for stable
solutions, we need only a subcritical or critical growth near zero. Our approach to get the integral
estimate from the stability makes use of delicate analysis with appropriate test functions and
weighted seminorms. At last, we discuss the case of half-space which is essential to prove the
relevant Bahri-Lions L∞ bounds for solutions in bounded domain via their Morse indices.
References
[1] A.Bahri and P.L.Lions, Solutions of superlinear elliptic equations and their Morse indices,
Comm. Pure Appl. Math. 45 (1992), 1205-1215.
In Memory of My Great Professor Abbas Bahri, with Gratitude
2
Characterization of the Palais-Smale sequences for the Conformal Dirac-Einstein
equation and applications
Ali Mâalaoui
[email protected]
Abstract
In this presentation, we study the behavior of the Palais-Smale sequences of the conformal Dirac-Einstein equation, in 3-dimensional compact Riemannian manifolds. The problem
originates from the super-symmetric model of coupling gravity with fermionic energy. Since
the energy functional is critical, bubbling occurs. We give here a precise characterization of
the violation of the (PS) condition and provide an Aubin type inequality guarantying the
existence of solutions. Along the proof we see that this problem is in tight relation with the
classical Yamabe problem and the Spinorial version of the Yamabe problem.
Opérateurs de Dirac équivariants et théorie géométrique des invariants
différentielle
Paul Emile Paradan
University of Montpellier, France
[email protected]
Abstract
Soit M une variété compacte munie d’une structure Spin invariante par l’action d’un
groupe compact. Si L est un fibré en droites équivariant sur M, nous donnons une formule
géométrique pour les multiplicités de l’indice équivariant de l’opérateur de Dirac sur M
twisté par L.
Stabilité via observabilité pour une classe de systèmes dissipatifs
Ahmed Bchatnia
Faculté des Sciences de Tunis, Tunisie
[email protected]
Abstract
Nous considérons dans cet exposé des équations d’évolution du second ordre dissipées
non-linéairement. Nous présenterons tout d’abord une caractérisation du taux de décroissance de l’énergie des solutions en utilisant une estimation d’observabilité pour le problème
non amorti associé. Nous considérons aussi le cas du domaine non-cylindrique et nous
démontrerons une estimation d’observabilité pour l’équation des ondes unidimensionnelle
posée sur un domine dépendant du temps.
3
Cohomology of osp(1|2) acting on the space of n-ary differential operators on R1|1
Mabrouk Ben Ammar
Departement de Mathématiques, Faculté des Sciences de Sfax , Tunisie
[email protected]
Abstract
We consider the µ-densities spaces Fµ with µ ∈ R, we compute the space H1diff (aff(1), Dλ,µ )
where λ = (λ1 , . . . , λn ) ∈ Rn and Dλ,µ is the space of n-ary differential operators from
Fλ1 ⊗ · · · ⊗ Fλn to Fµ . We also compute the super analog spaces H1diff (aff(1|1), Dλ,µ ),
H1diff (osp(1|2), aff(1|1), Dλ,µ ) and H1diff (osp(1|2), Dλ,µ ).
Convoluted Brownian motion: a semimartingale approach
Pierre Vallois
Elie Cartan Institute, University of Lorraine, France
[email protected]
Abstract
We analyse semimartingale properties of a class of Gaussian periodic processes, called
convoluted Brownian motions, obtained by convolution between a deterministic function and
a Brownian motion. A classical example in this class is the periodic Ornstein-Uhlenbeck process. We compute their characteristics and show that in general, they are never Markovian
nor satisfy a time-Markov field property. Nevertheless, by enlargement of filtration and/or
addition of a one-dimensional component, one can in some case recover the Markovianity.
We treat exhaustively the case of the bidimensional trigonometric convoluted Brownian motion and the multidimensional monomial convoluted Brownian motion. (joint work with S.
Roelly)
Théorèmes de type Hardy-Littlewood avec bornes indépendantes de la dimension
Luc Deleaval
[email protected]
Abstract
4
En 1982, Stein établit le résultat spectaculaire que l’opérateur maximal de HardyLittlewood peut être borné indépendamment de la dimension sur Lp (Rd ), 1 < p ≤ +∞,
ce qui a initié le programme de borner indépendamment de la dimension des opérateurs
maximaux de type Hardy-Littlewood. Dans cet exposé, nous donnerons un aperçu de résultats significatifs obtenus dans cette direction par Stein, puis par Bourgain, Carbery et
Müller. Nous présenterons ensuite un résultat récent obtenu avec C. Kriegler sur une
extension du résultat de Stein dans un cadre vectoriel.
Variété des représentations du groupe d’un nœud
Leila Ben Abdelghani
[email protected]
Abstract
Les noeuds ont commencé à intéresser les mathématiciens depuis une centaine d’années.
Pour leur classification, on utilise le concept d’« invariant de noeud.
Un des invariants les plus connus d’un noeud est son groupe dont l’étude directe est en
général très délicate. Une méthode permettant de contourner cette difficulté consiste en
l’étude de la variété des représentations du groupe du noeud dans un groupe qu’on maîtrise.
Des travaux récents prouvent l’existence d’un lien étroit entre la théorie des représentations
des groupes fondamentaux des variétés de dimension trois et la géométrie et la topologie
de ces variétés. Dans cet exposé, je présenterai un panorama général des résultats récents
concernant la structure locale de la variété des représentations du groupe du noeud tout en
mentionnant des contributions personnelles.
On functional Alexandroff spaces
Sami Lazâar
University of Sciences of Gafsa, Tunisia
[email protected]
Abstract
Alexandroff topological spaces (or principal topological spaces) are topological spaces
in which arbitrary intersections of open sets are open. They were introduced by Alexandroff [1] in 1937 and have been studied extensively. Because every finite topological space
is Alexandroff and computer arithmetic and graphics are based on fintie sets of machine
numbers or pixels, computer applications have driven much interest in Alexandroff spaces.
If X is a set and f : X −→ X is a function, the set {A ⊆ X : f (A) ⊆ A} of f-invariant
subsets of X form the closed sets of an Alexandroff topology P (f ) on X. Alexandroff
topologies which are realized as P (f ) for some function f : X −→ X are called functionally
Alexandroff spaces [6], or primal spaces [2]. Further properties of functionally Alexandroff
5
spaces have been studied in [4]. In a functionally Alexandroff space (X; P (f )), if x ∈ X,
the closure of {x} is the orbit {f n (x) : n ∈ N}. The smallest open neighborhood of x ∈ X
is Vf (x) = {y ∈ X : f n (y) = x for some n ∈ N}.
In this talk we introduce and study some properties of this particular class of Alexandroff
spaces.
References
[1] P. Alexandroff, Diskrete Räme, Mat. Sb. (N.S.) 2 (1937) 501-518.
[2] O. Echi, The category of flows of Set and Top, Topology Appl. 159 (2012) 2357-2366.
[3] A. Haouati and S. Lazaar. Primal spaces and quasihomeomrphisms, Appl. Gen. Topol. 16
(2015) 109-118.
[4] S. Lazaar, T. A. Richmond, T. Turki, Enumerating maps giving the same primal space, Q.
M. (to appear).
[5] T. A. Richmond, Quasiorders, Principal topologies, and Partially Ordered Partitions, Internat. J. Math. & Math. Sci. 21 (1998), no. 2, 221-234.
[6] F. Ayatollah Zadeh Shirazi and N. Golestani, Functional Alexandroff spaces, Hacettepe
Journal of Mathematics and Statistics. 40 (2011), 515-522.
Convexifying positive polynomials and a proximity algorithm
Krzysztof Kurdyka
University of Savoie MontBlanc, France
[email protected]
Abstract
We prove that if f is a positive C 2 function on a convex compact set X then it becomes
strongly convex when multiplied by (1 + |x|2 )N with N large enough. For f polynomial we
give an explicit estimate for N , which depends on the size of the coefficients of f and on
the lower bound of f on X. As an application of our convexification method we propose
an algorithm which for a given polynomial f on a convex compact semialgebraic set X
produces a sequence (starting from an arbitrary point in X) which converges to a (lower)
critical point of f on X. The convergence is based on the method of talweg which is a
generalization of the Lojasiewicz gradient inequality. (Joint work with S. Spodzieja).
Nonexistence results of solutions for biharmonic equations with supercritical
Saïma Khenissy
6
[email protected]
Abstract
We present results of nonexistence of solutions for biharmonic equations with supercritical nonlinearities as source terms. These results were proved by Donato Passaseo in 1983
and 1984 in the case of the Laplacian using the identity of Pohozeav. Precisely, Passaseo
constructs appropriate vector fields h for use in Pohozaev multipliers. We show these results
in the case of the biharmonic operator (elasticity operator). These results of nonexistence of
solutions are much awaited but the challenge is to build, like Passaseo, vector fields suitable
for the biharmonic operator. Moreover, we give a classification of domains based on some
classifying number (introduced by Renate Schaaf for the case of the Laplacian) for which
these results are valid: the Dirichlet h-starshaped Domains and the Navier h-starshaped
Domains according to the Dirichlet or Navier boundary conditions. Thus, we show that our
results of nonexistence of solutions are valid for domains having a nontrivial topology in
the sense of Bahri-Coron (homotopies with spheres) or having a rich geometry (dumbbell
shaped for example), but for large dimensions. Moreover, results of uniqueness of solutions for these same domains and nonlinearities are given when a bifurcation parameter is
introduced. These results are the subject of three publications.
Comment les métriques peuvent-elles expliquer la géométrie de certaines variétés
complexes?
Hervé Gaussier
Institut Joseph Fourier, 100 rue des Maths, St Martin d’Heres, Grenoble, France
[email protected]
Abstract
Nous expliquons comment la géométrie du bord de certains variétés complexes non compactes peut être décrite par le comportement des géodésiques pour certaines métriques
invariantes.
On the cortex of some solvable Lie algebra
Béchir Dali
Departement de mathématiques, Faculté des Sciences de Bizerte, 7021 Zarzouna, Bizerte
[email protected]
Abstract
The notion of cortex was first introduced by A. M. Vershik and S. I. Karpushev in their
study about cohomology groups of a locally compact group G with coefficients in a unitary
representation. It is defined as
b which cannot be Hausdorff-separated from 1G }.
cor(G) = {π ∈ G,
7
When G is connected and simply connected exponential Lie group with Lie algebra g, the
Kirillov theory enables us to identify cor(G) with a subset of g∗ where g∗ is the dual space
of the Lie algebra g of G and hence an analogous subset corG denoted by cor(g∗ ) has been
defined, more precisely
Cor(g∗ ) = { lim Ad∗xn (`n ), (xn )n ⊂ G, (`n )n ⊂ g∗ , lim `n = 0}.
n→∞
n→∞
∗
In this talk, we deal with cor(g ) where g is an exponential solvable Lie algebra. For this
class of Lie algebras, we exhibit some results, we detail some examples, and we give an
explicit description and a concrete algorithm for the determination of the cortex of some
nilpotent Lie algebras. Finally, we give some perspectives.
Graph packing problems
Kheddouci Hamamache
[email protected]
Abstract
In this talk, I review, classify and discuss several known conjectures, recent advances and
results obtained on graph packing problems. In particular, we give bounds and algorithms
for these problems and some of their applications in computer science.
Opérateurs de transmutation et transformations de Fourier de type Dunkl et leurs
applications
Fraj Chouchen
University of Sousse, Tunisia
[email protected]
Concentration at submanifolds for an elliptic Dirichlet problem near high critical
exponents
Fethi Mahmoudi
CMM, Universidad de Chile
[email protected]
Abstract
8
Let Ω be a open bounded domain in Rn with smooth boundary ∂Ω. We consider the
n−k+2
equation ∆u + u n−k−2 −ε = 0 in Ω, under zero Dirichlet boundary condition, where ε is a
small positive parameter. We assume that there is a k-dimensional closed, embedded minimal submanifold K of ∂Ω, which is non-degenerate, and along which a certain weighted
average of sectional curvatures of ∂Ω is negative. Under these assumptions, we prove existence of a sequence ε = εj and a solution uε which concentrate along K, as ε → 0+ , in the
sense that
n−k
2
δK
|∇uε |2 * Sn−k
as ε → 0
where δK stands for the Dirac measure supported on K and Sn−k is an explicit positive
constant.
Work in Collaboration with Shengbing Deng and Monica Musso
9
PARALLEL SESSIONS
Mathematical analysis of a three-step model of anaerobic digestion with different
removal rates
Nahla Abdellatif
a
a,b
University of Manouba, ENSI, Campus Universitaire de Manouba, 2010 Manouba, Tunisie.,
University of Tunis El Manar, ENIT, LAMSIN, BP 37, Le Belvédère, 1002 Tunis, Tunisie.
[email protected]
b
Abstract
Anaerobic digestion is a biological process in which organic matter is transformed into
methane and carbon dioxide (biogas) by microorganisms in the absence of oxygen. The
search for models simple enough to be used for control design is of prior importance today
to optimize the process and solve important problems such as the development of renewable
energy from waste. The anaerobic digestion is generally considered as a three step process: hydrolysis and liquefaction of the large, insoluble organic molecules by extracellular
enzymes, acid production by an acidogenic microbial consortium and a methane production
stage realized by a methanogenic ecosystem. Several mathematical models describing these
phenomena have been proposed in the literature. However, they are usually too complex
to be used for control synthesis [1-3,5]. For control purposes, the most appropriate way to
model the hydrolysis is still an open problem. The biologists believe that hydrolytic bacteria
has an important role in the whole process. Thus, we consider that the substrate compartment is divided into two parts: slowly degradable substrates X0 and readily biodegradable
substrate S1 which may be, for keeping the model simple enough, considered to be degraded
by the same biomass, e.g. a unique compartment grouping "hydrolitic and acidogenesis"
bacteria.
In this work, we analyze a mathematical model of the chemostat with enzymatic degradation of an organic substrate under a solid form, including positive mortality terms. This
work is an extension of [4]. The three step model reads:

Ẋ0




 Ṡ1
Ẋ1



Ṡ

 2
Ẋ2
= D (X0in − X0 ) − µ0 (X0 )X1 ,
= D (S1in − S1 ) + k0 µ0 (X0 )X1 − k1 µ1 (S1 )X1 ,
= (µ1 (S1 ) − (D + a1 )) X1 ,
= D (S2in − S2 ) + k2 µ1 (S1 )X1 − k3 µ2 (S2 )X2 ,
= (µ2 (S2 ) − (D + a2 )) X2 ,
(2)
where X0 (t) the concentration of the slowly biodegradable substrate (typically the solid
Chemical Oxygen Demand) at time t, with X0in the concentration in the input. Sj (t)
denote the concentrations of the substrates in the effluent, j = 1, 2 (the easily biodegradable
COD and the Volatile Fatty Acids, respectively), at time t; with Sjin the input substrate
concentrations j. Xi (t) denote the concentrations of the ith population of microorganisms,
i = 1, 2, at time t (the hydrolitic and acidogens on the first side, and the methanogens on the
other side). D denotes the dilution rate of the chemostat and ai , i = 1, 2 is a non-negative
parameter giving rise to death rate ai Xi which is due to decay.
We study the existence and the stability of equilibrium points of the model assuming a
10
monotonic growth rate for µi , i = 0, 1. The analysis of the 3-step model is derived from a
smaller order sub-model. We show that the considered sub-model may exhibit bistability.
The analysis of the 3-step model shows the existence of at most four positive equilibrium
and we prove that introducing decay in the model preserves the stability of the equilibria.
These results can be extended for a non monotonic growth rate and to the case of a density
dependence growth rate. Density dependence refers to the case where the growth rate
depends not only on the substrate concentration but also on the biomass, a case which
can be of interest to model non heterogeneous environments in the chemostat. We present
numerical experiments which are in perfect coherence with the analysis.
References
[1] B. Benyahia, T.Sari, B.Cherki and J. Harmand. Equilibria of an anaerobic wastewater
treatment process and their stability. JR. Banga, P. Bogaerts, J. Van Impe, D. Dochain and
I. Smets, editors, 11th International Symposium on Computer Applications in Biotechnology
(CAB 2010), pages 371-376. International Federation of Automatic Control, Leuven: Belgique,
2010.
[2] B. Benyahia, T.Sari, B.Cherki and J. Harmand. Sur le modèle AM2 de digestion anaérobie.
E. Badouel, A. Sbihi and I. Lokpo, editors, CARI’10, Proceedings of the 10th African Conference on Research in Computer Science and Applied Mathematics, pages 453-460, INRIA, 2010.
[3] G. Bastin and D. Dochain. On-line estimation and adaptive control of bioreactors, Dynamics
of Microbial Competition. A. and N.Y., Elsevier Science Publishers, 1991.
[4] R. Fekih-Salem, N. Abdellatif, T. Sari and J. Harmand. Analyse mathématiques d’un modèle
de digestion anaérobie à trois étapes. ARIMA Journal, volume 17, pages 53-71, 2014.
[5] I. Simeonov and S. Stoyanov. Modelling and dynamic compensator control of the anaerobic
digestion of organic wastes. Chem. Biochem. Eng. Q, volume 17(4), pages 285-292, 2003.
Impulsive Fractional Differential Equations with State-Dependent Delay
Khalida Aissani
Departement of Mathematics, University of Bechar, Algérie
aissani− [email protected]
Mouffak Benchohra
Department of Mathematics, University of Sidi Bel Abbès, Algérie
[email protected]
Abstract
In this work, we consider the existence of mild solutions for a class of impulsive fractional
11
equations with state-dependent delay described by
Dtα x(t) = Ax(t) + f (t, xρ(t,xt ) , x(t)),
∆x(tk ) =
t ∈ Jk = (tk , tk+1 ], k = 0, 1, . . . , m,
Ik (x(t−
k )),
k = 1, 2, . . . , m,
(3)
t ∈ (−∞, 0],
x(t) = φ(t),
where Dα is the Caputo fractional derivative of order 0 < α < 1, T > 0, A : D(A) ⊂ E → E
is the infinitesimal generator of an α-resolvent family (Sα (t))t≥0 , f : J × B × E −→ E
is a given function and ρ : J × B → (−∞, T ] are appropriated functions. Here, 0 =
t0 < t1 < . . . < tm < tm+1 = T, Ik : E → E, k = 1, 2, . . . , m, are given functions,
−
+
−
∆x(tk ) = x(t+
k ) − x(tk ), x(tk ) = lim x(tk + h) and x(tk ) = lim x(tk − h) denotes the right
h→0
h→0
and the left limit of x(t) at t = tk , respectively. We denote by xt the element of B defined
by xt (θ) = x(t + θ), θ ∈ (−∞, 0]. Here xt represents the history of the state from −∞ up to
the present time t. We assume that the histories xt belongs to some abstract phase space
B, to be specified later, and φ ∈ B.
References
[1] S. Abbas, M. Benchohra and G.M. N’Guérékata, Advanced Fractional Differential and Integral Equations, Nava Science Publishers, New York, 2015.
[2] R. P. Agarwal, B. De Andrade, and G. Siracusa, On fractional integro-difierential equations
with state-dependent delay, Comput. Math. Appl., 62 (2011), 1143-1149.
[3] K. Aissani and M. Benchohra, Semilinear fractional order integro-differential equations with
infinite delay in Banach spaces, Arch. Math., 49 (2013), 105-117.
Mathematical modeling and energy estimation of anaeorobic bioreactors
Norelhouda Azzizi
[email protected]
Abstract
The simulation of anaerobic bioreactors of batch type for the methane production according to the AM2 mathematical model reveals a strong sensitivity of the results (variation
of the concentrations of substrates, bacterial evolution and methane production ) with respect to the variations of the model parameters.
To highlight this sensitivity, we undertook an extensive literature review that has actually shown a dispersion of the model parameters as given and estimated by different
authors. This is due, probably, to the complexity of biotechnology phenomena, to the variety in the composition of the substrate and to other factors influencing the experimental
conditions (pH, temperature, etc.). An "average" estimate of the model parameters based
on the literature was determined and was used to simulate the operation of the bioreactors.
A comparative analysis of this model is performed enabling to show the variability of the
system parameters and its influence on the methane production.
Keywords: Anaerobic digestion, AM2 model, Simulation of anaerobic bioreactors,
Biotechnology.
12
References
[1] V.A. Vavilin, L. Ya. Lokshina, S. V. Rytov, "The <Methane> Simulation Model As The
First Generic User-Friend Model Of Anaerobic Digestion", vestnik moskovskogo universiteta.
Khimiya . Vol. 41, No. 6, (2000).
[2] J. Reynard , "Modélisation, optimisation dynamique et commande d’un méthaniseur par
digestion anaérobie", Rapport de projet de fin d’études, Esisar, (2006/2007).
[3] G.kiely et al, "physical and mathematical modling of anaerobic digestion of organic wastes",
War. Res. Vol. 31, No. 3: 534-540, (1997)
[4] T.G. Muller et al, "Parameter identification in dynamical models of anaerobic waste water
treatment", Mathematical Biosciences 177 & 178:147–160, Elsevier (2002).
Lie triple systems and invariant scalar products
Amir Baklouti
[email protected]
Abstract
A bilinear form on a Jordan (resp. Lie) triple system which is symmetric invariant and
non degenerate is called, with misuse of language, invariant scalar product. A Jordan (resp.
Lie) triple system endowed with such a bilinear form is said to be pseudo-Euclidean (resp.
quadratic). we aim to:
• show that the set of pseudo-Euclidean Jordan (resp. quadratic Lie) triple systems
contains strictly solvable and semi-simple Jordan (resp. Lie ) triple systems
• give two new characterizations of semisimple Jordan triple systems
• describe quadratic Lie triple systems by using the notion of double extensions.
Laplaciens non auto-adjoints sur un graphe orienté
Marwa Balti
[email protected]
Abstract
Dans ce travail, on considère un graphe pondéré orienté G avec un poids d’arêtes non
symétrique et on introduit un Laplacien non auto-adjoint associé à G. On s’intéresse aux
différentes propriétés spectrales de ces Laplaciens en s’appuyant sur l’étude d’autres opérateurs auto-adjoints speciaux pour obtenir des résultats sur le spectre. En outre, on établit
13
des inégalités isopérimétriques relatives à l’image numérique de Laplacien non symétrique.
Elles servent à montrer l’absence de spectre essentiel de notre Laplacien sur des graphes
lourds à l’infini.
Asymptotic behaviour for second-order differential equations with nonlinear
slowly time-decaying damping and integrable source
Mounir Balti
Faculte des sciences de Tunis université Tunis-el-manar Tunisie, IPEST
[email protected]
Abstract
On étudie le comportement asymptotique d’une solution de l’équation différentielle:
u00 (t) + γ(t)u0 (t) + 5F (u(t)) = g(t), t ≥ 0,
(4)
où F : H → R est une fonction de classe C1 , convexe, minorée sur un espace de Hilbert
H, γ : R+ → R+ une fonction positive qui a un comportement équivalent à tcα avec c > 0
et α ∈ [0, 1[. On obtien des conditions suffisantes sur le second terme g, qui assure la
convergence faible d’une solution u : R+ →H de (1) quand t → +∞ vers un point critique
de F s’il existe.
References
[1] F. Alvarez, On the minimizing properties of a second order dissipative system in Hilbert spaces.
SIAM J. Cont. Optim. 38 (4)(2000) 1102-1119..
[2] H. Attouch and Z. Chbani, Fast inertial dynamics and FISTA algorithms in convex optimi-
sation. Perturbation aspect. arXiv: 1507.01367v1. (2105).
[3] A. Cabot and P. Frankel, Asymptotics for some semilinear hyperbolic equations with non-
autonomous damping. J. Differential Equations 252 (2012) 294-322.
[4] A. Haraux, M.A. Jendoubi, On a second order dissipative ODE in Hilbert space with an
integrable source term. Acta Mathematica Scientia 32B(1)(2012) 155-163.
[5] M.A Jendoubi and R. May, Asymptotics for a second-order differential equation with non-
autonomous damping and an integrable source term. Applicable Analysis 94(2)(2015) 435-443.
[6] R. May, Long time behavior for a semilinear hyperbolic equation with asymptotically van-
ishing damping term and convex potential. J. Math. Anal. Appl. 430 (2015) 410-416.
Meixner Polynomials and 1D Para-bose Oscillator
Hanen Ben Mansour
14
Department of mathematics, University of Carthage, Faculty of Sciences of Bizerte 7021
Tunisia.
Department of mathematics, University of Tunis El Manar, Faculty of Sciences of Tunis, Tunis
2092, Tunisia.
[email protected]
Abstract
In this paper, the matrix elements of the para–squeeze operator in the eigenstate basis of
the para-bose oscillator are computed explicitly. These matrix elements are seen to involve
the Meixner polynomials. The underlying framework allows for the systematic derivation of
the recurrence relations, difference equation, lowering and rising operators and generating
functions, that these polynomials satisfy.
Jacobi spectral approximations
Abdellatif Bentaleb
Department of Mathematics and computer sciences, University Moulay Ismaïl, Faculty of
Sciences, BP 11201 Zitoune, Meknès, Morocco
[email protected]
Abstract
The aim goal of this note is to study the heat Jacobi semigroup generated by the operator Lf (x) := (1 − x2 )f 00 + [(β − α) − (α + β + 2)x]f 0 , α, β > −1, acting on the Hilbert space
L2 ((−1, 1), µ) with weight the normalized beta-type measureon (−1, 1), µ(dx) = cα,β (1 −
n
Q
x)α (1+x)β dx. We use some basic properties of the semigroups exp(t
(L − k(k + α + β + 1))
k=0
n ≥ 0, to analyze a large family of geometric inequalities that does not exist in the literature
and with which reinforced the (integral) Poincaré inequality. We also analyze the special
case when these inequalities are restricted to functions with are orthogonals to the space of
polynomials with degree less m.
Keywords and phrases. Jacobi operators, heat semigroup, Spectral gap, Poincaré
inequality, Superior order Poincaré-type inequality.
Inférence statistique dans un modèle à hasards proportionnels stratifié: test
d’ajustement
Rim Ben Elouefi
Monastir University, Tunisia.
[email protected]
Jean-François Dupuy
15
,
t≥0
IRMAR-INSA, Rennes, France.
[email protected]
Abstract
Dans cet exposé, on va présenter le test d’ajustement pour le modèle de régression à
hasards proportionnels stratifié. On va proposer une nouvelle statistique de test et investir
théoriquement et numériquement ses propriétés asymptotiques.
keyword: Processus de comptage, Processus Gaussien, Résidus des martingales, Distribution asymptotique, Données de survie.
Le modèle à hasards proportionnels startifié (Voir Martinussen et Scheike [1], par exemple) généralise le modèle de régression de Cox à hasards proportionnels pour des données
de survie partagées sur des différents groupes (référés pour nous par des strates) d’une population sous étude, en admettant des fonctions de hasards de base distinctes. Pécisement,
la strate divise l’échantillon en J groupes disjoints avec des différentes fonctions de hasards
de base α0,j et une valeur de paramètre de régression commune.
La fonction de risque instantanné est donnée par
>
αj (t) = α0,j (t)eβ0 Z ,
j = 1, . . . , J,
(5)
où Z est un p-vecteur de covariables, β0 est un p-vecteur de paramètre de régression inconnu
et α0,j ,j = 1, ..., J sont des fonction de hasards de base inconnues.
L’estimateur de maximum de vraisemblance βbn de β0 est consistant et asymptotiquement
gaussien (Voir Martinussen et Scheike [1]). Dans ce travail, on étudie l’adéquation du modèle
(5), basée sur un échantillon des données censurées à droite (Xi , ∆i , Zi , Si ) i = 1, ..., n. Soit
Z
Mi (t) = Ni (t) −
t
Yi (s)
0
J
X
>
α0,j (s)eβ0 Zi 1{Si =j} ds, i = 1, . . . , n
j=1
les martingales basées sur les processus de comptage. Nous définissons la somme cumulative
des résidus des martingales comme
n
1 Xc b
Mi (t, βn )1{Si =j} 1{Zi ≤z} ,
Qjn,z (t, βbn ) := √
n i=1
(6)
ci la version esavec z = (z1 , . . . , zp )> ∈ Rp , Si indique la strate de ième individus, M
timée de Mi dans [0, τ ] la période d’observation. Alors on définit notre statistique de test
d’ajustement comme
Sn := max max |Qjn,zq (τ, βbn )|.
1≤j≤J q∈Q
(7)
Dans cet exposé, on va étudier la distribution asymptotique de Sn sous l’hypothèse nulle
où le modèle (5) est bien correct. Nous utilisons l’approche basée sur la méthode de bootstrapp pour déterminer les valeurs critiques de notre test puis on va présenter les propriétés
asymptotiques de Sn à travers les simulations (en particulier on va évaluer son niveau et sa
puissance sous l’hypothèse nulle contre d’autres différentes alternatives).
Dans une deuxième partie on considère le modèle stratifié à partier d’un certain seuil ω ∗
inconnu
(
>
α0,1 (t)eβ0 Z if W ≤ ω ∗
α(t|Z, W ) =
(8)
>
α0,2 (t)eβ0 Z if W > ω ∗
Une version modifiée de l’estimateur de Maximum de vraisemblance est proposée. On définit
par la suite la somme partielle des résidus en considérant le cas de deux strates seulement.
16
Sous le modèle (8) le somme spécifiée pour chaque strate est donnée par
n
1 Xc
b
Q(1)
Mi,ω 1{Wi ≤ω} 1{βb> Zi ≤r}
r (βn,ω , ω) := √
n,ω
n i=1
et
n
1 Xc
b
Q(2)
Mi,ω 1{Wi >ω} 1{βb> Zi ≤r} ,
r (βn,ω , ω) := √
n,ω
n i=1
On se propose d’estimer le seuil ω ∗ par une valeur ω̂, une étude de simulation est proposé
avec des configurations variées et des résultats numériques et graphiques sont élaborées.
Spectral analysis of non-uniform Timoshenko beam acting on shear force with
feedback controller
Kaouther Boulehmi
Faculté des Sciences de Bizerte, Université de Carthage, Jarzouna 7021, Tunisie.
[email protected]
Abstract
A spectral method based on eigenfrequency method is applied to a non-uniform Timoshenko beam with a new thermoelastic coupling on shear force to get the uniform stabilization by a feedback controller. Firstly, we prove the well-posedness of the considered
problem and that the operator has compact resolvent and generates a C0 −semigroup in an
appropriate space. Then, we prove the existence of a sequence of generalized eigenvectors
that forms a Riesz basis. The asymptotic expressions of the spectrum obtained by a detailed
spectral analysis allow us to prove the exponential stability of the problem independently of
the wave speeds of propagation and to deduce the spectrum determined growth condition.
However, it no longer decays exponentially if there is no feedback controller. As application,
we consider the uniform problem to improve some already existing results and to find the
best choice of the feedback ensuring the exponential stability.
Laplacien discret sur une triangulation pondérée
Yassine Chebbi
[email protected]
Abstract
Nous introduisons la notion de face orientée et plus particulièrement de triangle sur
un graphe pondéré connexe orienté localement fini. Ce cadre nous permet alors de définir
l’opérateur de Laplace sur cette structure de 2-complexe simplicial et d’en étudier le caractère essentiellement auto-adjoint.
17
The distribution law of divisors on a sequence of integers
Mohamed Saber Daoud
[email protected]
Abstract
Le travail se place dans le cadre de la théorie probabiliste des nombres et de l’étude de
la répartition en moyenne des diviseurs d’un entier n strictement positif.
Deshouillers,Dress et Tenenbaum ont estimé la probabilité d’un diviseur de n d’être dans
l’intervalle [1, nv ] pour v ∈ [0, 1] . Plus précisément, ils définissent, pour chaque entier n, la
log d
variable aléatoire Dn , prenant les valeurs log
n , lorsque d parcourt l’ensemble des diviseurs
1
de n, avec une probabilité uniforme τ (n) , où τ (n) est le nombre total de ces diviseurs.
La fonction de répartition Fn de Dn est définie par:
Fn (v) := P rob(Dn ≤ v) =
X
1
1 ,
τ (n) d/n
v ∈ [0, 1].
d≤nv
Ils montrent que
Théorème Uniformément pour x ≥ 2 et 0 ≤ v ≤ 1, on a:
√
2
1X
1
Fn (v) = arcsin v + O( √
)
x
π
log x
n≤x
Notre but est de donner une généralisation du résultat précédent. Ainsi, pour une classe
de fonctions multiplicatives bien définie, nous montrons
Théorème :Soit (f, g) ∈ M(α, δ). Uniformément pour x > 1 et 0 ≤ v ≤ 1, on a
X
1
F (n, nv )
1
g(n)
= B(v, δ − α, α) + O(
),
inf(α,δ−α)
G(x) n≤x
F (n)
(log x)
avec
G(x) :=
X
g(n),
n≤x
F (n, v) =
X
f (d)
,
F (n, n) = F (n),
v ≥ 0,
d|n
d≤v
et B(v, a, b) désigne la fonction de répartition de la loi béta de paramètres a > 0 etb > 0
définie par
Z v
du
1
B(v, a, b) =
, v ∈ [0, 1].
B(a, b) 0 u1−a (1 − u)1−b
On donnera par la suite une généralisation de notre résultat sur les entiers friables
18
Around Multivariate Credibility: Properties of the Covariance Matrix
Ahlem Djebar
LaPS laboratory, Badji-Mokhtar University BP12, Annaba 23000-Algeria
[email protected]
Mohamed Riad Remita
LaPS laboratory, Badji-Mokhtar University BP12, Annaba 23000-Algeria
[email protected]
Halim Zeghdoudi
LaPS laboratory, Badji-Mokhtar University BP12, Annaba 23000-Algeria
[email protected]
Abstract
The credibility formula of Frees (2003) considers the data from both the claim number
and claim amounts processes. In this paper we will add some properties of the covariance
matrix which has a significant role theoretically and practically.
Keywords: Credibility theory, Irreducibility of a sequence of random variables, Positive
definite matrix, Frees formula.
Combined effects in fractional boundary value problems
Safa Dridi
Laboratoire de Modélisation mathématique, Analyse harmonique et Théorie du potentiel de la
faculté des sciences de Tunis
[email protected]
Abstract
This paper deals with existence and uniqueness of a positive solution for the fractional
boundary value problem
(
Dα u(x) = −a1 (x)uσ1 − a2 (x)uσ2 , x ∈ (0, 1),
lim Dα−1 u(x) = 0, u(1) = 0,
x→0+
where 1 < α ≤ 2, σ1 , σ2 ∈ (−1, 1) and a1 , a2 are nonnegative continuous functions on (0, 1)
that may be singular at x = 0 or x = 1. We also give the global behavior of a such solution.
19
Size and partition function of a fragmentation process
Samia Elji
[email protected]
Abstract
A random fragmentation of a segment Ix of length x ≥ 1 is considered. Ix splits into b
pieces of lengthPxV1 , . . . , xVb where V1 , . . . , Vb are random variables satisfying the dissipative
b
condition 0 ≤ j=1 Vj ≤ 1 almost surely. The process continue for all subinterval of length
≥ 1. We investigate the asymptotic behavior of the limiting partition function of the
corresponding tree. A phase of transition is occurred under the condition of the existence
of a Malthusian parameter β ∗ > 0 for the associated age dependent branching process. We
appeal for contraction method, limit theorems for random characteristics and martingale
theory.
Practical uniform input to state stability of perturbed triangular systems
Ines Ellouze
Department of Mathematics, Faculty of Sciences of Sfax , Tunisia
[email protected]
Abstract
In this paper, we present a practical uniform input-to-state stability result for perturbed
triangular systems depending on a parameter. We present sufficient conditions for which
each of these notions is preserved under cascade interconnection.
Distributions splines généralisées
Faïza Fourati
[email protected]
Abstract
Les distribtions splines généralisées ont été introduites pour étudier les projections de
mesures orbitales relatives à une action d’un groupe de Lie compact sur un espace vectoriel.
Leur définition fait intervenir la transformation de Markov-Krein. Une formule explicite
a été obtenue pour la densité des distributions splines généralisées. Les limites de ces
distributions font intervenir des distributions de Thorin-Bondesson.
20
Minimum Modulus, Perturbation for Essential Ascent and Descent of a Closed
Linear Relation in Hilbert Spaces
Zied Garbouj
Université de Monastir, Faculté des Sciences de Monastir
[email protected]
Abstract
The notion of ascent (resp. descent, essential ascent, essential descent) of a linear operator
was studied in several papers. In recent years some work has been devoted to extend these
concepts to the case of linear relations. In [E. Chafai and M. Mnif, Extracta Mathematicae,
29, 117-139, 2014], some properties related to descent (resp. essential descent) of linear operators are extended to linear relations (usually with additional conditions). We prove that
some results of [E. Chafai and M. Mnif, Extracta Mathematicae, 29, 117-139, 2014] related
to the stability of the essential descent and descent of a linear relation T everywhere defined
such that T (0) ⊆ ker(T ) by a finite rank operator F commuting with T remain valid when
F is an everywhere defined linear relation and without the assumption that T (0) ⊆ ker(T ).
We studied also the stability of the essential g-ascent and the essential ascent under a finite
rank relation. Motivated by the recent work of T. Álvarez and A. Sandovici (Complex Anal.
Oper. Theory 7, 801-812, 2013), we extend to a closed linear relation, the well known notion
of minimum modulus of a linear operator [H. A. Gindler and A. E. Taylor, Studia. Math.
22, 15-41, 1962]. Also, the new notion of minimum g-modulus is introduced, which allows
us to establish the connection between ascent and minimum g-modulus of a closed linear
relation.
Key words and phrases : Closed linear relations, Spectrum, Ascent, Essential ascent,
Descent, Essential descent, Minimum modulus, Semi-Fredholm relations.
A note on focusing coupled fourth-order nonlinear Schrödinger equations
Radhia Ghanmi
Département de Mathématiques, Faculté des Sciences de Tunis, Tunisie.
[email protected]
Abstract
Dans ce travail, on considère le système de Schrödinger
m
i
X
∂
uj + ∆2 uj −
ajk |uk |p |uj |p−2 uj = 0,
∂t
(9)
uj (0, .) = ψj,0 ,
(10)
k=1
de donnée initiale
où ajk = akj ≥ 0 et u := u(t, x) : R+ × RN → C.
En premier lieu, on montre un résultat d’existence d’état fondamental de (9) et on montre
l’existence globale par la méthode de potentiel well. En second lieu, on exprime quelques
constantes optimales pour les injections de Sobolev en fonction de l’état fondamental. Enfin,
on prouve l’existence globale pour des données petites dans le cas masse critique .
21
References
[1] N. V. Nguyen, R. Tian, B. Deconinck and N. Sheils, Global existence for a
coupled system of Schrödinger equations with power-type nonlinearities, J. Math. Phys.
54, 011503, (2013).
[2] T. Saanouni, A note on fourth-order nonlinear Schrödinger equation, Ann. Funct.
Anal. 6, No. 1, 249-266, (2015).
[3] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,
Commun. Math. Phys. 87, 567-576, (1983).
Ce travail est en collaboration avec Tarek Saanouni.
Entire solution of a nonlinear elliptic problem in Harmonic space N A and
Enclidean space Rd
Zeineb Ghardallou
Université de Tunis El Manar, Faculté des Sciences de Tunis, LR11ES11 Laboratoire
d’Analyse Mathématique et Applications. 2092, Tunis, Tunisie.
[email protected]
Abstract
We study entire solutions on either the harmonic space N A or the euclidean space Rd
of the nonlinear elliptic problem
Lu ± ϕ(·, u) = 0, in the sense of distributions,
(11)
We give sufficient and necessary conditions for the existence of bounded or large solutions
of (12) under the hypothesis that the oscillation sup ϕ(x, ·) − inf ϕ(x, ·) tends to zero as
|x|=r
|x|=r
r tends to infinity at specified rate.
Inégalités intégrales et applications
Taoufik Ghrissi
[email protected]
Abstract
1. Nouveau critère pour la stabilité asymptotique de systèmes variant dans le temps :
Dans ce travail on va donner des conditions suffisantes pour assurer la stabilité asymptotique pratique globale uniforme de certains systèmes dynamiques , ces conditions
sont exprimées comme relation entre la fonction de Lyapunov et de fonctions spéciales
du type K et du type KL.
22
2. Quelques généralisations du lemme de Gronwall : On considère de nouvelles formes
d’inégalités du type Gronwall-Bellman , avec deux applications : une application a la
stabilité des systèmes dynamiques et une autre au contrôle.
3. Quelques inégalités intégrales a retard : On considère des différents types d’inégalités
intégrales , comme application on cherche une estimation de la fonctions considérée qui
aura un effet sur l’étude qualitative des inégalités intégrales : on montre par exemple
que sous certaines conditions la fonction inconnue sera bornée.
4. Quelques inégalités du type Gronwall en utilisant les opérateurs de Picard : En considérant une inégalité intégrale , le second membre apparait sous la forme d’un opérateur
, en utilisant la théorie des opérateurs on trouve une estimation de la fonction inconnue
comme étant le point fixe de l’opérateur considéré. Comme application on étudie la
stabilité pratique et la stabilité asymptotique de certains systèmes dynamiques perturbés.
La transformation de Fourier à fenêtre associée à l’opérateur de Riemann-Liouville
Aymen Hammami
Université de Tunis El Manar, Faculté des Sciences de Tunis, Tunisie
[email protected]
Abstract
On définit et on étudie la transformation de Fourier à fenêtre associée à des opérateurs
aux dérivées partielles singuliers définis sur le demi-plan ]0, +∞[×R.
On établira plusieurs résultats d’analyse harmonique associés à cette transformation.
En particulier, on montrera un théorème de Plancherel et une formule d’inversion que nous
utilisons pour établir le principe d’incertitude de Heisenberg relatif à cette transformation.
Ensuite, on étudiera la transformation de Fourier à fenêtre sur des sous-ensembles de
2
[0, +∞[×R de mesure finie. Cela nous permettra d’établir le principe d’incertitude de
Heisenberg-Pauli Weyl pour cette transformation (de magnitude quelconque).
Enfin, nous établirons le principe d’incertitude local et nous donnons de bonnes applications.
A Twisted waveguide with a Neumann window
Hiba Hammedi
[email protected]
Abstract
23
In this presentation we are dealing with tubular twisted waveguides. In deed we study the
spectral properties of the Laplace operator with Dirichlet boundary conditions every where
on the boundary of the tube except on a bounded part where we consider the Neumann
boundary conditions. We show that the existence of bound states depends on the measure
of the Neumann window.
Wavelets and generalized windowed transforms associated with the Dunkl-Bessel
Laplace operator on IRd × IR+
Amina Hassini
Faculté des Sciences de Tunis, Tunisie
[email protected]
Abstract
In this presentation we study Wavelets and the generalized windowed transform associated with the Dunkl-Bessel Laplace operator on IRd × IR+ and we prove for this transform
Plancherel and inversion formulas.
Cohomologie of supergroup K(m) and Lie superalgebra K(m), m ≤ 3
Raja Hattab
Université de Gabes, ISSIG
[email protected]
Abstract
1|1
Let S
be the supercircle of dimension (1, 1), endowed with the contact 1-form α =
dx + θdθ and let K(1) be the Lie superalgebra of contact vector fields, K(1) the group of
contactomorphisms and Fλ the module of tensor densities on S 1|1 . We compute the second
cohomology spaces H 2 (K(1), Fλ ) and H 2 (K(1), Fλ ) and then classify the abelian extensions
of K(1) and K(1) by Fλ .
We give also an explicit construction of the 1-cocycles of the group of contactomorphisms
on the supercircle S 1|3 , with coefficients in the space of differential operators acting on
tensor densities on S 1|3 . We show that they satisfy properties similar to those of the superSchwarzian derivative.
L’opérateur maximal de Bochner-Riesz sur l’hypergroupe Ξq
Khadija Houissa
[email protected]
24
Abstract
Dans cet éxposé nous définissons
• les moyennes de Bochner-Riesz, à l’aide desquelles nous donnons une caractérisation
p,l
des espaces de Bessel-Besov B.Bα,µ
sur l’hypergroupe Ξq .
• les espaces de Bessel-Hardy Hµp par les atomes et nous établissons une version Hµp − Lp
pour l’opérateur maximal de Bochner-Riesz sur l’hypergroupe Ξq .
Logarithmic stability of the determination of the surface impedance and the
problem of detecting corrosion, respectively, of an obstacle from the scattering
amplitude and by a single electric measurement
Aymen Jbalia
Department of Mathematics, Faculty of Sciences of Bizerte, 7021 Jarzouna Bizerte, Tunisia
[email protected]
Abstract
We prove a stability estimate of logarithmic type for the inverse problem consisting in
the determination of the surface impedance of an obstacle from the scattering amplitude.
Then, we establish a logarithmic stability estimate for the problem of detecting corrosion
by a single electric measurement. We present a simple and direct proof, which is essentially
based on an elliptic Carleman inequality. The key idea consists in estimating accurately a
lower bound of the local L2 − norm at the boundary of the solution of the boundary value
problem.
Nouveaux résultats sur le développement asymptotique des intégrales
Nesrine Kamouche
Laboratoire d’application des mathématiques à l’électronique et à l’informatique,Université El
Hadj Lakhdar 05000 Batna
[email protected]
Abdallah Benaissa
Laboratoire d’application des mathématiques à l’électronique et à l’informatique,Université El
Hadj Lakhdar 05000 Batna
[email protected]
Abstract
Dans ce travail, nous considérons le développoment asymptotique d’intégrales doubles
de type de Laplace au cas de points minimaux non stationnaires, situés sur la frontière du
25
domaine d’intégration. Nous allons surtout exposer les derniers résultats sur ce sujet publiés
récemment dans "proceedings of the American mathematical society", dans un article de
Kamouche et Benaissa. On démontre dans ce travail que l’ordre de contact entre la courbe du
frontière du domaine d’intégration et la courbe niveau de la phase à travers le point minimum
gère le développemet asymptotique en question. Cette idée permettra de construire des
développements asymptotiques complèts dans des contextes plus généraux. En particulier,
le problème sera complètement résolu si la phase et la courbe du frontière sont analytiques
proche du point minimum.
Considérons l’intégrale du type de Laplace
Z
I (λ) =
g (x1 , x2 ) e−λf (x1 ,x2 ) dx1 dx2 ,
(12)
D
2
où D est un domaine borné de R , f et g sont deux fonctions à valeurs réelles, de classe
C ∞ sur la fermeture D de D, et λ un grand paramètre positif.
Key Words: Intégrale de type de Laplace, développement asymptotique, lemme de
Watson.
References
[1] Nesrine Kamouche & Abdallah Benaissa Asymptotic expansion of double laplacetype integrals : the case of non-stationary minimum points. Proceeding of the American
Mathematical Society. http://dx.doi.org/10.1090/proc/13064, May 4, 2016.
[2] A. Benaissa & C. Roger, Asymptotic expansion of multiple oscillatory integrals with
a hypersurface of stationary points of the phase, Proc. R. Soc. A 469 : 20130109.
http://dx.doi.org/10.1098/rspa.2013.0109, (2013).
[3] R. Wong, Asymptotic approximations of integrals, Academic Press, Boston, 1989.
[4] W. B. Fulks & J. O. Sather, II. Laplace method for multiple integrs, Pacific J. Math.
11, pp. 185-192, (1962).
Les variations spatiales de la solution de l’équation des ondes dirigée par un bruit
blanc en temps et en espace
Marwa Khalil
Faculté des sciences de Monastir et Université de Lille 1
[email protected]
Abstract
Lors de cet exposé on s’intéressera à étudier le comportement asymptotique de la variation quadratique spatiale de la solution de l’équation linéaire stochastiaque des ondes dirigée
par un bruit additif de type blanc en temps et en espace. En faite, on présentera un résultat de normalité asymptotique faible pour l’estimateur de la variation quadratique en
appliquant essentiellemnt la méthode de Stein combinée avec le calcul de Malliavin pour les
intégrales multiples d’Itô-Wiener. Également, la convergence normale presque sûre de cet
estimateur sera aussi vérifiée.
26
Concentration on Lines for a Neumann-Ambrosetti-Prodi type problem in
two-dimensional Domains
Zied Khemiri
Département de Mathématiques, Faculté Des Sciences De Tunis, Tunisie.
[email protected]
Abstract
In this presentation, we discuss the conjecture related to the existence of concentrating
solution of a superlinear Ambrosetti Prodi problem. Let Ω ⊂ R2 be a bounded domain with
smooth boundary and consider the problem

2
p

 −ε ∆u = |u| − Φ2 in Ω


∂u
=0
∂ν
on ∂Ω
where p > 1, ε > 0 is a small parameter, Φ2 is an eigenfunction of - ∆ with Neumann
boundary condition corresponding to the second eigenvalue λ2 , and ν denotes the outward
normal of ∂Ω. Let Γ be a curve intersecting orthogonally with ∂Ω at exactly two points and
dividing Ω into two parts. Moreover,
Γ satisfies stationary and non-degeneracy conditions
R
with respect to the functional Γ Φσ2 , where σ = p+3
2p . We prove the existence of a solution
uε concentrating along the whole of Γ, exponentially small in ε at any positive distance from
it, provided that ε is small and away from certain critical numbers.
On the class semigroup of a numerical semigroup
Faten Khouja
[email protected]
Abstract
The class semigroup of a numerical semigroup is the semigroup S(S) of the classes of
the non zero relative ideals of S. Our aim is to find some properties of S(S). Using the
Hasse Diagram of the poset G(S), the set of gaps of S, we show that S(S) is finite and we
can also compute its cardinality for some cases.
Similarly to ring context, the reduction number r(I) is well defined for a class of ideals I.
It turns out that r(I) is strictly bounded by e, the multiplicity of S and for any integer r,
1 ≤ r ≤ e − 1, there exists a class of ideals I such that r(I) = r.
The results of the talk are in a joint paper with Valentina Barucci.
27
Derivations on archimedean almost f -algebras and commutative Banach `-algebras
Naoual Kouki
[email protected]
Abstract
En 1955, Singer et Wermer ont montré que l’image d’une dérivation continue sur une
algèbre de Banach commutative est incluse dans le radical de Jacobson. Simultanément, ils
ont conjecturé que l’hypothèse de continuité est superflue. Il a fallu plus de 30 ans avant
que la conjecture a été confirmée par Thomas.
Il n’est pas étonnant que le problème d’inclusion de l’image des dérivations est lié à la
théorie des idéaux maximaux uniformement fermés. En s’inspirant par ce point de vue,
nous donnerons une nouvelle version du théorème de Thomas qui résout partiellement la
conjecture non-Banach de Singer-Wermer pour les dérivations agissant sur les f -algèbres
universellement complètes.
S−prime ideals over S−Noetherian ring
Achraf Malek
Faculty of sciences of Monastir, Tunisia
[email protected]
Abstract
Let A be a commutative ring with identity and S ⊆ A a multiplicative subset. In this
paper we introduce the concept of S−prime ideal which is a generalization of prime ideal.
Let I be an ideal of A disjoint with S. We say that I is an S−prime ideal if there exists
s ∈ S such that for all a, b ∈ A with ab ∈ I then sa ∈ I or sb ∈ I. In this work we show
that S−prime ideals enjoy analogs of many properties of prime ideals and we study them
over S−Noetherian rings.
Stabilisation des solutions des équations d’onde et de Klein-Gordon dans les
domaines non bornés
Mohamed Malloug
[email protected]
Abstract
28
Lors de cette communication, je m’intéresse à la décroissance de l’énergie (norme de la
solution) pour l’équation d’onde amortie et l’équation de Klein-Gordon amortie à l’extérieur
d’un domaine régulier compact avec un amortisseur localisé prés des rayons captifs. Sous une
condition géométrique appelée "condition du contrôle géométrique extérieur " on montrera
que la résolvante associée à la solution est uniformément bornée, ce qui entraîne une décroissance polynomiale de l’énergie locale. Je m’intèresse aussi à la décroissance de l’énergie pour
l’équation d’onde dans un guide d’onde avec un amortisseur localisé à l’infini. On met en évidence le phénomène de diffusion. De plus, dans ce cas la condition de contrôle géométrique
n’est pas satisfaite, on montrera que la décroissance de l’énergie a eu lieu qu’avec une perte
d’une certaine régularité de la condition initiale.
Estimation semi paramétrique pour un processus autorégressif à variance infinie
Tawfiq Fawzi Mami
Université d’Ain Témouchent, Algérie
[email protected]
Hakim Ouadjed
Université de Mascara, Algérie
[email protected]
Abstract
Nous proposons un nouvel estimateur basé sur la théorie des valeurs extrêmes de l’indice
d’autorégression d’un processus AR(1) α-stable à valeurs positives.
Key words and phrases: Théorie des valeurs extrêmes, processus autorégressif α-stable,
estimation de l’indice de queue.
AMS (MOS) Subject Classifications: 60G70, 62G32.
1
Introduction
La variable aléatoire α-stable X ∼ S(α, β, σ, µ) est définie par sa fonction caractéristique
ϕX (t) (voir Samorodnitsky et Taqqu [4]) donnée par
t
ϕX (t) = exp iµt − σ α |t|α 1 + iβ w(t, α)
(13)
|t|
avec

απ


 tan( 2 ) si α 6= 1
w(t, α) =


 2 ln |t| si α = 1,
π
où α ∈]0, 2], β ∈ [−1, 1], σ > 0 et µ ∈ R.
La famille des lois stables S(α, 1, 1, µ) avec 0 < α < 1, µ ≥ 0 définit des variables
aléatoires positives avec support (µ, ∞[, telles distributions sont devenues un outil standard
29
dans la modélisation de données à queue lourde dans des domaines aussi divers que la
finance, ingénierie et actuariat.
La plupart des applications statistiques nécessitent une dépendance temporelle. Soit le
processus AR(1)
Xt = a Xt−1 + εt ,
(14)
P∞ jδ
où 0 < a < 1 et j=0 a < ∞ pour 0 < δ < α et {εt } ∼ i.i.d. suivant une loi stable
positive S(α, 1, 1, µ), 0 < α < 1, µ ≥ 0.
Par suite ona l’estimateur de a
1/b
αHill
α
bπ n −
2
d
αX
α) sin( ) Xn−k,n
b
an = 1 − Γ(b
(15)
π
2 k
où X1,n ≤ X2,n ≤ . . . ≤ Xn,n étant la statistique d’ordre, et
"
α
bHill
k
1X
log Xn−i+1,n − log Xn−k,n
=
k i=1
#−1
est l’estimateur de Hill[2] pour α.
La normalité asymptotique de b
an est établie dans le théorème suivant.
Theorem 1.1. Supposons le processus (14) alors


√
 α2 (1 + aα )a2−2α (1 − aα )3 
k
D


(b
an − a) −→ N 0,

2
log (n/k)


2
απ
Γ(α) sin( )
π
2
:
On a comparé notre estimateur de a (basé sur α
bHill ) avec ceux obtenus par l’application
1. De α
bt−Hill (Fabián and Stehlík [1]),
2. De α
bP ickands (Pickands [3])
où nous avons obtenu un résultat meilleur en terme de bias.
References
[1] Fabián, Z. and Stehlík, M. (2009). On robust and distribution sensitive Hill like method.
IFAS Research Paper Series 43(4), online at http://www.jku.at/ifas/.
[2] Hill, B. M. (1975). A simple approach to inference about the tail of a distribution. Ann.
Statist. 3, 1136-1174.
[3] Pickands, J. (1975). Statistical inference using extreme order statistics. Annals of Statistics
3, 119-131.
[4] Samorodnitsky, G., Taqqu, M. (1994). Stable Non-Gaussian Random Processes, New York:
Chapman and Hall.
30
Asymptotic profiles for a class of perturbed Burgers equation in one space
dimension
Bechir Mannoubi
[email protected]
Abstract
We study the large time behavior of solutions tothe Burgers equation with some class of
perturbations uτ τ + uτ = (a(ξ)uξ )ξ − u2 + N (u) ξ . This equation is posed in one space
dimension depending on a positive parameter . Using various energy functionals rewritten in the variable √ξτ and log τ , we prove that the self-similar solutions of the well-known
Burgers equation uτ = uξξ − u2 ξ , are also asymptotically stable self-similar solutions of
the above hyperbolic equation in appropriate weighted Sobolev spaces.
Keywords: Burgers equation, Self-similar variables, Asymptotic behavior, Self-similar
solutions.
Image restoration by the evolution equations of Perona-Malik modified
Messaoud Maouni
Laboratory LAMAHIS, Department of Mathematics, Faculty of Science, Université 20 août
1955 Skikda, Algeria
[email protected]
Souilah Fairouz
Laboratory LAMAHIS, Department of Mathematics, Faculty of Science, Université 20 août
1955 Skikda, Algeria
Abstract
Image restoration by use of equations of evolution has aroused and continues to generate significant interest because of the opportunities it offers to smooth an image while
preserving its discontinuities. Since the seminal work of Perona and Malik and ROF, many
enhancements have been made to best meet the requirements of the restoration (removal of
noise without altering the contours for example), especially for greyscale images. Practical
applications of these methods of restoration are numerous and affect various fields (photography, medical, etc.). In this paper, we propose a new model of evolution equations for
image restoration based on a functional energy different from that found in the literature.
The resulting PDE is then characterized by the possible integration in the diffusion process
of a priori information on specific structures of the image you wish to restore. We present
some results obtained by this model on different images.
keywords: Image restoration, evolution equations.
31
References
[1] A. Chambolle. Algorithm for total variation minimization and applications. J. ath. Imaging
Vision. 20 (2004), pp. 89-97.
[2] L. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms.
Physica D, (1992), pp. 259-268.
[3] M.L. Hadji, M. Maouni and F.Z. Nouri. A Wavelet Inpainting by a Toxitrop Model. iv, 2010
14th International Conference Information Visualisation, 2010, IEEE ISBN: 978-0-7695-4165-5,
doi.ieeecomputersociety.org/10.1109/IV.2010.96, pp. 559-563.
[4] T. Le, R. Chartrand and T. J. Asaki. A variational approach to constructing images corrupted by Poisson noise. UCLA CAM Report 05-49, Sept. 2005. (In print: J. Math. Imaging
and Vision, 27, 2007.
[5] M.Maouni and F.Z.Nouri. Image Restoration by Partial Differential Equations. IEEE GMAI.
06, pp 250-254, 2006.
[6] M.Maouni , F.Z.Nouri and D.Meskine. Image Restoration by Non-standard Diffusion. IEEE
Geometric Modeling and Imaging Modern Techniques and Applications Volume-Issue 09-11,
July 2008, pp 98-101.
[7] P. Perona and J. Malik. Scale-space and edge detection using anistropic diffusion. IEEE
Transcations on Pattern Analysis and Machine Intelligence, 12(7):629-639, 1990.
Contribution des recherches linéaires inexactes dans la convergence de quelques
méthodes du gradient conjugué non linéaire
Romaissa Mellal
Université de Guelma. Algérie.
[email protected]
Mohamed Haiour
Université de Badji Mokhtar, Annaba. Algérie
[email protected]
Abstract
Ce travail est consacré à une étude numérique comparative des performances des méthodes du gradient conjugué non linéaire de HS, FR, PRP, PRP+, CD, LS, DY et HZ, avec les
recherches linéaires inexactes de Wolfe forte et d’Armijo avec rebroussement. Ces méthodes
sont testées (Scilab) sur un ensemble de six fonctions testes choisies des articles de (Andrei,
[1,2008]) et (Moré et Hillstrom, [5,1981]).
Mots clés: Gradient conjugué non linéaire, recherche linéaire inexacte.
32
References
[1] N. Andrei, An unconstrained optimization test functions collection, Advenced Modeling and
Optimization 10(1): 147-161, (2008).
[2] L. Armijo, Minimzation of function having lipschitz continous first partial derivatives, Pacific
Journal of Mathematics, Vol. 16(1), pp.1-3, (1966).
[3] Y.H. Dai and Y. Yuan, A non linear conjugate gradient with a strong global convergence
property, SIAM J. Optimization, Vol. 10(1), pp.177-182, (1999).
[4] R. Mellal and M. Haiour, Numerical simulations of some nonlinear conjugate gradient methods with inexact line searches, Proceedings of International Conference: Mathematical Science
and Applications, Abu Dhabi, UAE. Universal Journal of Mathematics and Mathematical Sciences, Volume 4, Number 1, Pages 63-84, (2013).
[5] J.G. B. Moré and K. Hillstrom, Testing unconstrained optimization software, ACM Transactions on Mathematical Software7: 17-41,( 1981).
[6] M.J.D. Powell, On the convergence of the variable metric algorithm; J. Inst. Math. Appl, 7,
pp. 21-36, (1971).
[7] P.Wolfe, Convergence conditions for ascent methods, SIAM Review, 11, pp. 226-235, (1969).
Estimations logarithmique optimales dans les espaces de Hardy-Sobolev H k,p du
disque et de l’anneau
Ameni Massoudi
[email protected]
Abstract
On s’interesse dans ce travail à établir des estimations logarithmique de type-optimale
dans les espaces de Hardy-Sobolev H k,p ; k ≥ 1 et 1 ≤ p ≤ ∞ du disque unité D et de
l’anneau Gs . Le cas Hilbertien p = 2 a été établi par I.Feki et al dans [3], le cas uniforme
p = 1 a été établi par S.Chaabane et I.Feki dans [1] et par I.Feki dans [2] pour le cas de
l’anneau.
References
[1] S. Chaabane, I. Feki, Optimal logarithmic estimates in Hardy-Sobolev spaces Hk;1, C. R.
Acad. Sci. Paris, Ser. I 347(2009) 1001-1006.
[2] Estimates in Hardy-Sobolev spaces on annular domains : Stability results. Czecholovak
Mathematical Journal, June 2013, Volume 63, Issue 2, pp 481-495.
33
[3] Optimal logarithmic estimates in the Hardy-Sobolev space of the disk and stability results.
Measure pseudo almost automorphic solutions for some differential equations with
reflection of the argument
Mohsen Miraoui
University of Kairouan, Tunisia
[email protected]
Abstract
The aim of this work is to study the new concept of the (µ, ν)−pseudo almost automorphic functions for some non-autonomous differential equations. We suppose that the linear
part has an exponential dichotomy. The nonlinear part is assumed to be (µ, ν)−pseudo
almost automorphic. We show some results regarding the completness and the invariance
of the space consisting in (µ, ν)−pseudo almost automorphic functions. Then we propose
to study the existence of (µ, ν)−pseudo almost automorphic solutions for some differential
equations involving reflection of the argument.
On the perturbed Hermite polynomials and application
Tahar Moumni
Department of Mathematics, University of Carthage, Faculty of Sciences of Bizerte, Jarzouna,
Tunisia
[email protected]
Abstract
In this talk, we introduce a new set of functions, which generalizes the Hermite’s polynomials in some sense. This family of functions, called perturbed Hermite polynomials, are
nothing but the eigenfunctions of an integral transform Fc . We show that the perturbed
Hermite polynomials, PHP for short, are the solutions of a differential operator which commutes with Fc . As an application, we use the PHP to solve the one dimensional inverse heat
conduction problem.
References
[1] A. Karoui, I. Mehrzi, T. Moumni, Eigenfunctions of the Airy’s integral transform: Properties, numerical computations and asymptotic behaviors, J. Math. Anal. Appl. 389 (2012)
989-1005.
[2] D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty–IV: Extensions to many dimensions; generalized prolate spheroidal functions, Bell System Tech.
J. 43 (1964), PP.3009-3057.
34
[3] P. J. Forrester, Log-Gases and Random Matrices, Princeton University Press, 2010.
[4] H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and
uncertainty-III; The dimension of the space of essentially time and band-limited signals.,
Bell System Tech. J. 41 (1962), PP.1295-1336.
[5] H.J. Landau, H.O. Pollak,
Prolate spheroidal wave functions, Fourier analysis and
uncertainty-II, Bell System Tech. J. 40 (1961), PP.65-84.
[6] D. Slepian and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and
uncertainty-I, Bell Syst. Tech. J. Vol. 40 (1961),PP. 43-64.
[7] L. L. Wang and J. Zhang, A new generalization of the PSWFs with applications to spectral
approximations on quasi-uniform grids, Appl. Comput. Harmon. Anal. 29 (2010) PP.303329.
[8] C. Niven; On the conduction of heat in ellipsoids of revolution, Phylosophical Tran. r. Soc.
Lond 171 (1880), pp: 117-151.
Characterizations of the normal variance-mean mixture by a stable mixing
Farouk Mselmi
Laboratory of Probability and Statistics. Sfax Faculty of Sciences. B.P. 1171, CP 3000.
Tunisia
[email protected]
Abstract
In this work, we characterize the normal variance-mean mixture by a stable mixing by
two approach. The first one is given in terms of natural exponential families. We explicit
the variance function and we determine the associated Lévy measure. The second approach
is given in terms of stochastic processes. We characterize the normal variance-mean mixture
process by some independence conditions and by using an exponential random time.
Homeomorphisms of regular curves
Issam Naghmouchi
University of Carthage, Faculty of Sciences of Bizerte, Department of Mathematics, Jarzouna,
7021, Tunisia.
[email protected] / [email protected]
Abstract
35
Regular continuum is a continuum such that for each x ∈ X and each open neighborhood
V of x, there exists an open neighborhood U of x such that U ⊂ V and the boundary set
∂(U ) of U is finite. Besides continuum theory and fractals, regular continua appear in
complex dynamics, for example Sierpinsky’s triangle can be realised as the Julia set of some
rational map. G. T. Seidler [Proc. Amer. Math. Soc. 108 (1990), no. 4, 1025–1030;
MR0946627] proved that every homeomorphism of a regular continuum has zero entropy.
In this talk we will prove further rigidity features of homeomorphisms of regular curves by
showing for instance that any ω-limit set is minimal.
Construction of blow-up solutions for complex Ginzburg-Landau equation in the
critical case
Nejla Nouaili
CEREMADE, Université Paris Dauphine, Paris Sciences et Lettres.
[email protected]
Abstract
We consider the following complex Ginzburg-Landau equation
ut = (1 + iβ)∆u + (1 + iδ)|u|p−1 u,
u(., 0) = u0 ∈ L∞ (IRN , C)
(16)
The complex Ginzburg Landau equation (CGLE) describe many nonequilibrium phenomena, such as the generation of spatiotemporal dissipative structures in lasers, binary
fluid convection, phase transition.
We construct a solution for the complex Ginzburg-Landau equation in the critical case,
which blows up in finite time T only at one blow-up point. We also give a sharp description
of its profile. The proof relies on the reduction of the problem to a finite dimensional one,
and the use of index theory to conclude.
Keywords: Blow-up profile, Complex Ginzburg-Landau equation.
Sur le comportement extrémal du processus max-autorégressif à queue lourde
Hakim Ouadjed
Université de Mascara, Algérie
[email protected]
Tawfiq Fawzi Mami
University d’Ain Témouchent, Algérie
[email protected]
36
Abstract
En se basant sur la théorie des valeurs extrêmes, on propose un estimateur semi paramétrique
asymptotiquement normal de l’indice extrêmal θ pour un processus ARMAX(1) à variance
infinie.
Key words and phrases: Théorie des valeurs extrêmes, processus max-autorégressifs,
estimation de l’indice de queue.
AMS (MOS) Subject Classifications: 60G70, 62G32.
2
Introduction
Le paramètre de l’indice extrémal caractérise le degré de dépendance locale dans les extrêmes d’une série chronologique stationnaire et a des applications importantes dans un
certain nombre de domaines tels que l’hydrologie, les télécommunications et les finances.
Ce paramètre est la clé pour étendre les résultats de la théorie des valeurs extrêmes de i.i.d.
à des séquences stationnaires.
Leadbetter et al [3] ont met une condition de mélange D(un ) sur la base de la probabilité
de dépassements d’un seuil elevé un , elle limite le degré de dépendance à long terme de la
séquence, assurant une indépendance asymptotique entre les observations extrêmes.
Definition 2.1. Une série stationnaire X1 , X2 , . . . satisfait la condition D(un ) au seuil un
si, pour tous entiers i1 < . . . < ip < j1 < . . . < jq avec j1 − ip > l,
P Xi1 ≤ un , . . . , Xip ≤ un , Xj1 ≤ un , . . . , Xjq ≤ un
− P Xi1 ≤ un , . . . , Xip ≤ un P Xj1 ≤ un , . . . , Xjq ≤ un ≤ δ(n, l),
où δ(n, ln ) → 0 pour une suite ln = o(n) telle que ln /n → 0 quand n → ∞.
Theorem 2.2 ([3]). Soit X1 , X2 , . . . , Xn un processus stationnaire et X̃1 , . . . , X̃n une suite
de variables aléatoires i.i.d. de même distribution marginale F que le processus stationnaire.
On définit Mn = max(X1 , . . . , Xn ) et M̃n = max(X̃1 , . . . , X̃n ). S’il existe des constantes
an > 0, bn ∈ R telles que
P[a−1
n (M̃n − bn ) ≤ x] → G(x),
(17)
quand n → ∞ et si le processus stationnaire vérifie d’une part la condition D(un ) avec
un = an x + bn pour tout x tel que G(x) > 0, et d’autre part
P[a−1
n (Mn − bn ) ≤ x] → H(x),
(18)
quand n → ∞, alors H(x) = [G(x)]θ où 0 < θ ≤ 1 est appelé indice extrémal.
Un processus ARMAX(1) à variance infinie est défini par
Xi = max (λ Xi−1 , Zi ) , 1 ≤ i ≤ n
où 0 < λ < 1, Z1 , . . . , Zn i.i.d et FZ (x) = exp(−x
θ = 1 − λα .
−α
(19)
), 0 < α < 2. Dans ce cas on a
Par suite on obtient
n −b
αX
θbn = Xn−k,n
k
où
X1,n ≤ X2,n ≤ . . . ≤ Xn,n
37
(20)
est la statistique d’ordre, et
"
α
bX
k
1X
=
log Xn−i+1,n − log Xn−k,n
k i=1
#−1
est l’estimateur de Hill[2] pour α.
La normalité asymptotique de θbn est établie dans le théorème suivant.
Theorem 2.3. Supposons le processus(19) et k = kn telles que k → ∞, k/n → 0, alors
√
k
D
(θbn − θ) −→ N 0, σ 2 ,
log (n/k)
où
σ 2 = α4 θ3 (2 − θ).
(21)
Nous comparons, en termes de biais et RMSE, notre estimateur avec celui de Ferro et
Segers [1] où on a établi la performance de notre estimateur pour α ∈ [0, 1] puis, avec celui
de Olmo [4] où on a montré son éfficacité toujours pour α ∈ [0, 1].
References
[1] Ferro, C. A. T., and Segers, J. (2003). Inference for Clusters of Extreme Values, Journal of
the Royal Statistical Society, Ser. B, 65, 545-556.
[2] Hill, B. M. (1975). A simple approach to inference about the tail of a distribution. Ann.
Statist. 3, 1136-1174.
[3] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties
of Random Sequences and Processes, Springer, New York.
[4] Olmo, J. (2015). A New Family of Consistent and Asymptotically-Normal Estimators for
the Extremal Index. Econometrics, 3, 633-653.
Monte Carlo simulation of Matrix Models
Ahlem Rouag
Department of Physics, Faculty of Sciences, Annaba University, Annaba, Algeria.
[email protected]
B. Ydri
Department of Physics, Faculty of Sciences, Annaba University, Annaba, Algeria.
Abstract
We study a three matrix model (bosonic IKKT Yang- Mills matrix models) with quartic
term, We have used Monte Carlo method to simulate. We have calculated the phase diagram,
we remark two phases: matrix phase and fuzzy sphere phase. The dynamically emergent
2
geometry, which is given by a fuzzy two-sphere SN
, in the 3-dimensional IKKT matrix
models, is found to be stable for all values of the deformation parameter M. The sphere-tomatrix transition line is pushed to 0 and only one phase survives.
38
A bifurcation problem associated to an asymptotically linear function
Soumaya Sâanouni
University of Tunis El Manar, Faculty of science of Tunis, Tunisia.
[email protected]
Abstract
We study the existence and the uniqueness of positive solutions to a two-order semilinear
elliptic problem with Dirichlet boundary condition
−div(c(x)∇u) = λf (u) in
Ω,
(Pλ )
u = 0
on ∂Ω.
Where Ω ⊂ IRn ; n ≥ 2 is a smooth bounded domain, f is a positive, increasing and
convex source term and c(x) is a smooth bounded positive function on Ω. We also prove
the existence of critical value and claim the uniqueness of extremal solutions. We take into
account the types of problems of bifurcation for a class of elliptic problems we also establish
the asymptotic behavior of the solution around the bifurcation point.
(*) S. Sâanouni and N. Trabelsi. A bifurcation problem associated to an asymptotically linear
function, Acta Mathematica Scientia 2016, 36B(6):1-16
(1) A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and
applications, J. Funct. Anal. 14 (1973), 349-381.
(2) P. Mironescu and V. Rădulescu, The study of a bifurcation problem associated to an asymtotically linear function, Nonlinear Analysis 26 (1996), 857-875.
(3) V. Rădulescu, Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations,
Contemporary Mathematics and Applications, Vol. 6 (Hindawi Publ. Corp., 2008).
(4) L. Hörmander, The Analysis of Linear Differential Operators I, (Springer-Verlag, Berlin,
1983).
Modèle des lois puissances à changement de régimes et calcul de la Value-at-Risk
Khaled Salhi
Université de Lorraine, Nancy, France
[email protected]
Abstract
39
L’exposé s’inscrit dans le cadre de la gestion du risque financier en s’appuyant sur la
Value-at-Risk (VaR), comme mesure de risque. Nous construisons un modèle d’évolution
de prix que nous confrontons à des données réelles issues de la bourse de Paris (Euronext
PARIS). Notre modèle prend en compte les probabilités d’occurrence des pertes extrêmes et
les changements de régimes observés sur les données. Notre approche consiste à détecter les
différentes périodes de chaque régime par la construction d’une chaîne de Markov cachée et
à estimer la queue de distribution de chaque régime par des lois puissances. Nous montrons
empiriquement que ces dernières sont plus adaptées que les lois normales et les lois stables.
L’estimation de la VaR est validée par plusieurs backtests et comparée aux résultats d’autres
modèles classiques sur une base de 56 actifs boursiers.
Mots-clefs : Value-at-Risk, lois puissances, modèles de Markov cachés, changements
de régimes.
Cycles limites d’une équation différentielle de sixième ordre à un paramètre
Nabil Sellami
Université 8 Mai 1945, Guelma. Algérie
[email protected]
Amar Makhlouf
Université Badji Mokhtar, Annaba. Algérie
[email protected]
Abstract
Dans ce papier, on étudie les cycles limites de l’équation différentielle de sixième ordre
.....
....
...
... .... ..... ......
x − 4λ x + (1 + 6λ2 ) x − 4(λ + λ3 ) x + (6λ2 + λ4 )ẍ − 4λ3 ẋ + λ4 x = εF x, ẋ, ẍ, x , x , x , x
(22)
......
avec
ẋ =
dx
d2 x ... d3 x .... d4 x ..... d5 x ...... d6 x
, ẍ = 2 , x = 3 , x = 4 , x = 5 , x = 6 ,
dt
dt
dt
dt
dt
dt
où λ est non nul, ε est suffisamment petit et F ∈ C 2 est 2π-périodique en t. En utilisant la
théorie de la moyennisation, la recherche des cycles limites de l’équation (22) est réduite à
la recherche des zéros d’un système de deux équations non linéaires avec deux inconnus.
Mots clés : Cycle limite , Théorie de la moyennisation, Equation différentielle de sixième
ordre.
References
[1] J. Llibre, N. Sellami and A. Makhlouf. Limit Cycles for a Class of Fourth Order Differential
Equations. Applicable Analysis, 88 (12): 1617-1630, 2009.
40
[2] J. Llibre, J. Yu, and X. Zhang. Limit Cycles for a Class of Third Order Differential Equations. Rocky Mountain J Math 40 Nno 2: 581-594, 2010.
[3] J. A. Sanders, and F. Verhulst. Averaging Methods in Nonlinear Dynamical Systems. Applied Mathematical Sciences 59, Springer, New York, 1985.
[4] N. Sellami and A. Makhlouf. Limit cycles for a class of fifth-order differential equations.
Annals of differential equations. 28 (2). 2012.
Discrétisation spectrale d’un modèle non linéaire pour un écoulement dans un
milieux poreux partiellement saturé
Moncef Touihri
Ecole de l’aviation de Borj El Amri, B.P. 1142, Tunisia.
[email protected]
N. Abdellatif
ENSI, Campus universitaire de Manouba, 2010 Manouba, Tunisia
& LAMSIN, ENIT, BP 137, Le Belvédère 1002, Tunis, Tunisia
[email protected]
C. Bernardi
Laboratoire Jacques-Louis Lions, C.N.R.S. et Univ Pierre et Marie Curie, Paris, France.
[email protected]
Driss Yakoubi
The Fields Institute for Research in Mathematical Sciences, Toronto, Canada.
[email protected]
Abstract
On considère l’équation donnée par:

α∂t u + ∂t b(u) − ∇ · (∇u + k ◦ b(u)ez )



u
(∇u
+
k
◦
b(u)e
)
·
n

z


u|t=0
=0
= uD
=f
= u0
dans Ω×]0, T [
sur ΓD ×]0, T [
sur ΓF ×]0, T [
dans Ω.
(23)
Cette équation régit l’écoulement de l’eau dans un milieu poreux souterrain, situé juste
sous la surface. Grâce à la présence de l’air sous la surface, le milieu poreux est uniquement
partiellement saturé par l’eau. Ω est un ouvert borné connexe de IRd , d = 2 or 3, dont la
frontière ∂Ω est Lipschitzienne, n est le vecteur unitaire normal sur ∂Ω, ∂Ω = ΓD ∪ ΓF ,
ΓD ∩ ΓF = ∅ et ΓD est de mesure strictement positive. T est un nombre réel positif, ez
est le vecteur unitaire de sens opposé à la gravité, b et k sont des fonctions données, α est
41
une constante positive. L’inconnue est la quantité u et les données sont uD , u0 et f définies
respectivement sur ΓD , Ω and ΓF . Plusieurs formulations variationnelles, pour ce problème,
ont été proposées dans la littérature. Nous optons pour une formulation mixte qui fait intervenir la variable q := −∇u − k ◦ b(u)ez , représentant le flux. Plusieurs discrétisations,
en éléments finis, voir par exemple [2], et en volumes finis [3], ont été proposées pour cette
formulation, nous proposons dans ce travail une discrétisation en espace par méthodes spectrales. Pour la discrétisation en temps, nous utilisons un schéma d’Euler implicite. L’étude
théorique de ce modèle est détaillée dans [1]. Nous mettons l’accent ici sur la résolution
numérique du problème (1) discrétisé en temps et en espace.
Nous proposons pour la résolution numérique un algorithme itératif. Pour définir le
schéma, on se donne un réel K ≥ supξ b0 (ξ) et on pose: bα (ξ) = αξ + b(ξ). On fixe un entier
j−1
j ∈ {1, · · · , J} et on initialise l’algorithme par uj,0
N = uN . Pour i > 0 donné, en supposant
j−1
j,i−1
j−1
j,i−1
que uN et uN
∈ XN et que qN et qN
∈ YN sont calculés, on considère le problème
linéaire suivant:
j,i
Trouver uj,i
tels que :
N ∈ XN et qN ∈ YN
j,i
ΓF
qN · n = −iN −1 f (·, tj ) sur ΓF
∀wN ∈ XN , ∀ϕN ∈ YN F ,

j,i
j,i
j,i−1
j,i−1


K(u
+ bα (uj−1
), wN )M ,

N , wN )M + τj (∇ · qN , wN )M = (KuN
N ) − bα (uN

j,i
j,i
j−1
j
(qN , ϕN )M − (uN , ∇ · ϕN )M = −(IN −1 (k ◦ b(uN )ez ), ϕN )M − (uDN , ϕN · n)ΓMD .
(24)
où uDN = iΓND u(., tj ), 1 ≤ j ≤ J avec iΓND , iΓNF et IN sont des opérateurs d’interpolation
polynômiale appropriés et où on remplace les intégrales dans la formulation variationnelle
du problème (1) par le produit discret (·, ·)M basé sur une formule de quadrature de GaussLobatto. Les espaces discrets XN , YN et YN F sont des sous-espaces de polynômes inclus
dans les espaces variationels des solutions de (23). On montre que ce problème est bien posé,
j,i−1
voir [1]. L’algorithme s’arrête quand uj,i
devient plus petit qu’une certaine tolérance.
N −uN
On prouve que l’algorithme est convergent. Des simulations numériques ont été effectuées
sur un domaine convexe dans un premier temps puis ont été étendus à des domaines plus
complexes. Pour le schéma en temps, nous avons utilisé un schéma d’Euler implicite avec un
pas uniforme τn = δt = 0.1. Les systèmes linéaires obtenus ont été résolus par l’algorithme
itératif GMRES (Generalized Minimal Residual) préconditionné, On considère le domaine
s3
et la solution exacte:
Ω =] − 1, 1[2 . En prenant b(s) = 2
s +1






u(x, y; t) = cos(πx) sin(πy) t,
(25)
on représente la courbe d’erreur entre la solution discrète et la solution exacte en normes
L2 (Ω et H 1 (Ω), au temps final T = 1 en variant le degré polynomial de N = 5 à N = 25.
Fig 1. Courbe d’erreur de u en fonction
de N
Fig 2. Courbe d’erreur du flux q en
norme L2 (Ω)
42
Fig 3. Solution exacte (à gauche) et solution discrète pour N = 15 (à droite).
Pour traiter des géométries plus complexes, on utilise la méthode des éléments spectraux
qui combine la décomposition de domaines avec la haute précision des méthodes spectrales.
On suppose que Ω est une réunion sans recouvrement d’un nombre fini de rectangles (d = 2)
ou de parallélipipèdes (d = 3) Ωk , 1 ≤ k ≤ K, et que l’intersection de deux sous-domaines
est soit un sommet, soit un côté soit une face commune. Dans la figure 3, on présente les
isovaleurs des solutions exacte et spectrale en prenant T = 1 et un degré polynomial N
égal à 15 dans chaque sous-domaine. Ces résultats sont en cohérence avec les estimations
théoriques.
References
[1] N. Abdellatif, C. Bernardi, M. Touihri, D. Yakoubi – A priori error analysis of an Euler implicit, spectral discretization of Richards equation,https://hal.archives-ouvertes.fr/hal01334559, submitted.
[2] E. Schneid, P. Knabner, F. Radu – A priori error estimates for a mixed finite element discretization of the Richards’equation, Numer. Math. 98 (2004), 353-370.
[3] P. Sochala, A. Ern, S. Piperno – Numerical methods for subsurface and overland flows,
Comput. Methods Appl. Mech. Engrg. 198 (2009), 2122-2136.
A new class of discounted non-linear optimal multiple stopping times problems
43
Faouzi Trabelsi
Monastir university, Tunisia
[email protected]
Noureddine Jilani ben naouara
Monastir university, Tunisia
Abstract
This paper is devoted to study a new discounted non-linear optimal multiple stopping
times problem with discounted factor β > 0 and infinite-horizon. Under the right-continuity
of the underlying process, we firstly show that the problem can be reduced to a sequence of
ordinary optimal stopping problems. Next in the Markovian case, we characterize the value
function of the problem in terms of β-excessive functions. Finally, in the special case of a
diffusion process, we give explicit expressions for the value function of the problem as well
as the optimal stopping strategy. As explicit example in finance, we apply our theoretical
results to manage a new generalized swing contract which gives its holder n rights to mark
the price X of a stock, where the payment is only allowed at the final exercise time rather
than at each exercise time as in the classical swing contact.
keyword: optimal multiple stopping, discounted factor, Markov process, β-excessive functions, diffusion process, generalized swing option.
Caractérisations des sous-anneaux maximaux non-locaux et non-PVD
Salma Trabelsi
[email protected]
Abstract
Soit R ⊂ S une extension d’anneaux. On dit que R est un sous-anneau maximal nonlocal (respectivement non-PVD) de S si R n’est pas local (respectivement n’est pas PVD)
et tout autre anneau intermédiaire entre R et S est local (respectivement est PVD). Notre
objectif est de caractériser ces types d’anneaux.
Some essential spectra of unbounded operator matrix and application
Ines Walha
Faculty of Sciences of Sfax, Department of Mathematics, BP 1171, Sfax 3000, Tunisia
[email protected]
Abstract
44
In this talk, we investigate some essential spectra of unbounded block operator matrix
defined on a Banach space with domain consisting of vectors satisfying certain relations
between their components. The abstract results are illustrated by an example for two-group
transport equations.
Graphe d’indécomposabilité du produit cartésien de deux graphes
Mouna Yaich
Faculty of Sciences Sfax, Tunisia.
[email protected]
Nadia Amri
Faculty of Sciences Sfax, Tunisia.
[email protected]
Imed Boudabbous
Preparatory Engineering Institute, Sfax, Tunisia.
[email protected]
Abstract
Given a graph G, a subset M of V (G) is a module of G if for a, b ∈ M and x ∈ V (G) \ M ,
xa ∈ E(G) if and only if xb ∈ E(G). A graph G with at least three vertices is prime if ∅, the
single-vertex sets, and V (G) are the only modules of G. A vertex x of a graph G is critical if
G − x is not prime. With each graph G is associated its indecomposability graph defined on
V (G) by: given x 6= y ∈ V , {x, y} is an edge of =(G) if G[V (G) \ {x, y}] is indecomposable.
Let G1 G2 be a Cartesian product of two connected graphs G1 and G2 . In this paper, we
give a complete description of the indecomposability graph of G1 G2 . Second, we prove
that each vertex of the Cartesian product of two connected graphs is not critical. Finally,
we establish that G1 G2 is prime.
Keywords: Graphs, prime, module, indecomposability graph, Cartesian graphs.
3
Introduction
Our work lies within the frame work of graph theory with a special focus on the decomposition and the product graphs problems. Among the good methods to construct large
graphs from small ones is the product graphs, it also has many application in the design of
interconnection networks (see [24]). There are many ways to define product of two graphs,
the most widely used one may be the Cartesian product, first introduced by Sabidussi [18].
In the same paper, Sabidoussi also proposed another kind of product, the strong product.
It has been known for a large time that the connectivity and the edge connectivity of the
Cartesian product of two graphs are at least the sum of the connectivity and the edgeconnectivity of two factor graphs, respectively (). Recently, the authors () have determined
the connectivity and the edge-connectivity of the Cartesian product of two graphs in terms
of the minimum degree, connectivity, edge connectivity and vertex number of the factor
45
graphs (). In this paper, we are interested the notion of decomposability in the Cartesian
product of two graphs. The concept of indecomposability has become fundamental in the
study of finite structures. In fact, each digraph is indecomposable or can be decomposed
on some indecomposable digraphs. Several papers along these lines have then appeared
e.g ([3, 7, 9, 11, 13, 14, 15, 19]) and are now presented in a book by Ehrenfeucht, Harju
and Rozenberg [9]. Properties of indecomposable substructures of a given indecomposable
structures were developed by Schmerl and Trotter (1993) in their fundamental paper [20].
Recently, Boussaïri, Chaïchaâ and Ille [6] introduce the indecomposability graph as follows:
With each digraph G, is associated its indecomposability graph, denoted by I(G), which is
the graph defined on the set V (G) in the following way. Given v 6= w ∈ V (G), vw ∈ E(I(G))
if G − {v, w} is indecomposable. Notice that the indecompsability graph is the major tool of
many research studies on the indecomposability. In [5], using the indecompsability graph,
Boudabbous and Ille gave a new approach to characterize the critical digraphs which were
be described by Schmerl and Trotter in [20]. In [4] Boudabbous and Ille used the indecomposability graph to study the critical and infinite directed graphs. Moreover in [2], basing
on this graph, Belkhechine, Boudabbous and Elayach described the "(-1)-critical graphs".
In the present paper, first we prove that the Cartesien product of two graphs is indecomposable. Second, we establish that the Cartasien product of two graphs minus a vertex remains
indecomposable. in another word we prove that each vertex of a Cartesian product of two
graphs is not critical. Finally, we give a complete description of the indecomposability graph
of the Cartesien product of two graphs. By this last result we improbe that was obtained
Ehrenfeucht, Harju and Rozenberg [9]. In fact, we characterize exactly the two vertices of
the Cartesian product of two graphs that their deletion give an indecomposable subdigraph.
4
Prerequisites
Throughout this paper, a graph G = (V, E) always means a finite undirected graph where
V = V (G) is the vertex-set and E = E(G) is a family of pairs of vertices called an edge-set.
We denote the edge by uv. Two distinct vertices u and v are adjacent if uv ∈ E; otherwise
v signifies that uv ∈ E, and u. . . . v signifies
u and v are non-adjacent. The notation u
that uv ∈
/ E. For each two disjoint subsets I and J of V , we denote by I
J whenever
for each (x, y) ∈ I × J, x
y. Similarly, for each x ∈ V and for each Y ⊆ V \ {x}, x
Y
(resp. x. . . . Y ) signifies that x
y (resp. x. . . . y) for each y ∈ Y . Furthermore, x ∼ Y
means x
Y or x. . . . Y . The negation is denoted by x 6∼ Y . Given a graph G = (V, E),
for each subset X of V , the subgraph
of G induced by X, G[X], is the graph whose vertex
set is X and edge set is E ∩ X
.
If
X
is a proper subset of V , G[V \ X] is also denoted by
2
G − X, and by G − {x} whenever X = {x}. The notion of isomorphism is in the following
manner. Let G = (V, E) and H = (V 0 , E 0 ) be two graphs. A one-to-one correspondence f
from V onto V 0 is an isomorphism from G onto H provided that for x 6= y ∈ V , xy ∈ E if
and only if f (x)f (y) ∈ E 0 . We denote by G ' H, if there is an isomorphism from G onto H.
With each graph G = (V, E) associate its complement G = (V, E) defined as follows. Given
x 6= y ∈ V , xy ∈ E if and only if xy ∈
/ E. The neighbourhood of a vertex v of V , denoted by
NG (v), is the set of all vertices adjacent to v. The non-neighbourhood of a vertex v of V ,
denoted by N G (v), is V \ (NG (v) ∪ {v}). The degree of a vertex x of V , denoted by dG (x),
corresponds to the number of vertices adjacent to x. The minimal degree of vertex of G,
denoted by δ(G).
Given a graph G = (V, E), a subset M of V is a module [21] (or a clan [9, 10], or an
interval [13, 20]) of G provided that for all a, b ∈ M and x ∈ V \ M , xa ∈ E if and only if
xb ∈ E. For example, ∅, {x}, where x ∈ V , and V are modules of G called trivial modules.
A two-element module of G is called a duo of G. The graphs which do not admit any duo,
are called duo-free graphs. A graph is indecomposable [20] if all its modules are trivial. An
indecomposable graph with at least three vertices is a prime graph [8]. All graphs with
at most two vertices are indecomposable. However, all the 3-vertex graphs are not prime.
Notice that the graphs G and G share the same modules. Thus, G is prime if and only if
46
G is prime. Given n ≥ 4, the simplest prime graph with n vertices is the path Pn defined
on {1, . . . , n} by for i, j ∈ {1, . . . , n}, ij is an edge of Pn if |i − j| = 1. Given a graph
G = (V, E). A nonempty subset C of V is a connected component of G if for x ∈ C and
y ∈ V \ C, xy ∈
/ E and if for x 6= y ∈ C, there is a sequence x = x0 , . . . , xn = y of elements
of C such that xi xi+1 ∈ E for 0 ≤ i ≤ n − 1. A vertex x of G is isolated if {x} constitutes
a connected component of G. The graph G is connected if it has at most one connected
component of G. Otherwise, it is called non-connected. For example, for each prime graph
is connected.
The indecomposability graph of G = (V, E) is the graph I(G) whose vertices are those of
G and edges are the unordered pairs of distinct vertices xy such that the induced subgraph
G[V \ {x, y}] is prime. Let G = (V, E) be a prime graph. A vertex x of V is called a critical
vertex of G if G − {x} is not prime. Let k be a positive integer. The graph G is (−k)-critical
[2] when it has exactly k non-critical vertices.
In the last few years graph products became again a flourishing topic in graph theory.
A graph product G ∗ H of graphs G and H most commonly means a graph on the vertexset V (G) × V (H), while its edges are determined by a function on the edges of the factors.
There are many such products but only four of them are really important: the direct product
(known also as the tensor product, the categorical product [23], the Kronecker product, the
cardinal product [17], the conjunction, the weak direct product, or just the product), the
Cartesian product [1, 12], the strong product (known also as the strong direct product or the
symmetric composition) and the lexicographic product [16] (known also as the composition
or the substitution).
In this paper, the revival of interest seems to the Cartesian product of graphs. The
Cartesian product GH of graphs G = (V1 , E1 ) and H = (V2 , E2 ) is the graph with vertexset is V1 × V2 and (a, x)(b, y) ∈ E(GH) whenever ab ∈ E1 and x = y or a = b and xy ∈ E2 .
The notation is due to Nešetřil. The operation is commutative as an operation on
isomorphism classes of graphs, and more strongly the graphs GH and HG are naturally
isomorphic, but it is not commutative as an operation on labeled graphs. A Cartesian
product of two graphs is connected if and only if both factors are connected, and this
fact is easily provable. Let G1 = (V1 , E1 ) and G2 = (V2 , E2 ) be two non-trivial graphs
where V1 = {x1 , . . . . , xn } and V2 = {y1 , . . . . , ym }. For a ∈ {x1 , . . . . , xn }, we denote the set
{(a, y1 ), . . . . , (a, ym )} by Ca . For a ∈ {y1 , . . . . , ym }, we denote the set {(x1 , a), . . . . , (xn , a)}
by La .
In the present work, first we show that the Cartesian product of two connected graphs is
prime. Second, we prove that each vertex of the Cartesian product of two connected graphs
is not critical. Finally, we give a complete description of the indecomposability graph of the
Cartesian product of two connected graphs.
5 The indecomposability graph of the Cartesian product
of two graphs
The next theorem specifies where the critical and non critical pairs of the Cartesian product
of two graphs occur.
Theorem 5.1. Let G1 = (V1 , E1 ) and G2 = (V2 , E2 ) be two connected graphs where |V1 | ≥ 3
and |V2 | ≥ 3. Consider a = (xa , ya ), b = (xb , yb ) ∈ V (G1 G2 ) such that a 6= b.
ab ∈
/ E(I(G1 G2 )) if and only if {(xa , ya ), (xb , yb )} ∈ S1 ∪ S2 ∪ S3 ∪ S4 where the description
of each set as follows.
• S1 = {{(xα , yα ), (xβ , yβ )} ⊂ V (G1 G2 ) such that there is i 6= j ∈ {α, β} such that
NG1 (xi ) = {xj } and NG2 (yj ) = {yi }}.
• S2 = {{(xα , yα ), (xβ , yβ )} ⊂ V (G1 G2 ) such that there is i 6= j ∈ {α, β}, ν ∈ V1
and γ ∈ V2 such that NG1 (ν) = {xi , xj }, NG1 (xi ) = {ν}, NG2 (γ) = {yi , yj } and
NG2 (yj ) = {γ}}.
47
• S3 = {{(xα , yα ), (xβ , yβ )} ⊂ V (G1 G2 ) such that there is i 6= j ∈ {α, β}, ν ∈ V1 ,
γ ∈ V2 such that NG1 (xi ) = {ν, xj }, NG1 (ν) = {xi }, NG2 (yj ) = {γ, yi } and NG2 (γ) =
{yj }}.
• S4 = {{(xα , yα ), (xβ , yβ )} ⊂ V (G1 G2 ) such that (there exists ν ∈ V1 such that
NG1 (ν) = {xα }, xα = xβ and {yα , yβ } is a module of G2 ) or (there exists γ ∈ V2 such
that NG2 (γ) = {yα }, yα = yβ and {xα , xβ } is a module of G1 )}.
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49