Sciencesconf.org · This implies that such a system is solvable by quadratures if, and only if its coecient matrix is conjugated to a triangular one (via a ... on Lie groups or Riemannian

La Faculté des Sciences Semlalia- Université Cadi AYYAD Marrakech organise, en collaboration avec

La Société Mathématique du Maroc

pour l’enseignement et la recherche : SMM(e, r),

le 2ème Colloque

des Mathématiciens Marocains à l’étranger

Le 27, 28 et 29 Septembre 2018 à Marrakech

https://cmme2-marrakech.sciencesconf.org

Conférenciers Marocains à l'Etranger invités Mostapha ADIMY (Lyon, France) Samir ADLY (Limoges, France) Brahim AMAZIANE ( Pau, France) Moulay BARKATOU ( Limoges, France) Said BENAYADI (Lorraine, France) El Amine KAIDI (Almeria, Espagne) Mohamed KHAMSI ( Texas, USA) Khalid KOUFANY (Nancy, France) EL Haj LAAMRI ( Lorraine, France) EL Maati OUHABAZ ( Bordeaux, France) Hassan OUKHABA (Besançon, France) Mohammed SEAID (Durham, UK) Nazha SELMAOUI (Noumea, Nouv. Caledonie, France) Jawad SNOUSSI ( Université Mexico, Mexique) Ahmed ZERIAHI (Toulouse, France)

Responsables du Colloque : -Pr. Allami BENYAICHE, [email protected]; SMM(e, r) ; Département de Mathématiques, F.S. Kénitra- Maroc. GSM : 06 60 07 95 13 -Pr. Youssef EL FROM, [email protected]; Département de Mathématiques, F.S. Marrakech Semlalia- Marrakech- Maroc. GSM : 06 61 67 98 93

2ieme Colloque des Mathématiciens Marocains à l'Etranger

Comité d'honneur :

- Ministre Délégué chargé des Marocains Résidant à l’Étranger et des Affaires de la Migration.- Secrétaire Perpétuel de l'Académie Hassan II des Sciences et Techniques- Président de l'Université Cadi Ayyad- Doyen de la faculté des Sciences Semlalia, Marakech

Conférenciers Marocains à l'Etranger invités

Mostapha ADIMY (Lyon, France)Samir ADLY (Limoges, France) Brahim AMAZIANE ( Pau, France) Moulay BARKATOU ( Limoges, France) Said BENAYADI (Lorraine, France) El Amine KAIDI (Almeria, Espagne) Mohamed KHAMSI ( Texas, USA) Khalid KOUFANY (Nancy, France) EL Haj LAAMRI ( Lorraine, France) EL Maati OUHABAZ ( Bordeaux, France) Hassan OUKHABA (Besançon, France) Mohammed SEAID (Durham, UK) Nazha SELMAOUI (Noumea, Nouv. Caledonie, France) Jawad SNOUSSI ( Université Mexico, Mexique)Ahmed ZERIAHI (Toulouse, France)

Comité d’organisation:

AKHLIDJ. M (Casa), AMOUCH. M (El Jadida), BAALAL. A (Casa), BOULITE. S (Marrakech), BENYAICHE. A (Kénitra), CHARIFI. A (Kénitra), EL FROM.Y (Marrakech), EL GOURARI. A(Kénitra), El OMARY. M. A (Settat), RAOUJ A. (Marrakech), IZELGUE L. (Marrakech), TAJMOUATI. A (Fès).

Comité d’organisation local:

RAOUJ Abdelaziz

BAROUNE Mahmoud

BOULITE Said

Bouziane Driss

KHOUDRAJI Abdelhaq

LALAOUI My Hicham

MAKKI NACIRI Abderrahim

OUAALI Mustapha

SADIK Brahim

TARIK Khalla

ESSAMAOUI Najoua

ESSOUFI Hasna

Comité scientifique :

AKKOUCHI. M (Marrakech), BOUCETTA. M (Marrakech), BOUSSEJRA. A(Kénitra), CHARKANI.M (Fès), EL ALAOUI Talibi M. (Marrakech), EL KAHOUI. M (Marrakech), EL FALLAH. O (Rabat), EZZINBI K. (Marrakech), KABBAJ. S (Kénitra), KOUFANY. Kh (Nancy), MANIAR L. (Marrakech), OUKNINE.Y (Marrakech ), SODAIGUI. B ( Valencienne), ZERIAHI. A (Toulouse),ZEROUALI. E. H (Rabat).

Responsables du Colloque

Youssef EL FROM

[email protected]

Allami BENYAICHE

[email protected]

Le Le 22èmeème CColloque desolloque desMMathématiciensathématiciens

MMarocains à l’arocains à l’EEtrangertranger

MarrakechMarrakech

Le 26-29 Septembre 2018Le 26-29 Septembre 2018

https://cmme2-marrakech.sciencesconf.org/

RÉSUMÉS CONFÉRENCES

DES INVITS

Le 2ème colloque des mathématiciens marocains à l’étranger Marrakech, Sept. 2018

Conférencier(e)s Invité(e)sNom Prénom Ville, pays Titre de la conférence

ADIMY Mostapha Lyon, France Modèle mathématique de la régulation du cy-cle cellulaire

ADLY Samir Limoges, France La prox-régularité dans tous ses étatsAMAZIANE Brahim Pau, France Homogenization and Numerical Simulation

of Multiphase Flow in Porous MediaBARKATOU Moulay Limoges, France Smallness of the formal exponents of an ir-

regular linear differential system, with an ap-plication to solvability by quadratures

BENAYADI Said Loraine, France Sur les Connexions linéaires bi-invariantessur les groupes de Lie et les algèbres de Leib-niz symétriques

KAIDI El Amine Almeria, Espagne Approche aux Espaces de Banach par leursOpérateurs

KHAMSI Mohamed Texas, USA Introduction to Variable Exponent Spaces:Some Recent Results

KOUFANY Khalid Nancy, France Covariant Bi-Differential Operators for Dif-ferential Forms

LAAMRI EL Haj Lorraine, France Existence globale pour des systèmes deréaction-diffusion avec contrôle de la masse: un panorama général

OUHABAZ EL Maati Bordeaux, France Estimations de Poisson du noyau de lachaleur de l’opérateur Dirichlet-to-Neumann

OUKHABA Hassan Besançon, France Sur quelques équations diophantiennes encaractéristique positive

SEAID Mohammed Durham, UK Recent advances in mathematical modellingand numerical simulation of radiative heattransfer.

SELMAOUI Nazha Nouvelle Calédonie Quelques domaines de motifs pour l’analysede données spatio-temporelles

SNOUSSI Jawad Mexico, Mexique Modifications, sections hyperplanes etcourbes polaires sur les singularités dessurfaces complexes

ZERIAHI Ahmed Toulouse, France Enveloppes pluri-sousharmonique et super-solutions.

6


Modèle mathématique de la régulation du cycle cellulaireMostafa Adimy

INRIA Grenoble Rhône-Alpes,[email protected]

Abstract

Nous vous proposons un modèle mathématique décrivant la dynamique dela population de cellules souches hématopoïétiques au cours du cycle cellulaire.Nous prenons deux populations de cellules en compte, au repos et proliférerun, et on note la différence entre la division des cellules qui entrent directe-ment à la phase de repos et en divisant les cellules qui retournent à la phaseproliférante de diviser à nouveau. Le modèle mathématique résultant est unsystème de deux structuré âge équations aux dérivées partielles. En intégrantce système sur l’âge, nous réduisons à un système différentiel de différence deretard, et nous étudions la stabilité des états stables. En construisant unefonction de Lyapunov, l’état d’équilibre trivial, décrivant la mort de cellulesout, est avérée être globalement asymptotiquement stable quand il est le seuléquilibre. L’analyse de la stabilité de l’état d’équilibre positif unique le plus bi-ologiquement significative, et l’existence d’une bifurcation Hopf permettent ladétermination d’une zone de stabilité, qui est liée à une équation caractéristiquedépendant du retard. Des simulations numériques illustrent nos résultats surle comportement asymptotique des états stables et montrent une dynamiquetrès riche de ce modèle. Cette étude peut aider à comprendre la proliférationincontrôlée des cellules sanguines dans certains troubles hématologiques.

Keywords : régulation du cycle cellulaire. Age-structured partial differential equa-tions, delay differential-difference system, stability switch, Hopf bifurcation, Lya-punov function, cell dynamic.AMS subject classifications : 34D20, 34D23, 34K06, 92C37.

7


La prox-régularité dans tous ses étatsSamir ADLY

Université de Limoges, Laboratoire XLIM,[email protected]

Abstract

L’analyse convexe traite les propriétés des ensembles et fonctions convexes.Malheureusement, de nombreux problèmes dans les applications engendrent desensembles non-convexes. La notion de prox-régularité a fait récemment l’objetd’études approfondies et semble être un outil intéressant pour pallier au défautde convexité des ensembles. Même si elle partage avec la convexité quelquesbonnes propriétés, la prox-régularité possède quelques défauts telles que la sta-bilité par intersection ou bien l’image directe ou réciproque par une application.Nous étudions dans cet exposé, des conditions suffisantes assurant la préser-vation de la prox-régularité ensembliste avec des applications à l’optimisationsous-contraintes et au processus de rafle de Moreau non-convexe.

AMS subject classifications : 46N10, 49J52

8


Homogenization and Numerical Simulation of MultiphaseFlow in Porous Media

Brahim AMAZIANECNRS / Univ. PAU & Pays Adour/ E2S UPPA, PAU

,[email protected]

Abstract

The modeling of multiphase flow in porous formations is important for both themanagement of petroleum reservoirs and environmental remediation. Petroleum engi-neers need to model multiphase flow for production of hydrocarbons from petroleumreservoirs. Hydrologists and soil scientists are concerned with underground waterflow in connection with applications to civil and agricultural engineering, and, ofcourse, the design and evaluation of remediation technologies in water quality controlrely on the properties of underground fluid flow. More recently, modeling multiphaseflow received an increasing attention in connection with the disposal of radioactivewaste and sequestration of CO2. In this talk, we focus our attention on the mod-eling of immiscible compressible two-phase flow in heterogeneous porous media, inthe framework of the geological disposal of radioactive waste. The processes mod-elled are two-phase (water and hydrogen) immiscible compressible two-componenttransient flow in a heterogeneous porous medium under isothermal conditions. Thethree-dimensional (3D) model represents a module of a repository for high-level wastein a clay host rock. We will present a mathematical upscaling method combined toa vertex-centred finite volume discretization and numerical results for 2D [2] and 3D[1] benchmarks. This is a joint work with Etienne Ahusborde, Mustapha El Ossmaniand Mladen Jurak.

AMS subject classifications : 74F10 76M12 76S05 76T10 35B27

References[1] Ahusborde, E., Amaziane, B., Jurak, M. Three-dimensional numerical simula-

tion by upscaling of gas migration through engineered and geological barriersfor a deep repository for radioactive waste (2015) Geological Society SpecialPublication, 415 (1), pp. 123-141.

[2] Amaziane, B., El Ossmani, M., Jurak, M. Numerical simulation of gas migrationthrough engineered and geological barriers for a deep repository for radioactivewaste (2012) Computing and Visualization in Science, 15 (1), pp. 3-20.

9


Smallness of the formal exponents of an irregular lineardifferential system, with an application to solvability by

quadraturesMoulay Barkatou

University of limoges, Limonges (France)[email protected]

Abstract

In this talk, I will present a recent joint work with Renat Gontsov (RussianAcademy of Sciences).We prove that the formal exponents of a linear differential system with non-resonant irregular singular points whose coecient matrix is small, are alsosmall enough. This implies that such a system is solvable by quadraturesif, and only if its coecient matrix is conjugated to a triangular one (via aconstant conjugating matrix). This generalizes the corresponding theorem byIlyashenko-Khovanskii for Fuchsian systems.

Keywords : The formal exponents.AMS subject classifications : 34M03 34M15 34M35 12H05.

10


Sur les Connexions linéaires bi-invariantes sur les groupes deLie et les algèbres de Leibniz symétriques/ On bi-invariantlinear connections on Lie groups and symmetric Leibniz

algebrasSaïd Benayadi

Université de [email protected]

Abstract

Nous montrons qu’il y a une correspondance biunivoque entre la classe desconnexions linéaires bi-invariantes sans torsion sur un groupe de Lie connexe Gqui ont la même courbure que la connexion sur G donnée par ∇0

XY = 12 [X,Y ],

pour tout champs de vecteurs X,Y invariants à gauche sur G, (que nous ap-pelons connexions spéciales) et la classe des structures de Poisson sur l’algèbrede Lie G de G (une structure de Poisson sur G est une structure d’algèbreassociative commutative sur l’espace vectoriel sous-jacent à G pour laquelleadu, la multiplication gauche par u dans l’algèbre de Lie G, est une dérivation,pour tout u ∈ G). Ensuite, le calcul de l’algèbre de Lie de l’holonomie de cesconnexions linéaires, va nous permettre d’introduire des sous-classes de cetteclasse de connexions. Nous montrons que la classe des algèbres de Leibnizsymétriques (qui sont en particulier des algèbres de Poisson) apparaît d’unemanière naturelle en tant que structure de Poisson associée à une sous-classeimportante de la classe des connexions spéciales (une algèbre est dite Leibnizsymétrique si les multiplications gauche et droite Lu et Ru, par tout élément ude cette algèbre, sont des dérivations). Enfin, nous donnons des résultats surla structure et les représentations des algèbres de Leibniz symétriques.

Keywords : Lie group, Lie algebra, Bi-invariant linear connection, Poisson algebra,Symmetric Leibniz algebra.AMS subject classifications :17A32, 17B05, 17B30, 17B63, 17D25, 53C05.

11


APPROCHE AUX ESPACES DE BANACH PAR LEURSOPéRATEURS

El Amine KAIDIUniversité de Almeria, Espagne

[email protected]

AbstractNous allons présenter plusieurs propriétés « remarquables » des opérateurs

d’un espace de Banach et chercher de classes d’ espaces dont les opérateursvérifient de telles conditions, et si possible, les déterminer. Le cas des espacesd’Hilbert sera traité en détail.Exemples : Espaces de Banach «hopfiens» (resp. «co-hopfiens») Tout oper-ateur surjectif (resp. injectif) est bijectif, Espaces de Banach Dedekind finis(Tout opérateur inversible d’un côté est inversible) Espaces de Banach dontl’algèbre d’opérateurs est unitaire « semblable » á une C∗-algèbre . . . ..

Keywords : Espace de Banach, Espaces d’Hilbert, Espaces de Banach « hopfiens ».AMS subject classifications :47H07, 47L30.

References[1] A. Aviles and P. Koszmider : A Banach Space in which every injective operator

is surjective, Bull. London Math. Soc .45 (2013) 1065-1074.

[2] Spiros A. Argyros, Jordi Lopez-Abad, Stevo Todorcevic : A class of Banachspaces with few non-strictly singular operators, Journal of Functional analysis 22(2005), 306-384.

[3] Becerra, Burgos, Kaidi, Rodriguez : Algebras with large groups of unitaryelements, Quart. J. Math. 58 (2007) , 203-220.

[4] M. Burgos, A. Kaidi, M. Mbekhta, M. Oudghiri : The descent spectrum andperturbations , Journal of Operator Theory 56 (2006) 259-271.

[5] Becerra, Rodriguez, Wood : Banach spaces whose algebra of operators are uni-tary: A holomorphic Approach, Bull. London Math. Soc. 35 (2003) 218-224.

[6] W.T. Gowers, B. Maurey : The unconditional basic sequence problem , Journalof the American Mathematical Society 6 (1993), 851-874.

[7] A. Haily, A. Kaidi, A. Rodriguez : Algebra descent spectrum of operator, IsraelJ. of Math. 177 (2010), 349-368.

[8] A. Haily, A. Kaidi, A. Rodriguez : Centralizers in semisimple algebras, anddescent spectrum in Banach algebras, Journal of Algebra 347(2011) 214-223.

[9] A. Hmaimou, A. Kaidi and E. Sanchez : Generalized Fitting modules and rings,Journal of Algebra 308 (2007), 199-214.

12


Introduction to Variable Exponent Spaces: Some RecentResults

Mohamed Amine KhamsiDepartment of Mathematical Sciences,University of Texas at El Paso, USA

[email protected]

Abstract

On an intuitive level, the variable exponent space is obtained by replacingthe energy (also known as modular)∫

Ω|f(x)|pdx with

∫Ω|f(x)|p(x)dx;

where p(x) is a function defined on Ω. Lately variable exponent spaces haveattracted quite a bit of attention. Variable exponent spaces are connectedto variational integrals with nonstandard growth and coercivity conditions.These nonstandard variational problems are related to modeling of the so-called electrorheological fluids and also appear in some models related to imagerestoration.In this talk, we start by a simple introduction to variable exponent spaces.Then we move to discuss some recent results related to the modular geometryof such spaces.

Keywords : Variable Exponent Spaces.AMS subject classifications : Primary 47H09, Secondary 46B20, 47H10

13


Conformally covariant bi-differential operators for differentialforms

Khalid KoufanyInstitut Elie Cartan,

Université de Lorraine, Nancy, [email protected]

Abstract

The classical Rankin-Cohen brackets are bi-differential operators from C∞(R)×C∞(R) into C∞(R). They are covariant for the (diagonal) action of SL(2,R)through principal series representations. We construct generalizations of theseoperators, replacing R by Rn, the group SL(2,R) by the group SO0(1, n + 1)viewed as the conformal group of Rn, and functions by differential forms. Thisis a continuation of a previous construction (see Int. Math. Res. Notes(2018). https://doi.org/10.1093/imrn/rny082) of bi-differential operators, us-ing the source operator method.

Keywords : Rankin-Cohen brackets, .AMS subject classifications :17A15 .

14


Existence globale pour des systèmes de réaction-diffusionavec contrôle de la masse : un panorama général

El-Haj LAAMRIUniversité de Lorraine,

Institut Elie Cartan, Nancy, [email protected]

Abstract

Les systèmes de réaction-diffusion ont connu ces dix dernières années unnet regain d’intérêt du fait que ces systèmes apparaissent dans de nombreuxmodèles en biologie, en chimie, en biochimie, en dynamique des populations eten sciences de l’environnement en général.

De nombreux systèmes de réaction-diffusion évolutifs présentent naturelle-ment les deux propriétés suivantes :(P ) : la positivité des solutions est préservée au cours du temps ;(M) : la masse totale des composants est contrôlée (voire préservée) pour touttemps.

On peut penser que (P ) et (M) garantissent l’existence globale en temps.Mais, il s’avère que la réponse n’est pas si simple. En particulier, des explo-sions dans L∞ peuvent apparaître en temps fini si bien qu’on doit abandonnerl’idée de trouver des solutions globales uniformément bornées et s’intéresser àla recherche de solutions globales faibles qui peuvent sortir de L∞ de temps entemps, mais qui continuent à exister.Dans cet exposé, nous allons donner un panorama général des systèmes vérifi-ant (P ) et (M). Nous rappelerons tout d’abord les résultats d’existence connuset les techniques utilisées. Ensuite nous présenterons quelques nouveaux résul-tats, et enfin nous mentionnerons des problèmes ouverts.

Keywords : systèmes de réaction-diffusion, existence globale, explosion en tempsfini, solutions classiques, solutions faibles, méthode de dualité, technique L1, conver-gence vers l’équilibre.AMS subject classifications :35B40, 35K57, 35B45.

References[1] El-Haj Laamri : Global existence of classical solutions for a class of reaction-

diffusion systems, Acta Appl. Math. 115 (2011), no. 2, 153-165.

[2] Klemens Fellner, El-Haj Laamri : Exponential decay towards equilibrium andglobal classical solutions for nonlinear reaction-diffusion systems, J. Evol. Equ.16 (2016), 681-704.

[3] El-Haj Laamri, Michel Pierre : Global existence for reaction-diffusion systemswith nonlinear diffusion and control of mass. Ann. Inst. H. Poincaré Anal. NonLinéaire 34 (2017), no. 3, 571–591.

15


Poisson bounds for the heat kernel of theDirichlet-to-Neumann

Elmaati OuhabazUniversité Bordeaux 1,

Institut de Mathématiques de [email protected]

Abstract

During the last decades there was an increasing interest in understandingupper and lower bounds for heat kernels in several settings. The most signif-icant bounds are either Gaussian or of Poisson type. Such bounds have beenproved to be true for a wide class of operators such as elliptic operators onEuclidean domains with boundary conditions, on Lie groups or Riemannianmanifolds. These bounds have been used in a very successful way to solveproblems from harmonic analysis, evolution equations, spectral theory... Inthis talk we briefly review this subject and consider the problem of Poissonbounds for the Dirichlet-to-Neumann operator on domains of class C1+κ forsome κ > 0. This last operator appears in many branches of mathematics.

Keywords : Poisson bounds.AMS subject classifications :34C07, 35P15.

16


Sur quelques equations diophantiennes en caractéristiquepositive

OUKHABA HASSANUniversité de Franche-comté, [email protected]

Abstract

Je discuterai les solutions de l’équation linéairen∑i=1

aiλi = 0,

où les ai sont des éléments de certains anneaux comme par exemple les anneauxd’entiers de corps de nombres, ou des anneaux de fonctions régulières sur descourbes définies sur un corps fini. Alors que les λi sont soit des abssices depoints de torsion de courbes elliptiques définies sur un corps global, soit despoints de torsion de modules de Drinfeld.Les résultats que j’exposerai font partie d’un travail en cours, en collaborationavec Mohamed Elkati.

Keywords : Equations linéaires, points de torsion.AMS subject classifications :11D04, 11G09 et 11G05.

17


Mathematical modelling and numerical simulation ofmorphodynamic flow problems

Mohammed SeaidUniversity of Durham (United Kingdom)

[email protected]

AbstractThe aim of this contribution is to develop a well-balanced finite-volume

method for the accurate numerical solution of the equations governing sus-pended sediment and bed load transport in both two- and three-dimensionalshallow water flows. The modelling system consists of three coupled modelcomponents: (i) the single and multi-layer shallow water equations for the hy-drodynamical model; (ii) a transport equation for the dispersion of suspendedsediments; and (iii) an Exner equation for the morphodynamics. These coupledmodels form a hyperbolic system of conservation laws with source terms. Theproposed finite volume method consists of a predictor stage for the discretiza-tion of gradient terms and a corrector stage for the treatment of source terms.The gradient fluxes are discretized using a modified Roe’s scheme using thesign of the Jacobian matrix in the coupled system. A well-balanced discretiza-tion is used for the treatment of source terms. In this contribution, we alsoemploy an adaptive procedure in the finite volume method by monitoring theconcentration of suspended sediments in the computational domain during itstransport process. The method uses unstructured meshes and incorporates up-winded numerical fluxes and slope limiters to provide sharp resolution of steepsediment concentrations and bed load gradients that may form in the approx-imate solutions. Details are given on the implementation of the method, andnumerical results are presented for two idealized test cases, which demonstratethe accuracy and robustness of the method and its applicability in predictingdam-break flows over erodible sediment beds. The method is also applied to asediment transport problem in the Nador lagoon.

Most of the mathematical models and numerical methods presented in thiswork have been subject to intensive research published in [1, 2] among others.

Keywords : Morphodynamics, Hydrodynamics, Shallow water equations, Sedimenttransport, Finite volume method, Numerical Analysis.AMS subject classifications :97N40, 65M08, 93C20.

References[1] F. Benkhaldoun, I. Elmahi, A. Moumna, M. Seaid: A non-homogeneous Rie-

mann solver for shallow water equations in porous media. Applicable Analysis2016; 95:2181–2202.

[2] F. Benkhaldoun, I. Elmahi, S. Sari, M. Seaid: An unstructured finite volumemethod for coupled models of suspended sediment and bed load transport inshallow water flows. International Journal for Numerical Methods in Fluids 2013;72:967–993.

18


Quelques domaines de motifs pour l’analyse de donnéesspatio-temporelles

Nazha Selmaoui-FolcherISEA, Université de la Nouvelle Calédonie

[email protected]

Abstract

La science des données (Data Science) est une science nouvelle née del’explosion de la quantité d’informations collectées ces dernières années. Ellevise à apporter des méthodologies et technologies nouvelles pour traiter et anal-yser des données massives (Big Data) et complexes (multivariées, multi-échelles,spatiales et temporelles). Elle est à l’intersection de plusieurs domaines scien-tifiques : statistiques, informatique, mathématiques, intelligence artificielle,modélisation, fouille de données, apprentis- sage automatique, etc.Je parlerai de la fouille de données et plus particulièrement du problème del’extrac- tion de motifs et présenterai un projet [3] utilisant ce type d’approchespour faire du suivi environnemental en croisant série temporelle d’images satel-litaires et données collectées sur le terrain.

Keywords :Data Science, Big Data, Fouille de données spatio-temporelles, Extrac-tion de motifs, Fouille de graphes, Classification supervisée, Série temporelles d’imagessatellitaires.AMS subject classifications : 68Pxx,

References[1] http ://foster.univ–nc.nc

19


Modifications, sections hyperplanes et courbes polaires sur lessingularités des surfaces complexes

Jawad SNOUSSIMexico, Mexique

[email protected]

Abstract

Pour étudier les singularités locales des surfaces complexes, nous proposonsde comprendre le comportement de ces surfaces par sections hyperplanes. Lagéométrie de ces sections est liée au comportement de cette même surface pardeux types de modifications : l’éclatement de la singularité et la modificationde Nash. Nous décrirons alors ces deux modifications et leurs interactions.

References[1] J. Snoussi, Linear components of the tangent cone in the Nash modification of

a complex surface singularity, in Journal of Singularities, 3 (2011) 83–88.

[2] J. Snoussi, The Nash Modification and hyperplane sections on surfaces. Bull.Braz. Math. Soc. (NS) 36, No.3, 309-317 (2005).

[3] J. Snoussi, Limites d’espaces tangens à une surface normale. Comment. Math.Helv. 76 (2001), 61-88.

AMS subject classifications :14B05 et 32Sxx

20


Enveloppes plurisousharmoniques et sursolutionsAhmed Zeriahi

Institut de Mathématiques de Toulouse, [email protected]

Abstract

Les enveloppes de fonctions convexes ou sous-harmoniques sont des objetsclassiques en Analyse convexe et en Théorie du Potentiel. Elles interviennentpar exemple dans la résolution du problème de Dirichlet pour l’opérateur deLaplace par la méthode de Perron (1921). Elles fournissent également la so-lution dans certains problèmes d’obstacle ou à frontière libres pour certainesEDP non linéaires du second ordre.

Elles ont été introduites un peu plus tard en Analyse complexe par H.Bremmermann ( 1959) et J. Siciak ( 1969) notamment pour étudier des prob-lèmes analogues relatifs aux fonctions plurisousharmoniques. Elles ont été util-isées ensuite avec un grand succès par E. Bedford et B.A. Taylor ( 1976-1982)pour résoudre le problème de Dirichlet pour les équations de Monge-Ampèrecomplexes par la méthode de Perron. Cette méthode consiste à considérerl’enveloppe supérieure des sous-solutions du problème. Dans le contexte deséquations de Monge-Ampère complexes, de telles sous-solutions sont dites sous-solutions pluri-potentielles.

Par dualité on pourrait en principe considérer l’enveloppe inférieure des sur-solutions du problème. Cette notion est très délicate dans le cas non linéaireet souffre d’un manque de symétrie rendant cette dualité imparfaite.

Par ailleurs d’autres notions de sous/sur-solutions ont été introduites parP.L. Lions et son école ( 1980) sous le nom de sous/sur-solutions de vis-cosité pour étudier certaines EDP (non linéaires ) elliptiques dégénérées pourlesquelles la notion de solution faible ne fait pas sens. Toutes ces notionsont été adaptées récemment au cas complexe avec des applications significa-tives à des problèmes de Géométrie Kählérienne. Il s’avère que les notionsde sous-solutions coïncident dans les deux théories. Cependant le lien entreles sur-solutions pluri-potentielles et les sur-solutions de viscosité est demeurémystérieux en raison du caractère non linéaire de l’opérateur de Monge-Ampèrecomplexe.

Dans cet exposé nous montrerons comment la méthode des enveloppesavec obstacle permet de faire ce lien et d’obtenir des résultats nouveaux enGéométrie kählerienne grâce à la généralisation d’une méthode d’approximationoriginale dûe à R. Berman ( 2013). C’est un travail en collaboration avec ChinnH. Lu et Vincent Guedj à paraître dans JDG en 2018-2019 (voir arXiv:1703.05254).

Keywords :Fonctions convexes, fonctions plurisousharmoniques, Enveloppes plurisoushar-monique, Enveloppes sursolutions.AMS subject classifications :46N10, 65E05.

21

RÉSUMÉS CONFÉRENCES


ProfesseursNom Prénom Ville, pays Titre de la communication

BENZIADI Fatima Saida, Algérie The homeomorphic property of the stochas-tic flow of a natural equation in Multi-dimensional case

EL HAJIOUI Khalid Kenitra Sur la caractérisation de convergences varia-tionnelles des suites de fonctions convexes etde fonctions convexes-concaves

EL HARAMI Mohamed Meknes Pettis conditional expectation of random setsELQORACHI Elhoucien Agadir Stability of trigonometric functional equa-

tions on amenable groupsFADILI Ahmed Béni Mellal On the Complex Inversion Formula and Ad-

missibility for a Class of Volterra Systemswith non scalar kernel

KACHA Ali Kenitra Transcendence and approximation measureof continued fractions

MAMOUNI My Ismail RABAT Topological Data AnalysisMARAGH Fouad Agadir The Favard classes of k-regularized resolvent

families for Volterra equationsOUKHTITE Lahcen Fès New classes of endomorphisms and some

classification theoremsROCHDI Abdellatif Casablanca Algèbres réelles de division de dimension finieSABIRI Mohammed Errachidia Construction of some codes Quasi-Cyclic

CodesSAMMAD Khalil Kenitra On the measure of central disintegration for

the regular representation in non unimodulargroups

SEDDIK Abdelalim Casablanca Hereditarily Co-Hopfian Abelian GroupsTAJANI Chakir Larache A metaheuristic algorithm for solving an in-

verse problem for Laplace equation

23


The homeomorphic property of the stochastic flow of anatural equation in Multi-dimensional case

Fatima Benziadi, Abdeldjebbar KandouciDr. Moulay Tahar University, Saida (Algeria)

[email protected]

Abstract

The one-default models are widely applied in modeling financial risk andprice valuation of financial products such as Credit default swap. In this paper,we are interested essentially in the so-called natural model. This model isexpressed by a stochastic differential equation called equation introduced in[1]; this equation displays the evolution of the defaultable market. So, on thesame model and with some assumptions, we will try to prove a few propertiesof stochastic flow generated by the ]-equation but in a multidimensional caseand with some modifications. This is the main motivation of our research.

Keywords : Credit risk ; Stochastic flow ; Diffeomorphism.AMS subject classifications : Primary 60G17 ; Secondary 60H05.

References[1] Monique Jeanblanc, Shiqi Song, Random times with given survival probability

and their F-martingale decomposition formula, Stochastic Processes And theirApplica- tions 121(2010-2011).

[2] K. D. Elworthy, Stochastic dynamical systems and their flows, stochastic analysised. by A.Friedman and M. Pinsky, 79-95, Academic press, New York, 1978

[3] P. Malliavin, Stochastic calculus of variation and hypoelliptic operators, Kyoto,Confe- rence, 1976, Wiley 1978, 195-263.

[4] J. M. Bismut, Flots stochastiques et formula de Itô-Stratonovich généralisée,C.R. Acad. Sci. Paris, t. 290 (10 mars 1980).

[5] H. Kunita, On the decomposition of solutions of stochastic differential equations.

24


Sur la caractérisation de convergences variationnelles dessuites de fonctions convexes et de fonctions convexes-concaves

Khalid El HajiouiDépartement des Sciences Economiques et Gestion,

Université Ibn Tofail, Kenitra (Maroc)

Abstract

Cette communication consiste en une nouvelle caractérisation d’une classede convergences variationnelles qui jouent un rôle crucial en optimisation eten analyse variationnelle. Lorsque les fonctions considérées sont dans Γ(X) ouΓ(X∗) où X est un espace vectoriel normé, cette caractérisation est donnée entermes d’approximations inf-convolutives associées à des référentiels généraux.Lorsque les fonctions considérées sont à deux variables convexes-concaves, nousintroduisons une nouvelle convergence appelée t-épi/hypo-convergence que nouscaractérisons d’abord en termes de Lagrangiens augmentés généralisés puis entermes d’approximations inf-sup-convolutives généralisées associés aux trans-formées partielles de Legendre-Fenchel des composantes convexes des fonctionsconvexes-concaves initiales. Nos démonstrations et nos résultats sont originauxet s’insèrent dans le cadre de l’analyse variationnelle et la théorie de la dualité.Certains résultats qui existent déjà dans la littérature sont alors généralisés etcomplétés

Keywords : Fonction convexe (convexe-concave),convergences variationnelles, t-convergence, t-épi/hypo-convergence, référentiel, approximation inf (inf–sup)-convolutive,Lagrangien augmenté.AMS subject classifications : 49J45, 49J52(primary), and 49J40, 47N10(secondary).

References[1] H. Attouch, D. Azé and G. Beer, On some inverse stability problems for the

epigraphical sum, Nonlinear Anal.Theo. Meth. Appl. 16 (1991), 241-254.

[2] H. Attouch and R. J. -B. Wets, A convergence theory for saddle functions, Trans.Amer. Math. Soc. 280 (1983), 1-41.

[3] G. Beer, Lipschitz regularization and the convergence of convex functions, Nu-mer. funct. Anal. Optim. 15 (1994), 31-46.

[4] J. Lahrache, Stabilité et convergence dans les espaces non réflexifs, Sém. d’Anal.Convexe Montpellier, exposé 10, 1991.

[5] D. Mentagui et K. El Hajioui, Convergences des fonctions convexes et approx-imations inf-convolutives généralisées, Publ. Inst. Math., Nouvelle série, Tome72(86) (2002), 123- 136.

25


Pettis conditional expectation of random setsMohamed El Harami

University Moulay Ismail,Higher School of Technology, Meknes

[email protected]

Abstract

There are many works in the literature treating the existence of conditionalexpectation for Pettis integrable random variables (resp. random sets ), let usmention the works of authors [4, 3]. Recently, [2] established an extension ofthose results to the case where the random sets are only with closed convexand bounded values. In this work we are interested to the same question ofthe existence of this operator but for random sets with closed convex andunbounded values, which is more general and more complicated than the casewhere the random sets are with convex weakly compact (resp. closed convexand bounded) values. As application of the existence of EB(X), we extendthe Levy’s convergence theorem to the closed convex valued Pettis integrablerandom sets processing by linear topology, compare with the works in [5, 6,1] dealing with the Mosco’s topology and closed convex and bounded (resp.convex and weakly compact) random sets.

Keywords : Pettis Conditional expectation, Levy’s theorem, linear topology.AMS subject classifications : Primary: 58J65. Secondary: 60H05, 60H25.

References[1] F. Akhiat, C. Castaing and F. Ezzaki, Some various convergence results for

multivalued martingales, Adv. Maths. Ec, 13 (2010), 1-33.

[2] F. Akhiat, M. El Harami and F. Ezzaki, Pettis conditional expectation of closedconvex random sets in a banach space without RNP, Journal. Korean. Math.Soc., in the print (2018).

[3] M. EL Harami and F. Ezzaki, General Pettis conditional expectation and conver-gence theorems, International journal of mathematics and statistics, 11 (2012),91-111.

[4] H. Ziat, Convergence des suites adaptées multivoques application à la loi fortedes grands nombres multivoque, Thèse de doctorat Montpellier II, (1993).

[5] Zhen-PengWang, Xing-Hong Xue, On convergence and closedness of multivaluedmartingales, Trans. American. Math. Soc., 341 (1994), 807-827.

[6] D. Wenlong and W. Zhenpeng, On representation and regularity of continuousparameter multivalued martingales, Proc. American. Math. Soc., 126 (1998),1799-1810.

26


Stability of trigonometric functional equations on amenablegroups

Ajebbar Omar, Elqorachi Elhoucien,Ibn Zohr University, Faculty of Sciences,

Department of Mathematics, Agadir, [email protected]

Abstract

In this paper, we obtain the Hyers-Ulam-Rassias stability of the trigono-metric functional equation

f(xy) = f(x)g(y) + f(y)g(x) + h(x)h(y), x, y ∈ G,

where G is an amenable group.

Keywords : Hyers-Ulam stability, Group, Cosine equation, Sine equation, Multi-plicative function, Additive function.AMS subject classifications : 39B82(primary), and 39B32(secondary).

References[1] J.K. Chung, Pl. Kannappan and C.T. Ng, A generalization of the Cosine-Sine

functional equation on groups. Linear Algebra and Appl. 66 (1985), 259-277.

[2] D.H. Hyers, On the stability of linear functional equation, Proc. Nat. Acad. Sci.U.S.A., 27 (1941), 222-224.

[3] D.H. Hyers, G. Isac and Th. M. Rassias, Stability of functional equations inseveral variables, Birkhäuser, Boston, 1998.

[4] Pl. Kannappan, Functional Equations and Inequalities with Applications,Springer, 2009.

[5] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc.Amer. Math. Soc. 72 (1978), 297-300.

[6] L. Székelyhidi, The stability of the sine and cosine functional equations, Proc.Amer. Math. Soc. 110 (1990), 109-115.

[7] S.M. Ulam, A collection of Mathematical Problems, Interscience Publishers, NewYork, 1960.

27


On the Complex Inversion Formula and Admissibility for aClass of Volterra Systems with non scalar kernel

Ahmed FadiliUniversité de Sultan Moulay Sliman, PB 592 Beni Mellal, (Maroc)

Fouad Maragh, Hamid BounitUniversité Ibn Zohr, BP 8106, Agadir 80 000, (Maroc)

[email protected]

Abstract

This work studies Volterra integrodifferential evolution equations of convo-lution type from the point of view of complex inversion formula and the admis-sibility for the resolvent operators. We first present results on the validity ofthe inverse formula of the Laplace transform for the resolvent operators associ-ated to a non scalar Volterra integral equations of convolution type in Banachspaces, which extends and improves the results in [2] respectively, including thestronger version for a class of non scalar Volterra integrodifferential equationsof convolution type on unconditional martingale differences UMD spaces, pro-vided that the leading operator generates a C0-semigroup. Next, a necessaryand sufficient condition for Lp−-admissibility (p ∈ [1,∞[) of the system’s con-trol operator is given in terms of the UMD-property of its underlying controlspace for a wider class of Volterra integrodifferential equations when the leadingoperator does not necessarily be a generator, which provides a generalizationof a result known to hold for the standard Cauchy problem [1].

Keywords :C0-semigroup, Volterra equation, resolvent operators, UMD space, com-plex inversion, Laplace transform, admissibility.AMS subject classifications : 34K30, 35R15, 39A14, 32A70, 93C25, 93C20.

References[1] H. Bounit, A. Driouich, and O. El-Mennaoui. Admissibility of control operators

in UMD spaces and the inverse Laplace transform. Integr. Equ. Oper. Theory,68:451–472, 2010.

[2] I. Cioranescu and C. Lizama. On the inversion of the Laplace transform forresolvent families in UMD spaces. Arch. Math. (Basel), 81:182–192, 2003.

28


Transcendence and approximation measure of continuedfractions

Ali Kacha , Kacem BelhroukiaUniversité Ibn Tofail (Maroc)

[email protected]

Abstract

The transcendence of the continued fractions having partial quotients thatincrease rapidly have been studied by several authors such as P. Bundschuh[1], A. Durand, W. Lianxiang [2] , G. Nettler [3], T. Okano [4]. In this work,we give sufficient conditions on the elements of continued fractions A and Bwhich will assure us that A, B, A ± B, AB and A/B are all transcendentalnumbers. The used method also permits us to calculate approximation measureof a continued fraction A.

Keywords :Transcendence, continued fraction, approximation measure.AMS subject classifications : 40A15 and 15A60.

References[1] P. Bundschuh - Transcendental Continued Fractions, J. Number Theory, 18

(1984), 91-98.

[2] W. Lianxiang - p-adic continued fraction (II), Scientia Sinica Ser. A 28, No 10,(1985) 1018-1028.

[3] G. Nettler - Transcendental continued fractions, J. Number Theory ; 13 (1981),456-462.

[4] T. Okano - A note on the transcendental continued fractions ; Tokyo J. Math.vol 10, No. 1 (1987), 151-156.

29


Topological Data AnalysisMy Ismail Mamouni

UCRMEF Rabat, [email protected]

AbstractTopological Data Analysis (TDA) is a recent and fast growing eld providing

a set of new topological and geometric tools to infer relevant features for pos-sibly complex data. TDA emerged from various works in applied (algebraic)topology and computational geometry, it started with the pioneering works ofEdelsbrunner [EH] and was popularized in a landmark paper in 2009 Carlsson[C].Persistent homology, is the powerful tool in TDA for investigating the struc-ture of data. It produces a kind of diagram, called "the persistence diagram",that can show a great deal of information about a given point cloud such asclustering, which is usually necessary. It can also describe more complicatedstructure such as loops and voids that are not visible with other methods.Persistent homology has found success in the investigation of data from manydifferent domains; these include image processing, time series analysis, phylo-genetics, neuroscience, sensor networks, medecine, finance. In this talk, we willexpose the work in progress of a Moroccan research group to develop some realapplications in : Image Analysis ([CM]), Machine Learning ([BEM]), Neurology([BMM]), Civil ([BLM]) Engineering and Linguistic ([M]).

Keywords :topological data analysis, persistent homology, cubical homology, digitalimages, machine learning, clique graphs, complex networks, design project, writtensystem.AMS subject classifications : 55U05, 55U10, 68P05, 90C90, 55N35, 05C10,57M15.

References[BEM] H. Bouazzaoui, M. A. Elomary and M.I. Mamouni, Bongard problems : What

TDA can tell us (in progress).

[BMM] D. Bennis, A.M. Maganga and M.I. Mamouni, Cognitive dysfunction andradiation exposure : TDA can alert us (in progress).

[BLM] D. Bennis, G. Lebbar and M.I. Mamouni, A new approach for design projectfrom TDA (in progress).

[CM] H. Chawqi and M.I. Mamouni, From images to bare codes (submitted).

[C] G. Carslsson, Topology and Data, Bulletin (New Series) of the AMS, Volume 46(2009), Number 2, 255-308.

[EH] H. Edelsbrunner, J. Harer, Persistent homology : a survey. Contemporarymathematics 453(2008), 257-282.

[M] M.I. Mamouni, Topology of written systems (submitted).

30


The Favard classes of k-regularized resolvent families forVolterra equations

Fouad Maragh, Hamid BounitUniversité Ibn Zohr, BP 8106, Agadir 80 000, (Maroc)

Ahmed FadiliUniversité de Sultan Moulay Sliman, PB 592 Beni Mellal, (Maroc)

[email protected]

Abstract

In the present paper, we extend the definition and some well-known theo-rems of the Favard spaces for k-regularized resolvent family defined for resolventfamily in [1], Favard class for semigroups introduced and was developed as earlyas 1967 by B.L. Butzer and H. Berens presented in the monograph [2].

Keywords :Semigroups, Volterra integral equations, regularized resolvent families,Favard spaces.AMS subject classifications : 34K30, 35R15, 39A14, 32A70, 93C25, 93C20..

References[1] H. Bounit and A. Fadili. On the Favard spaces and the admissibility for Volterra

systems with scalar kernel. Electronic Journal of Differential Equations, (2015):N42 pp. 1–21, 2015.

[2] P.L. Butzer and H. Berens. Semi-Groups of Operators and Approximation.Springer-Verlag, New York, 1967.

31


New classes of endomorphisms and some classificationtheoremsL. Oukhtite

Department of Mathematics,Faculty of Science and Technology of Fez, Box 2202,

University S. M. Ben Abdellah Fez, [email protected]

Abstract

In the present work we introduce new classes of endomorphisms and studytheir connection with commutativity of prime rings with involution of the sec-ond kind. Moreover, we give a complete description and classification for someof these endomorphisms. Furthermore, we provide examples to show that thevarious restrictions imposed in the hypotheses of our theorems are not super-fluous.

Keywords :rime ring, involution, commutativity, endomorphisms.AMS subject classifications : 16N60, 16W10, 16W25.

References[1] H. E. Bell and M. N. Daif, On commutativity and strong commutativity preserving

maps, Canad. Math. Bull. 37 (1994), 443-447.

[2] H. E. Bell and W. S. Martindale III, Centralizing mappings semiprime rings,Canad. Math. Bull. 30 (1987), no. 1, 92-101.

[3] M. Bresar, Commuting traces of biaddiitive mapping, commutativity preservingmapping and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), no. 2, 525-546.

[4] M. Bresar and C. R. Miers, Strong commutativity preserving mappings ofsemiprime rings, Canad. Math. Bull. 37 (1994), 457-460.

[5] A. Mamouni, L. Oukhtite and B. Nejjar, On ∗-semiderivations and ∗-generalizedsemiderivations, J. Algebra Appl. 16 (2017), no. 4, 1750075, 8 pp.

[6] B. Nejjar, A. Kacha, A. Mamouni and L. Oukhtite, Commutativity theorems inrings with involution, Comm. Alg. 45 (2017), no. 2, 698-708.

[7] L. Oukhtite and A. Mamouni, Generalized derivations centralizing on Jordan ide-als of rings with involution, Turkish J. Math. 38 (2014), no. 2, 225-232.

[8] L. Oukhtite, Posner’s Second Theorem for Jordan ideals in rings with involution,Expo. Math. 29 (2011), no. 4, 415-419.

[9] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100.

32


Algèbres réelles de division de dimension finieA. ROCHDI

Université Hassan II, Faculté des Sciences Ben M’Sik,Casablanca (Maroc)

[email protected]

Abstract

Nous exposons l’histoire des algèbres réelles de division de dimension finiedepuis la découvertes des Quaternions H et des Octonions O. Nous rappelonsles résultats classiques et énonçons les plus récents.

Keywords :Algèbres non associatives (de division, absolument valuées), Quater-nions, Octonions, Procédé de Cayley Dickson, Isotopie.AMS subject classifications : (17A35, 17A36)

References[1] A. A. Albert, Absolute valued real algebras. Ann. Math. 48 (1947), 495-501.

[2] J. C. Baez, The octonions. Bulletin (New Series) of the AMS 39 (2) 145-205.

[3] A. Calder on, A. Kaidi, C. Mart in, A. Morales, M. Ram irez, and A. Rochdi,Finite dimensional absolute valued algebras. Israel J. Math. 184 (2011), 193-220.

[4] A. Chandid, and A. Rochdi, A survey on absolute valued algebras satisfying(xi, xj, xk) = 0. Int. J. Algebra, 2 (2008), 837-852.

[5] A. Cuenca Mira, On one-sided division infinite-dimensional normed real algebras.Publicacions Matem‘ atiques 36 (1992), 485-488.

[6] J. A. Cuenca, On composition and absolute valued algebras. Proceedings of theRoyal Society of Edinburgh, 136A, (2006) 717-731.

[7] J. A. Cuenca, R. De Los Santos Villodres, A. Kaidi, and A. Rochdi, Realquadratic flexible division algebras. Linear Algebra Appl. 290, (1999) 1-22.

[8] E. Darp o, and A. Rochdi, Classification of the four-dimensional power-commutative real division algebras. Proceedings of the Royal Society of Ed-inburgh, 141 A, 1207-1223, (2011)

33


Construction of some codes Quasi-Cyclic CodesMohammed. Sabiri

Moulay Ismail university.Faculty of Sciences and Technology

P.O. Box 509-Boutalamine, 52 000 [email protected]

Abstract

Quasi-cyclic codes over a finite commutative ring are viewed as cyclic codesover a noncommutative ring of matrices over a finite commutative ring. Thestudy of these codes permits to generalize some known results about quasi-cyclic codes over a finite fields and to propose a construction of some quasi-cyclic codes.

Keywords : Quasi-cyclic codes, Linear recurrent sequences, Rings of matrices.AMS subject classifications : 16R40 and 94B15.

References[1] Mac Williams and N. J. A. Sloane, The Theory of Error-Corecting Codes, Third

printing North-Holland Mathematics Library (1981), Volume 16.

[2] Pierre-Louis Cayrel, Christophe Chabot, Abdelkader Necer, Quasi-cyclic codesas codes over rings of matrices, Finite fields and their applications, 16 (2010)100-115.

[3] Christophe Chabot, Reconnaissance de codes,structure des codes quasi-cycliques,thesis N 29-2009.

34


On the measure of central disintegration for the regularrepresentation in non unimodular groups

Akkouchi Mohamed1 , Allal Bakali2 , Kabbaj Samir3 , and Sammad Khalil41Département de Mathématiques, Faculté des Sciences,

Université Cadi Ayyad, Marrakech, Maroc2Département de Mathématiques, Faculté des Sciences,

Université Ibn Tofail, Kénitra, Maroc3Département de Mathématiques, Faculté des Sciences,

Université Ibn Tofail, Kénitra, Maroc4Département de Mathématiques, Faculté des Sciences,

Université Ibn Tofail, Kénitra, [email protected]

[email protected]@yahoo.fr

[email protected]

Abstract

In the harmonic analysis context associated to Gelfand measures, we stud-ied the -non degeneracy of representations without mentioning some requiredconditions.In this paper we will give more details such as all closed subspace ofL2(G) invariant by left translation contains a unit, in the case of non unimod-ular groups which generalise the Gelfand pairs case.

Keywords :Topological locally compact group, Unitary irreducible representation,Gelfand measure, Unit element, Non degenerate unitary representation, disintegra-tion.AMS subject classifications : 22D10.

35


Hereditarily Co-Hopfian Abelian GroupsSeddik ABDELALIM,

Faculty of Sciences Ain Choc,Hassan II University of Casablanca, Morocco

[email protected]

Abstract

An abelian group A is Hereditarily Co-Hopfian if any subgroup of A is Co-Hopfian. In this paper We will characterize Hereditarily Co-Hopfian abeliangroup in the category of abelian group. It is clear that the subgroup of Hered-itarily Co-Hopfian abelian is also Hereditarily Co-Hopfian abelian group, Fi-nally we show that the quotien of two Hereditarily Co-Hopfian abelian is alsoHereditarily Co-Hopfian abelian.

Keywords : Abelian group, Hereditarily Co-Hopfian abelian group,AMS subject classifications :20K01

References[1] Seddik. Abdelalim Characterization The strongly Co-Hopfian abelian groups

in the Category of Abelian torsion Groups Journal of Mathematical analysisVolume 6 ISSUE 4(2015), PAGES 1-10.

[2] B. Goldsmith and K. Gong On super and hereditarily hopfian and co-hopfianAbelian groups. Arch. Math Springer. 99 (2012), 1–8

[3] A. Hmaimou, A. Kaidi and E. Sanchez Campos, Generalized Fitting modulesand rings, Journal of Algebra 308 (1) (2007), 199-214.

[4] A. kaidi et M. Sangharé,Une caractérisation des anneaux artiniens ideaux prin-cipaux, Lecture notes in Mathématique 1328 (1988) 245-254.

[5] L. Fuchs, Infinite Abelian Groups, vol. 1,2 Academic press New York, 1970.

36


A metaheuristic algorithm for solving an inverse problem forLaplace equation

Chakir TAJANIFaculté polydisciplinaire à Larache,

Université Abdelmalek Essaadi (Maroc)[email protected]

Abstract

We consider an inverse problem for the Laplace equation called data comple-tion problem, which aims at recovering missing conditions on some inaccessiblepart of the boundary from the overspecified boundary data on the remainingpart. This problem arises and can be encountered as challenge in several ar-eas of engineering, especially in the detection of corrosion problem where thecompleted data are used to calculate the Robin coefficient that represent thecoefficient damage and which is the quotient of these extended data. This prob-lem is ill-posed in the sense of Hadamard [1], since the existence, uniquenessand stability of the solution are not always assured. Solving this problem bydirect method is very difficult and leads to unstable solutions.Many performing numerical methods have been developed to overcome the ill-posed nature of this kind of problem. Among them we mention the method ofQuasi-reversibility, Thikhonov method and the iterative method... In this pa-per, the considered inverse problem is formulated as an optimization problemand we investigate the use of genetic algorithm for regularizing this ill-posedproblem. The Laplace equation is discretised using the Finit Element Method.Nmnerical results are presented and discussed for several test examples.

Keywords : inverse problem, Genetic algorithm, Metaheuristic algorithmAMS subject classifications : 65N21; 35J05; 65N30

References[1] J. Hadamard, Lectures on the Cauchy Problem in Linear Partial Diferential

Equations, Yale University Press, New Haven, 1923.

[2] Xin-She Yang, Metaheuristic algorithms for inverse problems, Int. J. InnovativeComputing and Applications, Vol. 5, No. 2, 2013, pp. 76-84.

[3] N. S. Mera, L. Elliott, D. B. Ingham, Numerical solution of a. boundary detec-tion problerm using genetic algorithrms, Enginiering Analysis with BoundaryElements, Vol. 28, No. 4, 2004, pp. 405-411.

37

RÉSUMÉS POSTERS


DoctorantsNom Prénom Ville, pays Titre du poster

ABDELLAOUI Mohammed Fès On some nonlinear parabolic equations with un-bounded function and measure data

AIT TOUCHENT Kamal Errachidia Optical solitary wave solutions to the(2 + 1)−dimensional Calogero-Bogoyavlenskii-Schiff equation with fractional conformablederivative

AIT ZEMZAMI Omar Fès On Center-Like Subsets in Prime RingsBOULAYAT Brahim Beni mellal Some factorization properties of the composite ringEL ALAOUI Haitham Fès Bézout-like properties in trivial ring extensionsEL AZHAR Hamza Rabat Idempotents and Moment problem of discrete mea-

sureEL HOUCH Atmane Casablanca Feedback stabilization and polynomial decay esti-

mate for time delay semilinear systemsESSAMAOUI Najoua Marrakech Stable tameness of coordinates in two variablesESSOUFI Hasna Marrakech The geometry of generalized Cheeger-Gromoll

metrics on the total space of transitive EuclideanLie algebroids

FARID Mohamed amine Casablanca Chatterjea fixed point theorem in a space withthree metrics

HAYNOU Mbarek Errachidia On the generators of the 2−class group of somepure quartic numbereld

IDRISSI My Abdallah Fès Classification of some special generalized deriva-tions

KHLIFI Ismail Kenitra Sobolev-Dirichlet problem for quasilinear ellipticequations in generalized Orlicz spaces

KREIT Karim Marrakech An alternating direction method of multipliers forthe total variation regularization

LABIHI Omar Marrakech Evaluation de certaines sommes courtesMARZOUGUE Mohamed Agadir BSDEs driven by normal martingale under general

assumptionsOMARI Youssef Rabat Complete Interpolating sequences for small Fock

SpacesROSSAFI Mohamed Kenitra Basic Properties of ? − K−Frames in Hilbert A-

modulesSOUKTANI Imane Casablanca Characterization of Uk−monomorphic hermitian

l2-structuresTAMOUSSIT Ali Marrakech A Class of Locally Free Modules Issued from Inte-

ger Valued Polynomials

39


On some nonlinear parabolic equations withunbounded function and measure data

M. Abdellaoui, E. AzroulLAMA, FSDM - USMBA of Fez (Morocco)

[email protected]

AbstractSuppose taht Ω is a bounded domain in RN with Lipschitz boundary ∂Ω, T

is a positive number. In this paper we study the following nonlinear parabolicproblem

b(x, u)t − div(a(t, x,∇u)) = µ in ΩT ≡ Ω× (0, T ),u = 0 in (RN\Ω)× (0, T ), b(x, u)(t = 0) = b(x, u0) in Ω.

(1)

where 1 < p < N such that b(x, u) is a unbounded function and−div(a(t, x,∇u))is the Leray-Lions operator which grows as |∇u|p−1 in ∇u. We assume thatµ ∈Mb(ΩT ) and b(·, u0) ∈ L1(Ω). Due to the lack of regularity of the solution,the distributional formulation is not strong enough to provide uniqueness. Toovercome this difficulty, it is reasonable to work with renormalized solutions,which need less regularity than weak solutions. The notion of renormalizedsolutions was first introduced by DiPerna and Lions for the study of Boltz-mann equation, it was then adapted to the study of some nonlinear elliptic andparabolic problems (see [1], [2]). In this work, we introduce some preliminaryresults about the parabolic p-capacity and measures, we obtain the a prioriestimates and we establish the existence result using auxillary functions andcut-off functions (see [3]).

Keywords : Measures and Capacity, Parabolic equations, Renormalized solutionsAMS subject classifications :35R06(primary), and 35A35(secondary)

References[1] F. Petitta, A. C. Ponce, A. Porretta, Diffuse measures and nonlinear parabolic

equations, Journal of Evolution Equations, 11 (2011), no. 4, 861–905.

[2] A. Marah, H. Redwane, Nonlinear parabolic equations with diffuse measure data,Journal of Nonlinear Evolution Equations and Applications Equations, Volume2017, Number 3, pp. 27–48.

[3] F. Petitta, Renormalized solutions of nonlinear parabolic equations with generalmeasure data, Annali di Matematica (2008) 187-563.

40


Optical solitary wave solutions to the (2 + 1)−dimensionalCalogero-Bogoyavlenskii- Schiff equation with fractional

conformable derivativeKamal AIT TOUCHENT, Zakia HAMMOUCH, Toufik MEKKAOUI

Université Moulay IsmailFst Errachidia

[email protected]

Abstract

In this article, we construct a new explicit solutions of nonlinear frac-tional (2 + 1)− dimensional Calogero−Bogoyavlenskii− Schiff equation withconformable fractional derivative by using a new technique, called the tan(φ(ξ)

2 )-expansion method. The obtained solutions are written in terms of exponentialfunctions, hyperbolic functions and trigonometric functions.

Keywords :tan(φ(ξ)2 )-expansion method, fractional conformable derivative.

AMS subject classifications : 35Q40, 44A10, 44A15, 44A35

References[1] Qi J, Zhang F, Yuan W, et al. Some new traveling wave exact solutions of the

(2 + 1)-dimensional Boiti–Leon–Pempinelli equations. Sci. World J. 2014;2010:9pages.

[2] K.Ait touchent and F.M. Belgacem, Nonlinear Fractional Partial DifferentialEquations Systems Solutions Through a Hybrid Homotopy Pertubation SumuduTransform Method, Nonlinear Studies, 22(4), 1-10 (2015)

41


ON CENTER-LIKE SUBSETS IN PRIME RINGSOmar AIT ZEMZAMI, Lahcen OUKHTITE

Department of Mathematics, Faculty of Science and TechnologyUniversity S. M. Ben Abdellah Fez, Box 2202, Morocco

[email protected]

Abstract

The aim of this talk is to show that certain subsets, which are defined bycommutativity conditions involving additive maps, coincide with the center inprime rings.

Keywords :Prime ring, derivation, endomorphism, center-like subset.AMS subject classifications :16W20, 16W25, 16U80, 16N60.

References[1] M. Chacron, A commutativity theorem for rings, Proc. Amer. Math. Soc. 59

(1976), no. 2, 211-216.

[2] H. E. Bell, M. N. Daif, Center-like subsets in rings with derivations or epimor-phisms, Bull. Iranian Math. Soc .42 (2016), no. 4, pp. 873-878.

[3] M. Brešar, Jordan Derivation On Semiprime Rings, Proc. Amer. Math. Soc. 104(1988), no. 4, 1003-1006.

[4] I. N. Herstein, A note on derivations II, Canad. Math. Bull. 22 (1979), no. 4,509-511.

42


SOME FACTORIZATION PROPERTIES OF THECOMPOSITE RING

B. Boulayat1, S. El Baghdadi1 and L. Izelgue2

1Sultan Moulay Slimane University2Cadi Ayyad University

[email protected]

Abstract

Let A ⊆ B denote an extension of commutative integral rings with identity,Γ a nonzero torsion-free (additive) grading monoid with Γ ∩ −Γ = 0 andΓ ∗ = Γ \ 0. B[Γ] is the semigroup ring of Γ over B.In this talk, we will discuss some factorization properties of the pullback A +B[Γ∗] = f ∈ B[Γ]|f(0) ∈ A

Keywords : Atomq, Primal elements, Schreier domainsAMS subject classifications : 13A15, 13F05, 13B30, 13C11, 13F20, 13G05

References[1] D.F Anderson, D.N. El Abidine, Factorization in integral domains, J. Pure Appl.

Algebra 135 (1999) 107-127.

[2] R.Matsuda, Note on Schreier Semigroup Rings, Math. J. Okayama University,(1997), 41-44.

[3] T. Dumitrescu, N. Radu and M. Zafrullah, Primes that Become Primal in aPullback. Trends in Commutative Rings research, Ayman Badawi, Nova Sciences(2003) 31-42.

43


Bézout-like properties in trivial ring extensionsHaitham EL ALAOUI and Hakima MOUANIS

Sidi Mohamed Ben Abdellah University Fez, [email protected]

Abstract

In this paper, we introduce a generalization of the well-known notion of aP -Bézout rings and a 2-Bézout rings, which we call a P -2-Bézout rings. Weestablish the transfer of this notion to trivial ring extensions, to homomorphicimages and in direct products. We conclude with a brief discussion of the scopeand limits of our results.

Keywords :P-Bézout rings, 2-Bézout rings, P-2-Bézout rings, homomorphic image,direct products, trivial rings extensions.AMS subject classifications :13A15, 13B10, 13F05, 13D02

References[1] S. Glas, Commutative Coherent rings, Lecture Notes in Mathematics, 1371,

Springer-Verlag, Berlin, 1989.

[2] H.Huckaba, Commutative rings with zero divisors, Marcel Dekker, New York,1988.

[3] J. J. Rotman, An Introduction to Homological Algebra, Academic Press, NewYork, 1979.

44


Idempotents and Moment problem of discrete measureH. El Azhar, A. Harrat

Université de Mohammed V de [email protected]

Abstract

We investigate the full moment problem for discrete measures using Vasilescu’sidempotent approach [1] based on multiplicative elements with respect to as-sociated square positive Riesz functional.Using this approach we give a new characterization of the density of polyno-mials in the space of square integrable functions.

Keywords :Idempotents, Λ-multiplicative, Discrete measures, Moment problem.AMS subject classifications :44A60(primary), and 47A57, 46C99(secondary)

References[1] F.-H. Vasilescu, An idempotent approach to truncated moment problems, Integral

Equations Operator Theory, 79,(2014), 301-335.

[2] H. El Azhar, A. Harrat , Idempotents and Moment problem of discrete measure,Preprint.

45


Feedback stabilization and polynomial decay estimate for timedelay semilinear systems

A. El Houch, A. Tsouli, Y. Benslimane, A. AttiouiUniversité Hassan II [email protected]

Abstract

In this work, we are concerned with the question of weak and strong stabi-lization for distributed semilinear systems with time delay using a continuousfeedback controls. In the case of strong stabilization, a polynomial decay rate ofthe stabilized state is estimated. Necessary and sufficient conditions for stabi-lization problems are given as well. Some illustrating applications to hyperbolicand parabolic like equations are considered.

Keywords : Time delay, Feedback stabilization, Polynomial decay estimate, Semi-linear systemsAMS subject classifications : 47N70, 58E25, 70G60 and 76F20

References[1] D.W.C. Ho, G. Lu, Y. Zheng, Global stabilization for bilinear systems with time

delay, IEE Proc., Control Theory Appl. 149, (2002), pp. 89–94.

[2] J. Ball and M. Slemrod. Feedback Stabilization of Distributed Semilinear ControlSystems. Applied Mathematics and Optimization, 5, (1979), pp. 169-179.

[3] J. Wu, Theory and Applications of Partial Functional Differential Equations,Springer, Berlin, (1996).

[4] M. Ouzahra, Strong stabilization with decay estimate of semilinear systems,Systems Control Lett., 57, (2008), pp. 813-815.

46


Stable tameness of coordinates in two variablesM’hammed El Kahoui, Essamaoui Najoua, Mustapha Ouali

Université Cadi Ayyad, [email protected]

Abstract

This is a joint work with my PhD project supervisors M’hammed El Kahouiand Mustapha Ouali.

Throughout, all considered rings are commutative with unity. Given a ringR we use R[n] to denote the polynomial ring in n variables over R. The groupof R-automorphisms of the R-algebra R[n] is denoted by GAn(R). An automor-phism σ ∈ GAn(R) is said to be tame if it belongs to the subgroup TAn(R),generated by the general linear group GLn(R) and EAn(R) the subgroup gen-erated by the elementary automorphisms. It is said to be stably tame if itbecomes tame after the addition of some variables. The famous Jung-van derKulk Theorem [3] asserts that all two-dimensional polynomial automorphismsover a field are tame. Recently, Berson, van den Essen and Wright [2] provedthat over any regular ring R the R-automorphims of R[2] are stably tame.

A weak version of the stable tameness Conjecture asserts that every coordi-nate of R[n] is stably tame, i.e., a component of an automorphism in TAn+m(R)for some m ≥ 0. In [1] Berson proved that, over a noetherian one-dimensionaldomain R containing the rationals, every coordinate of R[2] is stably tame. Inthis talk, we will show that over any ring R every stable coordinate of R[2] isstably tame .

Keywords : Coordinate, Stable coordinate, Stable tamenessAMS subject classifications : 14R10, 14R25 (primary), and 13B25 (secondary)

References[1] J. Berson. Two-dimensional stable tameness over Noetherian domains of dimen-

sion one. J. Pure Appl. Algebra, 178(2):115-129, 2003.

[2] J. Berson, A. van den Essen, and D. Wright. Stable tameness of twodimensionalpolynomial automorphisms over a regular ring. Adv. Math., 230(4-6):2176–2197,2012.

[3] W. van der Kulk. On polynomial rings in two variables. Nieuw Arch. Wiskunde(3), 1:33–41, 1953.

47


The geometry of generalized Cheeger-Gromoll metrics on thetotal space of transitive Euclidean Lie algebroids

Mohamed Boucetta, Hasna EssoufiUniversité Cadi-Ayyad

Faculté des sciences et techniquesBP 549 Marrakech [email protected]

[email protected]

Abstract

Natural metrics (Sasaki metric, Cheeger-Gromoll metric, Kaluza-Klein met-rics etc.. ) on the tangent bundle of a Riemannian manifold is a central topicin Riemannian geometry. Generalized Cheeger-Gromoll metrics is a family ofnatural metrics hp,q depending on two parameters with p ∈ R and q ≥ 0. Thisfamily has been introduced recently and possesses interesting geometric prop-erties. A transitive Euclidean Lie algebroid is a transitive Lie algebroid withan Euclidean product on its total space. In this paper, we show that naturalmetrics can be built in a natural way on the total space of transitive EuclideanLie algebroids. Then we study the properties of generalized Cheeger-Gromollmetrics on this new context. We show a rigidity result of this metrics whichgeneralizes so far all rigidity results known in the case of the tangent bundle.Atiyah Lie algebroids constitute an important class of transitive Lie algebroidsand we will show that natural metrics on the total space of Atiyah EuclideanLie algebroids have interesting properties. In particular, it permits us to builda large class of Riemannian metrics with positive scalar curvature.

Keywords :Lie algebroids, Natural metrics, Scalar curvature.AMS subject classifications :53E20, 53C07, 53A55, 53C24, 53C25

References[1] M. T. K. Abassi and S. Maati, On Riemannian g-natural metrics of the form

ags+bgh+cgv on the tangent bundle of a Riemannian manifold (M, g), Mediterr.J. Math. 2 (2005), 19-43.

[2] Michèle Benyounes, Eric Loubeau and Chris M. Wood, The geometry of gener-alized Cheeger-Gromoll metrics, Tokyo J. of Math. Vol. 32, No. 2, 2009.

[3] Mohamed Boucetta, Riemannian Geometry of Lie algebroids, Journal of theEgyptian Mathematical Society, Volume 19, 1, pp. 57-70 (2011).

48


Chatterjea fixed point theorem in a space with three metricsM.A.Farid, K.Chaira, El M.Marhrani, M.AamriUniversity Hassan II Casablanca Morocco

Faculty of Science Ben M’SikDepartment of Mathematics and Computer ScienceLaboratory of Algebra, Analysis and Applications

[email protected]

Abstract

In 1922, Banach proved the following famous fixed point theorem [1]. Laterin1972 Chatterjea had established a fixed point theorem for a C-contraction[2]. Recently, El-Miloudi Marhrani and Karim Chaira proved a generalizationof the Banach contraction fixed point theorem in a space with two metrics[3], The aim of this work is to show the fixed point theorem of Chatterjeacontraction in a space with three metrics.

Keywords :Fixed point, Complete metric space, Chatterjea contraction.AMS subject classifications :47H10 and 54H25.

References[1] S. Banach,(1922), Sur les operations dans les ensembles abstraits et leur appli-

cation aux equations integrales ; Fundamenta Mathematicae, vol.3, pp. 133-181.

[2] SK, Chatterjea(1972),Fixed point theorems, C. R. Acad. Bulgare Sci. vol.25,pp.727-730.

[3] EL Marhrani, K.Chaira(2015), Fixed point theorems in a space with two metrics; Adv. Fixed Point Theory, vol.5, no. 1,pp. 1-12..

49


On the generators of the 2-class group of some pure quarticnumber fieldMbarek HAYNOU

Moulay Ismail university.Faculty of Sciences and Technology

P.O. Box 509-Boutalamine, 52 000 Errachidia [email protected]

Joint work with: Mohammed Taous

Abstract

Let K = (Q( 4√pd2) be a pure quartic number field with quadratic subfield

k = Q(√p), such that p is a prime number and d is a positive square-freeinteger. In this work, we compute the 2-rank and 4-rank of the 2-class groupof some pure quartic number field K and determine its generators whenever itis the type (2, 2).

Keywords : 2-class groups, pure quartic field, generators.AMS subject classifications :11R29, 11R16.

References[1] A. Azizi, Sur la capitulation des 2-classes d’idéaux de Q(

√2pq, i) où p ≡ −q ≡

1 mod 4, Acta Arithmetica 94 (2000), p. 383-399.

[2] G. Gras, Class field theory, from theory to practice, Springer Verlag 2003.

[3] J . A. Hymo and C. J. Parry, On relative integral bases for pure quarticfields, Indian J. Pure Appl. Math., 23, 1992, 359-376.

[4] C. J. Parry, A genus theory for quartic fields, J. Reine Angew. Math. 314(1980), 40-71.

[5] C. J. Parry, Pure quartic number fields whose class numbers are even, J. ReineAngew. Math. 264 (1975), 102-112.

[6] M. Taous, Capitulation des 2-classes d’idéaux de certains corps Q(√

d, i) de type(2, 4), thèse, Université. Mohammed Premier Faculté des Sciences , Oujda, 2008.

[7] Q. Yue The generalized Rédei-matrix, Math. Z. 261 (2009), 23-37.

50


Classification of some special generalized derivationsMy Abdallah IdrissiWith Lahcen Oukhtite

Department of Mathematics,Faculty of Science and Technology of Fez, Box 2202.

University S. M. Ben Abdellah Fez, [email protected]

Abstract

The purpose of the present talk is to classify generalized derivations satis-fying more specific algebraic identities in a prime ring with involution of thesecond kind. Some well-known results characterizing commutativity of primerings by derivations have been generalized by using generalized derivation.

Keywords : Prime ring, involution, commutativity, derivation, generalized deriva-tion.AMS subject classifications :16N60; 16U80; 16W10; 16W25.

References[1] M. Ashraf, N. Rehman, S. Ali and M. R. Mozumder, On semiprime rings with

generalized derivations, Bol. Soc. Paran. Mat. 28 (2010), no. 2, 25-32.

[2] B. Nejjar, A. Kacha, A. Mamouni and L. Oukhtite, Certain commutativity cri-teria for rings with involution involving generalized derivations, Comm. Alg. 45(2017), no. 2, 698-708.

[3] M. R. Khan and M. M. Hasnain, On semiprime rings with generalized deriva-tions, Kyungpook Math. J. 53 (2013), 565-571.

[4] L. Oukhtite and A. Mamouni, Generalized derivations centralizing on Jordanideals of rings with involution, Turkish J. Math. 38 (2014), no. 2, 225-232.

[5] L. Oukhtite, Posner’s Second Theorem for Jordan ideals in rings with involution,Expo. Math. 29 (2011), no. 4, 415-419.

[6] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957),1093-1100.

51


Sobolev-Dirichlet problem for quasilinear elliptic equations ingeneralized Orlicz spaces

Allami BENYAICHE and Ismail KHLIFI.Ibn Tofail University,

Department of Mathematics, B.P: 133 , [email protected]

[email protected]

Abstract

The aim of this work is to solve, under natural assumptions, the obstacleproblem and the Sobolev-Dirichlet problem for the quasi linear elliptic equa-tions in the generalized Orlicz-Sobolev space setting. In particular, we obtainsuch results for the (p(x),q(x))-Laplacian, G-Laplacian, and the double phasesituation.

Keywords : Musielek-Orlicz space, Obstacle problem, Sobolev-Dirichlet problem,G(.)-laplacian.AMS subject classifications :35B65, 35J60.

52


An alternating direction method of multipliers for the totalvariation regularization

A. H. Bentbib1, A. Bouhamidi2 and K. Kreit1

1Université Cadi Ayyad, Marrakesh, Morocco2Université du Littoral, L.M.P.A, Calais, France

[email protected]@lmpa.univ-littoral.fr

[email protected]

Abstract

In this paper, we consider the problem of image restoration by consideringthe total variation regularization. We use alternating direction method of mul-tipliers to split our problem to two interactive problems. The novelty of ourpaper is the use of Krylov subspace methods for solving a generalized Sylvestermatrix equation . The convergence of this method is proved. Some numericalexamples are given to illustrate the effectiveness of the proposed method.

Keywords : Ill-posed problem, regularization, total variation, alternating directionmethod of multipliers, conditional gradient method, image restoration, Sylvester ma-trix equation, convex programming.AMS subject classifications :65F10, 65R30.

53


Evaluation de certaines sommes courtesOmar Labihi

Université Cadi Ayyad, [email protected]

Abstract

L’objet de l’exposé est de présenter un résultat général obtenu en collabora-tion avec A. Raouj, concernant l’évaluation asymptotique des sommes courtesde le forme :

∑x≤n<x+y

an où (an) est une suite de nombres réels ou complexes.

Des applications seront données pour le comportement asymptotique des moyennesde certaines fonctions arithmétiques usuelles sur des classes d’entiers.

Keywords :Théorie analytique des nombres, Méthode de Selberg-Delange, Régionsans zéros de la fonction zêta de Riemann.AMS subject classifications :11M41, 11M26, 11N37.

References[1] Z. Cui, J. Wu, The Selberg-Delange Method In Short Intervals with an applica-

tion, Acta Arithmetica, 2014, 163 (3), pp.247-260

[2] A. Ivić, The Riemann Zeta-function , Wiley Interscience publication, 1985.

[3] Karatsuba, A.A. and Voronin, S.M., The Riemann Zeta-Function, De GruyterExpositions in Mathematics, 1992.

[4] A.A. Sedunova, On the mean values of some multiplicative functions on the shortinterval, 2013, arXiv:1302.0471v1.

[5] G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres,Belin, 2008.

[6] E.C. Titchmarsh, The theory of the Riemann Zeta-Function, Oxford UniversityPress, Second Edition, 1986.

54


BSDEs driven by normal martingale under generalassumptions

Mohamed Marzougue, Mohamed El OtmaniUniversité Ibn Zohr (Maroc)

[email protected]

Abstract

We consider a backward stochastic differential equations (BSDEs in short)driven by a class of normal martingales (Mt)t≤T which verifying the structuralequation

d[M,M ]t = dt+ (ϕt + ψtMt−)dMt

where (ϕt)t≤T and (ψt)t≤T are two non negative progressively measurable pro-cesses. We prove the existence and uniqueness results of the solution under thestochastic Lipschitz condition on the coefficient.

Keywords :BSDEs, normal martingale, stochastic LipschitzAMS subject classifications :60H20, 60H30 (primary), and 91B70 (secondary)

References[1] S. Attal, A. C. R. Belton (2007) The chaotic-representation property for a class of

normal martingales, Probability Theory and Related Fields, Vol. 239, 543–562.

[2] C. Bender, M. Kohlmann (2000), Backward stochastic differential Equations withStochastic Lipschitz Condition

[3] M. Émery (1990), On the Azéma martingales, In: Séminaire de ProbabilitésXXIII, Vol. 1372 of Lecture Notes in Mathematics, 66–87.

55


Complete Interpolating sequences for small Fock SpacesYoussef OMARI

Université Mohammed 5 de Rabat (Maroc)Under the direction of Pr. O. El-Fallah

[email protected]

Abstract

Let α > 0 and ϕ(z) = α(log+ |z|

)2, z ∈ C. The associated Fock spaces Fpϕ

are the following

Fpϕ :=f ∈ Hol(C) : ‖f‖pp,ϕ :=

∫C

∣∣∣f(z)e−ϕ(z)∣∣∣p dm(z) <∞

, 1 ≤ p <∞,

where dm stands for the area Lebesgue measure in the complex plane C. Wegive a complete characterization of complete interpolating sequences for theFock spaces Fpϕ, 1 ≤ p <∞. Our results are analogue to the classical Kadets-Ingham’s 1/4−Theorem on perturbation of Riesz bases of complex exponen-tials.

Keywords :Fock spaces, Riesz bases, Sampling, Interpolation, Complete interpolat-ing sequences, uniquness set.AMS subject classifications :30H20 (primary), and 30E05, 32A15 (secondary)

References[1] M. Kadets. Exact value of the Paley-Wiener constant. Doklady Akademii Nauk

SSSR, 155(6):1253, 1964.

[2] A. Baranov, A. Dumont, A. Hartmann, and K. Kellay. Sampling, interpola-tion and Riesz bases in small Fock spaces. Journal de Mathématiques Pures etAppliquées, 103(6):1358–1389, 2015.

[3] Y. I. Lyubarskii and K. Seip. Complete interpolating sequences for Paley-Wienerspaces and Muckenhoupt’s (A_p) condition. Rev. Mat. Iberoamericana, page 2,1997.

56


Basic Properties of ∗-K-Frames in Hilbert A-modulesMohamed ROSSAFI, Samir KABBAJDépartement de Mathématiques

Faculté des Sciences de Ké[email protected]

Abstract

Frames generalize orthonormal bases and allow representation of all theelements of the space. Some results of ∗-K-frames are discussed. After givingsome basic definitions about these ∗-K-frames, we give some results about thecharacteristics of ∗-K-frames. We also study ∗-K-frames in two Hilbert A-modules with different C*-algebras.

Keywords :Frame, ∗-K-Frame, Hilbert A-modules.AMS subject classifications : 42C15 (primary), and 46L05 (secondary).

References[1] M. Frank, D. R. Larson, Frames in Hilbert C*-modules and C*-algebras, J. Oper.

Theory 48 (2002), 273-314.

[2] A. Najati, M. M. Saem, P. Gavruta, Frames and operators in Hilbert C*-modules, Oam, vol. 10, no 1 (2016), 73-81.

[3] M. Rossafi and S. Kabbaj, ∗-g-frames in tensor products of Hilbert C*-modules,Ann. Univ. Paedagog. Crac. Stud. Math. 17 (2018), 15-24.

[4] M. Rossafi and S. Kabbaj, ∗-K-g-frames in Hilbert C*-modules, Journal of Linearand Topological Algebra, Vol. 07, No. 01, 2018, 63- 71.

57


Characterization of Uk−monomorphic hermitian l2-structuresImane Souktani, Abderrahim Bousairi, Kawtar Attas

Université Hassan II,Faculté des Sciences Ain Chock Casablanca (Maroc)

[email protected]

AbstractA complex l2-structure g is a function from E2(V ) to a complex field C

where V is a nonempty set and E2(V ) = (x, y) : x 6= y ∈ E. The adjacencymatrix of a complex l2-structure g with respect to an ordering v1, v2, . . . , vn ofits vertices is the n × n complex matrix Mg = [mij ]1≤i,j≤n in which mij = 0if i = j and mij = g(vi, vj) otherwise. A complex l2-structure g on a set V ishermitian if Mg is hermitian or equivalently g(u, v) = g(v, u) for u 6= v ∈ V .A hermitian l2-structure g is k−monomorphic if all its substructures with kvertices are isomorphic. In other words for subset X and Y ,|X| = |Y | = kthere is a permutation matrix P such that P−1Mg[X]P = Mg[Y ]. In order togeneralize the notion of monomorphy, let k be a positive integer and denoteby GLk(C) the general linear group of degree k, that is of all k × k invertiblematrices with entries in the field C under matrix multiplication. Let G be asubgroup of GLk(C) containing all k× k permutation matrices. A complex l2-structures g is called G-monomorphic if for all X,Y, |X| = |Y | = k, there existsQ ∈ G such that Q−1Mg[X]Q = Mg[Y ]. In this work, we study the class ofwhere is Uk-monomorphic hermitian l2-structures, where Uk the unitary groupof degree k.

Keywords :Hermitian l2-structure, tournament, Hadamard tournament, General-ized total orderAMS subject classifications :05C20 and 06A05.

References[1] J. W. Moon,"Topics on Tournaments," Holt, Rhinehart and Winston,New

York,1968.

[2] A. Ehrenfeucht, T. Harju, G. Rozenberg, The Theory of 2-Structures, a Frame-work for Decomposition and Transformation of Graphs,World Scientific, Singa-pore, 1999.

[3] H. Huang, L. Oeding. Symmetrization of principal minors and cycle-sums. Linearand Multilinear Algebra 65 (2017), 1194-1219.

[4] Jianxi Liu, Xueliang Li, Hermitian-adjacency matrices and Hermitian energiesof mixed graphs, Linear Algebra Appl. 466 (2015) 182–207.

[5] M. Pouzet, Application d’une propriété combinatoire des parties d’un ensembleaux groupes et aux relations. Math. Z. 150 (1976), 117-134.

[6] .B. Reid, E. Brown, Doubly regular tournaments are equivalent to skewHadamard matrices, J. Combin. Theory Ser. A 12 (1972)332–338

58


A Class of Locally Free Modules Issued from Integer–ValuedPolynomials

L. Izelgue1, A. Mimouni2 and A. Tamoussit1

1Cadi Ayyad University2King Fahd University of Petroleum & Minerals (KSA)

[email protected]

Abstract

Let D be an integral domain with quotient field K. The polynomials withcoefficient in K that take values in D form a commutative D–algebra, denotedby Int(D). In [2], the authors asked whether Int(D) is always (locally) free, orflat, as a D–module. In this talk we give positive answers for certain domains.By the way we recover some well–known results on this problem. Particularly,we show that Int(D) is a locally free D–module for any almost Krull domainD.

Keywords : Integer–valued polynomials, Locally free modules, Locally essentialdomainsAMS subject classifications : Primary: 13F20, 13C11; Secondary: 13B25, 13F05,13B30

References[1] P.-J. Cahen and J.-L. Chabert, Integer-Valued Polynomials, Math. Surveys

Monogr., vol. 48, Amer. Math. Soc., (1997).

[2] P.-J. Cahen, M. Fontana, S. Frisch and S. Glaz, Open Problems in CommutativeRing Theory, Commutative Algebra: Recent Advances in Commutative Rings,Integer–Valued Polynomials, and Polynomial Functions, M. Fontana, S. Frischand S. Glaz (editors), Springer (2014), 293–305.

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LISTE DES PARTICIPANTS


Liste ParticipantsNom Prénom Ville, pays EmailABCHIR Hamid Casablanca [email protected] Mohammed Fes [email protected] Fatima-Ezzahrae Marrakech [email protected] Mbark Fes [email protected] Mostapha Lyon, France [email protected] Samir Limoges, France [email protected] Imane Casablanca [email protected] BENHASSI El Mustapha Marrakech [email protected] LAAMIM Maroua Settat [email protected] TOUCHENT Kamal Errachidia [email protected] ZEMZAMI Omar Fes [email protected] Omar Agadir [email protected] Brahim Pau, France [email protected] Kamal Oujda [email protected] Oualid Rabat [email protected] Brahim Kenitra [email protected] Moulay Limoges, France [email protected] Mahmoud Marrakech [email protected] Kacem Kenitra [email protected] Houda Kenitra [email protected] Mohsine Kenitra [email protected] Abdelhadi Rabat [email protected] Said Lorraine, France [email protected] Riham Kenitra [email protected] Abdelkarim Marrakech [email protected] Ghassane Casablanca [email protected] Abdellatif Fes [email protected] Meriem Casablanca [email protected] Allami Kenitra [email protected] Fatima Saida, Algérie [email protected] Najjla Marrakech [email protected] Mohammed Settat [email protected] Fatima Marrakech [email protected] Yousra Rabat [email protected] Fadoua Kenitra [email protected] Brahim Beni Mellal [email protected] Said Marrakech [email protected] Abed Agadir [email protected] Mohamed Casablanca [email protected] Marouane Rabat [email protected]

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Liste ParticipantsNom Prénom Ville, pays EmailCHAABELASRI Elmiloud AL Hociema [email protected] Ahmed Kenitra [email protected] Tarik El Jadida [email protected] Redouane Casablanca [email protected] Abdessamad Casablanca [email protected] Nadia Marrakech [email protected] ALAOUI Haithame Fes [email protected] ALAOUI TALIBI Mohamed Marrakech [email protected] AZHAR Hamza Rabat [email protected] BHIH Amine Casablanca [email protected] FROM Youssef Marrakech [email protected] HADFI Youssef Khouribga [email protected] HAJIOUI Khalid Kenitra [email protected] HAOUI Youssef Meknes [email protected] HARAMI Mohamed Meknes [email protected] HARTI Rachid Settat [email protected] HOUCH Atmane Casablanca [email protected] JAMALI Mohamed Agadir [email protected] KAHOUI M’hammed Marrakech [email protected] KASSIMI Mohamed Meknes [email protected] KOUFI Amine Casablanca [email protected] MIR Hajar Fes [email protected] MOUMNI Mostafa El Jadida [email protected] WASSOULI Fouzia Casablanca [email protected] Iz-iddine Kenitra [email protected] Youssef Rabat [email protected] Safaa Rabat [email protected] Aiad Kenitra [email protected] Nour-eddine Casablanca [email protected] Meryeme Rabat [email protected] Mhamed Oujda [email protected] Karima Marrakech [email protected] Youssef Rabat [email protected] Mohamed Abdou Settat [email protected] Elhoucien Agadir [email protected] Mohamed Marrakech [email protected] Najoua Marrakech [email protected] Hasna Marrakech [email protected] Ahmed Béni Mellal [email protected] Said Meknes [email protected] Mohamed amine Casablanca [email protected] Allal Rabat [email protected]

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Liste ParticipantsNom Prénom Ville, pays EmailCHAABELASRI Elmiloud AL Hociema [email protected] rachid Fes [email protected] Najib Fes [email protected] ALI Errachidia [email protected] Moulay Ahmed Errachidia [email protected] Moha Errachidia [email protected] Mohamed Marrakech [email protected] Sidi Lafdal Laayoune [email protected] Ayoub Rabat [email protected] Mbarek Errachidia [email protected] Mouna Marrakech [email protected] Najima Rabat [email protected] moulay Abdallah Fes [email protected] Bouchta Kenitra [email protected] Samir Kenitra [email protected] Ali Kenitra [email protected] EL Amine Almeria, Espagne [email protected] Kamal Marrakech [email protected] Mohamed Texas, USA [email protected] Abdelmjid Marrakech [email protected] Ismail Kenitra [email protected] Khalid Nancy, France [email protected] Karim Marrakech [email protected]’HAMRI Raja Rabat [email protected] EL Haj Lorraine, France [email protected] Imane Rabat [email protected] Omar Marrakech [email protected] Aziz Fes [email protected] Fouad Kenitra [email protected] rajae Meknes [email protected] Abdellah Errachidia [email protected] My Ismail Rabat [email protected] LAHCEN Marrakech [email protected] Fouad Agadir [email protected] Ibrahim Rabat [email protected] Mohamed Agadir [email protected] Mostafa Kenitra [email protected] Zakaria Marrakech [email protected] Mohamed Kenitra [email protected] Fouzia Rabat [email protected] Soufiane Marrakech [email protected]

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Liste ParticipantsNom Prénom Ville, pays EmailCHAABELASRI Elmiloud AL Hociema [email protected] Youssef Casablanca [email protected] Karim Oujda [email protected] Ayoub Casablanca [email protected] Youssef Rabat [email protected] Soukaina Casablanca [email protected] El Maati Bordeaux, France [email protected] Hassan Besançon, France [email protected] Lahcen Fes [email protected] Rida Casablanca [email protected] Abdelaziz Marrakech [email protected] Abdellatif Casablanca [email protected] Mohamed Kenitra [email protected] Mohamed Errachidia [email protected] Khalil Kenitra [email protected] Mohamed Durham, UK [email protected] Nazha Noumea, France <[email protected] Jawad Mexico, Mexique [email protected] Imane Casablanca [email protected] Chakir Larache [email protected] ABDELAZIZ Fes [email protected] Ali Marrakech [email protected] Nada Rabat [email protected] Youssef Fes [email protected] M. Mustapha Errachidia [email protected] Ahmed Toulouse, France [email protected] Mohammed Fes [email protected] Lhassane Agadir [email protected]

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Documents

Sciencesconf.org · This implies that such a system is solvable by quadratures if, and only if its coecient matrix is conjugated to a triangular one (via a ... on Lie groups or Riemannian