1. THE PARTIAL r-BELL POLYNOMIALS

Transcription

‫ﺗﻘﺮﻳﺮﻋﺎم ﻟﻤﺸﺮوع اﻟﺒﺤﺚ‬
1
Rapport général du projet PNR
‫وزارة اﻟﺘﻌﻠﻴــﻢ اﻟﻌﺎﻟــﻲ و اﻟﺒﺤــﺚ اﻟﻌﻠﻤــﻲ‬
Ministère de l’Enseignement Supérieur et de la Recherche Scientifique
‫اﻟﻤﺪﻳﺮﻳـﺔ اﻟﻌﺎﻣـﺔ ﻟﻠﺒﺤـﺚ اﻟﻌﻠﻤـﻲ‬ ‫و اﻟﺘﻄﻮﻳــﺮ اﻟﺘﻜﻨﻮﻟﻮﺟــﻲ‬
Direction Générale de la Recherche Scientifique et du Développement
Technologique
‫اﻟﺘﻌﺮﻳﻒ ﺑﺎﻟﻤﺸﺮوع‬-1
Organisme pilote
I-IDentification du projet:
PNR
Sciences fondamentales
ATRST (Ex. ANDRU)
Domiciliation du projet :
Faculté des Mathématiques USTHB
‫ﻋﻨﻮان اﻟﻤﺸﺮوع‬
Intitulé du projet
Théorie de la combinatoire énumérative et application des techniques
stochastiques en finance
Chercheurs impliqués dans le projet
Nom et prénom
‫اﻻﺳﻢ و اﻟﻠﻘﺐ‬
MIHOUBI Miloud
MAHDID Rachida
MAAMRA Mohammed Said
Grade
‫اﻟﺮﺗﺒﺔ‬
MCA
MAB
Doctorant
‫أﻋﻀﺎء اﻟﻤﺸﺮوع و اﻟﻤﺆﺳﺴﺔ اﻟﻤﺴﺘﺨﺪﻣﺔ‬
Etablissement employeur
‫اﻟﻤﺆﺳﺴﺔ اﻟﻤﺴﺘﺨﺪﻣﺔ‬
USTHB
USTHB
Banque Al-Salam
Observation
Déroulement du projet :
Développer des axes de recherche liés à la théorie de la combinatoire énumérative et leur
interprétation combinatoire en probabilité, en processus stochastique et en apprentissage
statistique.
Modélisation mathématiques des modèles de gestion des risques bancaires (risque de
crédit et risque opérationnel)
Développer de nouvelles méthodes de simulation adéquates aux techniques
d’atténuation des risques.
Réaliser des applications au sein de la banque algérienne afin de valider les modèles
conçus précédemment.
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TABLE DE MATIERES
INTRODUCTION
CHAPITRE 1 : PUBLICATIONS INTERNATIONALES
• Polynomials of multipartitional type and inverse relations
• The (exponential) bipartitional polynomials and polynomial
sequences of trinomial type: part I
• The (exponential) bipartitional polynomials and polynomial
sequences of trinomial type: part II
• The (r1 ,…, rp)-Stirling numbers of the second kind
• The inverse of power series and the partial Bell polynomials
• The (exponential) multipartitional polynomials and polynomial
sequences of multinomial type, part I
• The (exponential) multipartitional polynomials and polynomial
sequences of multinomial type, part II
• Generalization of universal partition and bipartition theorems
• The role of binomial type sequences in determination identities
for Bell polynomials
CHAPITRE 2: PREPUBLICATIONS INTERNATIONALES
• The partial r-Bell polynomials
• The (r1 , . . . , rp )-Bell polynomials
CHAPITRE 3: ARTICLES SOUMIS OU EN REVISION
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• Linear recurrences for r-Bell polynomials
• The (r1 , . . . , rp )-Bell polynomials
• Inequalities with several variables
• The s-degenerate r-Lah numbers
• Recursiveness properties for multipartitional polynomials
• The values of the high order Bernoulli polynomials at integers
and the r-Stirling numbers
• The partial r-Bell polynomials
• Some identities for Complete Bell polynomials related to the
divisors of an integer
CHAPITRE 4: CONFERENCES D’AUDIENCE
INTERNATIONALES
• The partial r-Bell polynomials (DIMACOS’12)
• The r-Lah numbers and their restrictions (DIMACOS’12)
• Tthe m-associated r-Stirling numbers of the second kind
(DIMACOS’12)
• Inequalities with several variables (RAMA08)
• The r-Lah numbers and their restrictions (suite) (RAMA08)
• Bell polynomials and random variables (RAMA08)
• The m-associated r-Stirling numbers of the second kind (suite)
(RAMA08)
• The (r1 ,…, rp)-Stirling numbers of the second kind (ISOR’11)
• On the reciprocals and compositional inverses of power series
(ISOR’11)
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• Approche multi-agents pour l’ordonnancement dynamique
d’atelier de production (MOSIM’12)
• The (exponential) multipartitional polynomials: interpretations
and recurssivness (5thSSCaudi science conference)
• The compositional inverses of remarquable power series
(DIMACOS’11)
• The polynomial sequences of multinomial type and their
applications
on
the
multipartitional
polynomials
(DIMACOS’11)
• Polynomials of multipartitionnal type and inverse relations
(JIAA’2011)
• The
multipartitional
polynomials
and
random
vectors
(ICPAM’2012)
CHAPITRE 5 : CONFERENCES D’AUDIENCE
NATIONALES
• Touchard
polynomials
and
partial
Bell
polynomials
(CMA’2012)
•
A generalized recurrence for r-Bell polynomials (CMA’2012)
CHAPITRE 6 : MEMOIRES SOUTENUS
CONCLUSION
REFERENCES
INFORMATION FINANCIERE
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INTRODUCTION
La
combinatoire
énumérative
englobe
plusieurs
aspects
mathématiques, dont les polynômes multivariés de Bell, les polynômes
multivariés de Touchard, les partitions, les multipartitions, etc.
Les
polynômes listés au dessus présentent des outils mathématiques très
intéressants et interviennent en combinatoire, probabilité, processus
alétoires, algèbre, analyse et possèdent aussi des applications dans d’autres
domaines tels que la physique. Les partitions sont aussi bien connues et
interviennent dans différents domaines, alors que les multipartitions, qui
généralisent les partitions ne sont pas très célèbres malgré leur importance,
en premier lieu, mathématiques.
Les méthodes de notation se rangent en deux grandes catégories : les
méthodes qualitatives et les méthodes quantitatives. Les systèmes de
notation des agences de rating et les systèmes experts utilisés dans les
banques reposent sur des méthodes principalement qualitatives. A l’inverse,
les modèles de score sont des outils de mesure des probabilités de défaut qui
reposent sur des méthodes statistiques. Les deux méthodes utilisent à la fois
des informations publiques « hard information », notamment comptables,
disponibles sous forme de nombres et de chiffres, et des informations
privées qualitatives « soft information » disponibles sous forme de
jugements, d’avis ou d’opinions.
Le projet consiste à se focaliser dans deux importants volets. L’un
théorique qui traite des problèmes de combinatoire énumérative et leur
interprétation dans le domaine stochastique et complexité algorithmique
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tandis que le deuxième volet présente un aspect applicatif dans la gestion des
risques bancaires.
Pour le premier volet, les chercheurs adhérent à ce projet essayent
d’élaborer quelques propriétés des polynômes de Bell et des polynômes
multipartitionnels, d’avoir des contributions algébriques et combinatoires
des polynômes de Touchard et d’interpréter des polynômes de Bell dans les
problèmes de classification (déterministe et stochastique).
Pour le deuxième volet, , les chercheurs étudient les modèles de
notation des entreprises pour octroi de crédit (classification), la cartographie
des risques opérationnels et les méthodes de simulation des crises et valeurs
extrêmes. Plus en détails, dans ce projet, on traite :
• Etude sur les polynômes de Bell et sur ses applications.
• Etude sur les polynômes de Touchard et sur ses applications.
• Etude sur les partitions et leurs liens avec la combinatoire, l’analyse,
etc.
• Extension de l’étude sur les partitions en l’étude sur les
multipartitions.
• Application de la combinatoire au profit du domaine des variables et
vecteurs aléatoires
• Analyse discriminante.
• Les modèles logit.
• Support Vector Machine.
• Les réseaux de neurones.
7
En effet, la stratégie suivie par les chercheurs est de développer des
axes de recherche liés à la théorie de la combinatoire énumérative et leur
interprétation combinatoire en probabilité, en processus stochastique et en
apprentissage statistique, modélisation mathématiques des modèles de
gestion des risques bancaires (risque de crédit et risque opérationnel),
Développer de nouvelles méthodes de simulation adéquates aux
techniques d’atténuation des risques, et, réaliser des applications au sein de
la banque algérienne afin de valider les modèles conçus précédemment.
8
CHAPITRE 1 : PUBLICATIONS
INTERNATIONALES
Ce chapitre consiste à lister toutes les publications internationales
réalisées par les membres l’équipe de recherche du projet.
Toutes ces publications rentrent dans les thèmes de recherche objet du
dit projet. Elles traitent exhaustivement la combinatoire énumérative et ses
applications. Ces thèmes englobent :
• Les polynômes de Bell et ses applications à la recherche de la fonction
réciproque d’une fonction ainsi que l’inverse d’une fonction
• Extension de ces polynômes à l’échelle bi-variée ainsi que multivariée en donnant plusieurs propriétés à ces polynômes telles que
leurs interprétations combinatoires et probabilistes
• Extension des nombres célèbres de Stirling de seconde espèce ainsi
que les polynômes de Bell uni-varié en prouvant leur utilité dans la
combinatoire énumérative tout en donnant plusieurs propriétés
combinatoires.
• La partition d’un entier ou d’un vecteur d’entiers
Nous donnons le résumé de chaque publication ainsi que ses
références et ses auteurs.
1. POLYNOMIALS OF MULTIPARTITIONAL TYPE AND INVERSE
RELATIONS
Miloud Mihoubi and Hacène Belbachir
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Discussiones Mathematicae, General Algebra and Applications 31 (2011)
185–200.
Abstract. Chou, Hsu and Shiue gave some applications of fàa di Bruno’s
formula to characterize inverse relations. Our aim is to develop some inverse
relations connected to the multipartitional type polynomials involving to
binomial type sequences.
References used: 4, 5, 11, 19, 22, 44, 46, 47, 58, 59.
2. THE (EXPONENTIAL) BIPARTITIONAL POLYNOMIALS AND
POLYNOMIAL SEQUENCES OF TRINOMIAL TYPE: PART I
#A18 INTEGERS 11 (2011)
http://www.integers-ejcnt.org/vol11.html
Abstract. The aim of this paper is to investigate and present the general
properties of the (exponential) bipartitional polynomials. After establishing
relations between bipartitional polynomials and polynomial sequences of
binomial and trinomial type, a number of identities are deduced.
References used: 17, 22, 44, 45, 47, 58, 59, 70.
3. THE (EXPONENTIAL) BIPARTITIONAL POLYNOMIALS AND
POLYNOMIAL SEQUENCES OF TRINOMIAL TYPE: PART II
Hacène Belbachir and Miloud Mihoubi
#A29 INTEGERS 11 (2011)
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Abstract. In a previous paper we investigated the (exponential) bipartitional
polynomials involving polynomial sequences of trinomial type. Our aim is to
give properties of bipartitional polynomials related to the derivatives of
polynomial sequences of trinomial type. Furthermore, we deduce identities
involving Bell polynomials.
References used: 11, 17, 22, 44, 45, 47, 49, 58, 59.
4. THE (r1 ,…, rp)-STIRLING NUMBERS OF THE SECOND KIND
Miloud Mihoubi and Mohammed Said Maamra
#A35 INTEGERS 12 (2012)
Abstract. Let R1, . . . ,Rp be subsets of the set [n] = {1, . . . , n} with |Ri| = ri
and Ri∩Rj =Ø for all i, j = 1, . . . , p, i "= j. The (r1, . . . , rp)-Stirling number
of the second kind, p # 1, introduced in this paper counts the number of
partitions of the set [n] into k classes (or blocks) such that the elements in
each Ri, i = 1, . . . , p, are in different classes (or blocks). Combinatorial and
algebraic properties of these numbers are explored.
References used: 12, 13, 14, 17, 20, 22, 23, 30, 32, 40, 41, 50, 58.
5. THE INVERSE OF POWER SERIES AND THE PARTIAL BELL
POLYNOMIALS
Miloud Mihoubi and Rachida Mahdid
Journal of Integer Sequences, Vol. 15 (2012).
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https://cs.uwaterloo.ca/journals/JIS/vol15.html
Abstract. Using the Bell polynomials, in this paper we give the explicit
compositional inverses and/or the reciprocals of some power series. We
illustrate the obtained results by some examples on Stirling numbers.
References used: 2, 3, 4, 11, 14, 21, 22, 26, 27, 28,39, 44, 45, 46, 47, 50, 58,
59, 60, 68.
6. THE (EXPONENTIAL) MULTIPARTITIONAL POLYNOMIALS AND
POLYNOMIAL SEQUENCES OF MULTINOMIAL TYPE, PART I
Arab journal of mathematical sciences (2013, article in press).
http://www.sciencedirect.com/science/article/pii/S1319516613000327
Abstract. We establish some formulas relating multipartitional polynomials
to multinomial polynomials. They appear, respectively, as a natural
extension of Bell polynomials and of polynomials of binomial type. Our
results are illustrated by some comprehensive examples.
References used: 10, 17, 22, 44, 47, 49, 58, 59.
7. THE (EXPONENTIAL) MULTIPARTITIONAL POLYNOMIALS AND
POLYNOMIAL SEQUENCES OF MULTINOMIAL TYPE, PART II
Arab journal of mathematical sciences (2013, article in press).
http://www.sciencedirect.com/science/article/pii/S1319516613000273
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Abstract. We establish recursiveness properties for multipartitional
polynomials and their connection with the derivatives of polynomials of
multinomial type. Various comprehensive examples are illustrated.
References used: 10, 11, 44, 47, 49, 53, 59.
8. GENERALIZATION OF UNIVERSAL PARTITION AND BIPARTITION
THEOREMS
#A59 INTEGERS 13 (2013).
Abstract. Let A = (ai,j) , i = 1, 2, . . . , j = 0, 1, 2, . . . , be an infinite matrix
with elements ai,j = 0 or 1; p (n, k;A) the number of partitions of n into k
parts whose number yi of parts which are equal to i belongs to the set Yi = {j
: ai,j = 1} , i = 1, 2, . . . . In this paper, we present a generalization of the
universal theorem on partitions. We show that this generalization remains
true when ai,j are indeterminate. We also take into account the bi-partite and
multi-partite situations.
References used: 11, 17, 22, 44.
9. THE ROLE OF BINOMIAL TYPE SEQUENCES IN DETERMINATION
IDENTITIES FOR BELL POLYNOMIALS
Miloud Mihoubi
Ars Combinatoria 111 (July 2013) pp. 323-337.
http://www.combinatorialmath.ca/arscombinatoria/vol111.html
13
Abstract. Our paper deals about identities involving Bell polynomials.
Some identities on Bell polynomials derived using generating function and
successive derivatives of binomial type sequences. We give some relations
between Bell polynomials and binomial type sequences in first part, and, we
generalize the previous results obtained by Mihoubi in second part.
References used: 1, 11, 22, 44, 58, 59, 65.
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CHAPITRE 2: PRE-PUBLICATIONS
INTERNATIONALES
Dans ce chapitre nous listons les pré-publications qui sont publiées en
ARXIV.
Ces pré-publications englobent:
• Un travail nouveau sur une extension du polynôme partiel de Bell et
ses applications à la combinatoire
• Un deuxième travail qui consiste à completer un travail déjà publié et
qui rentre dans le cadre des nombres de Stirling
Miloud Mihoubi and Mourad Rahmani
Preprint in ARXIV
http://arxiv.org/abs/1308.0863v1
Abstract. In this paper, we show that the r-Stirling numbers of both kinds,
the r-Whitney numbers of both kinds, the r-Lah numbers and the r-WhitneyLah numbers form articular cases of family of polynomials forming a
generalization of the partial Bell polynomials. We deduce the generating
functions of several restrictions of these numbers. In addition, a new
combinatorial interpretations ispresented for the r-Whitney numbers and the
r-Whitney-Lah numbers.
15
References used: 11, 14, 18, 20, 22, 43, 44, 57, 67.
2. THE (r1 , . . . , rp )-BELL POLYNOMIALS
Mohammed Said Maamra and Miloud Mihoubi
Preprint in ARXIV
http://arxiv.org/abs/1212.3191v1
Abstract. In a previous paper, Mihoubi et al. introduced the (r1, . . . , rp)Stirling numbers and the (r1, . . . , rp)-Bell polynomials and gave some of
their combinatorial and algebraic properties. These numbers and
polynomials generalize, respectively, the r-Stirling numbers of the second
kind introduced by Broder and the r-Bell polynomials introduced by Mez˝o.
In this paper, we prove that the (r1, . . . , rp)-Stirling numbers of the second
kind are log-concave. We also give generating functions and generalized
recurrences related to the (r1, . . . , rp)-Bell polynomials.
References used: 9, 14, 15, 16, 29, 31, 40, 41, 42, 51, 52, 61, 71, 72, 73.
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CHAPITRE 3 : ARTICLES SOUMIS OU EN
REVISION
Ces articles sont en fin de révision par des referees pour une publication
internationale.
Ils englobent :
• De nouvelles propretés sur les récurrences linéaires pour les
polynômes r-Bell. Ce qui a donné de nouvelles identités sur les
nombres r-Stirling.
• De nouvelles propriétés sur les polynômes (r1,..,rp)-Bell telle que la
log concavité.
• De nouvelles inégalités relatives à certains polynômes à plusieurs
variables qui généralisent plusieurs résultats connus
• Un travail sur des restrictions des nombres de Lah à travers lesquels
nous obtenons de nouvelles propriétés à ces nombres
•
Applications des polynômes multi-partitionnels dans la combinatoire
et aux vecteurs aléatoires
• Donner de nouvelles expressions aux polynômes de Bernoulli des
deux espèces à valeurs entières par lesquels nous déduisons plusieurs
identités et congruences liant les nombres r-Striling et les coefficients
binomiaux
1. LINEAR RECURRENCES FOR r-BELL POLYNOMIALS
Submitted.
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Abstract. Letting Bn;r the n-th r-Bell polynomial, it is well known that
Bn(x) admits specific integer coordinates in the two following bases {xi}
and {xBi(x)}
according respec-tively to Stirling numbers and binomial
coeficients. Our aim is to prove that the sequences Bn+m;r(x); Bn;r+s(x) and
Bn;r(x) admit a binomial recurrence coeficient in different bases of the Qvectorial space formed by polynomials of Q[X].
References used: 6, 7, 8, 9, 14, 15, 16, 22, 29, 40, 41, 42, 50, 51, 61, 63, 69.
2. THE (r1 , . . . , rp )-BELL POLYNOMIALS
Article in revision. (INTEGERS)
Abstract. In a previous paper, Mihoubi et al. introduced the (r1, . . . , rp)Stirling numbers and the (r1, . . . , rp)-Bell polynomials and gave some of
their combinatorial and algebraic properties. These numbers and
polynomials generalize, respectively, the r-Stirling numbers of the second
kind introduced by Broder and the r-Bell polynomials introduced by Mez˝o.
In this paper, we prove that the (r1, . . . , rp)-Stirling numbers of the second
kind are log-concave. We also give generating functions and generalized
recurrences related to the (r1, . . . , rp)-Bell polynomials.
References used: 9, 14, 15, 16, 29, 31, 40, 41, 42, 51, 52, 61, 71, 72, 73.
3. INEQUALITIES WITH SEVERAL VARIABLES
Submitted.
18
Abstract. Let m, n be natural numbers with m ≥3, n ≥ 2; and H : Rm →R; be
a symmetric function differentiable respect to each variable xi at xi = 0; i = 1,
… , m. Our aim is to establish, for all real numbers a1,a2,…,am that the sign
of a polynomial Pn (a1,…, am) depends only on the choice of the function H.
References used: 44, 47, 74, 75, 76, 77, 78, 79, 80.
4. THE s-DEGENERATE R-LAH NUMBERS
Miloud Mihoubi and Lilia Reggane
Submitted
Abstract. Recently, Belbachir and Belkhir give some recurrence relations
for the r-Lah numbers. In this paper, we give other properties for the r-Lah
numbers and we introduce and study a restricted class of these numbers.
5. RECURSIVENESS PROPERTIES FOR MULTIPARTITIONAL
POLYNOMIALS
Submitted.
Abstract. We establish some formulas relating multipartitional polynomials
to multinomial polynomials. They appear, respectively, as a natural
extension of Bell polynomials and of polynomials of binomial type. Our
results are illustrated by some comprehensive examples.
References used: 10, 11, 17, 22, 44, 47, 49, 53, 58, 59, 83, 84, 85.
19
6. THE VALUES OF THE HIGH ORDER BERNOULLI
POLYNOMIALS AT
INTEGERS AND THE R-STIRLING NUMBERS
Miloud Mihoubi and Meriem Tiachachat
Submitted.
Abstract. In this paper, we exploit the r-Stirling numbers of both kinds in
order to give explicit formulae for the values of the high order Bernoulli
numbers and polynomials of both kinds at integers. We give also some
identities and congruences linked the r-Stirling numbers and binomial
coefficients.
References used: 14, 22, 40, 52, 59, 86, 87, 88, 89, 90, 91, 92, 93, 94, 9(, 96,
97, 98, 99.
Miloud Mihoubi and Mourad Rahmani
Submitted.
Abstract. In this paper, we show that the r-Stirling numbers of both kinds,
the r-Whitney numbers of both kinds, the r-Lah numbers and the r-WhitneyLah numbers form articular cases of family of polynomials forming a
generalization of the partial Bell polynomials. We deduce the generating
functions of several restrictions of these numbers. In addition, a new
combinatorial interpretations ispresented for the r-Whitney numbers and the
r-Whitney-Lah numbers.
References used: 11, 14, 18, 20, 22, 43, 44, 57, 67.
20
8. SOME IDENTITIES FOR COMPLETE BELL POLYNOMIALS RELATED
TO THE DIVISORS OF AN INTEGER
Submitted
Abstract. In this paper, by using an (universal) Theorem for the integer
partitions, the Gauss-Jacobi identity, Ramanujan’s identity and other
identities, we deduce some identities for the complete Bell polynomials.
References used: 11, 17, 22, 44, 45, 47, 58, 102, 103, 104.
21
INTERNATIONALES
Ce chapitre consiste à lister toutes les conférences faites par les membres de
l’équipe dans des colloques internationaux et nationaux
Miloud Mihoubi
DIMACOS’12: International conference on discrete mathematics and
computer science, Beyrouth, Libanon, 13-17 Novembre 2012.
Abstract. It is well known that the Stirling numbers of both kinds and the
Lah numbers can be written as particular cases of the partial Bell
polynomials and count, respectively, the number of permutations of the set
[n] ={1,…,n} into k cycles, the number of partitions of a [n] into k blocks
and the number of of partitions of a [n] into k ordered blocks. Some
restrictions of the Stirling numbers, named the r-Stirling numbers of the .rst
and second kind, have introduced and studied by Broder [1984]. In this
paper, we show that the r-Stirling numbers of both kinds can be considered
as particular cases of family of combinatorial polynomials forming a
restriction of the partial Bell polynomials. We also give the generating
functions of several restrictions of the r-Stirling and r-Lah numbers and we
propose new combinatorial interpretations for the r-Whitney numbers of
both kinds and the r-Whiney-Lah numbers.
References used: 11, 14, 18, 20, 22, 43, 44, 57, 67.
22
2. THE r-LAH NUMBERS AND THEIR RESTRICTIONS
Lilia Reggane and Miloud Mihoubi
Abstract. The Lah number counts the number of partitions of the set [n]
={1,…,n} into k ordered blocks and the r-Lah number counts the number of
partitions of [n] into k ordered blocks such that the r first elements are in
di¤erent blocks. These numbers are studied by several authors and have
many applications in di¤erent frameworks, specially in combinatorics.
Properties of some restrictions of r-Lah numbers are unknown, such the massociated r-Lah number and the m-trucated r-Lah number, which count, the
number of partitions of [n] into k ordered blocks such that the r first
elements are in di¤erent blocks with the condition that the blocks are of
cardinality ≥ m for the first and ≤ m for the second. In this paper, we present
some properties of several restrictions of the r-Lah numbers, such generating
functions, congruences, recurrence relations.
References used: 11, 18.
3. THE m-ASSOCIATED R-STIRLING NUMBERS OF THE SECOND KIND
Meriem Tiachachat and Miloud Mihoubi
Abstract. The m-associated Stirling number of the second kind counts the
number of partitions of the set [n] ={1,…,n} into k blocks such that each
23
block is of cardinality at least m. These numbers have appeared firstly in two
papers and are studied by several authors and used later with the Stirling
numbers to develop such propertities of Euler and Bernoulli polynomials.
The m-associated r-Stirling number; introduced in this paper, counts the
number of partitions of [n] into k blocks such that the r first elements are in
different blocks and each block is of cardinality at least m: In this paper, we
present generating functions, recurrence relations and some applications of
these numbers.
References used: 22, 81, 100.
4. INEQUALITIES WITH SEVERAL VARIABLES
RAMA08, Algeries 26-29 November 2012.
Abstract. In this work, to generalize an inequality given in different ways
by Neuman (1986) and by Belbachir (2008), we define a class of
polynomials (in m(≥ 3) variables) having a constant sign. After that, we
exploit these polynomials to determine inequalities with several variables.
References used: 74, 75, 76, 77, 78, 79, 80.
5. THE r-LAH NUMBERS AND THEIR RESTRICTIONS (suite)
Miloud Mihoubi and Lilia Reggane
Abstract. This work is motivated by the study by several authors of the well
known Lah and r-Lah numbers. Properties of some restrictions of r-Lah
numbers are unknown, such the m-associated r-Lah number and the m-
24
truncated r-Lah number. In this paper, we present some properties of these
restrictions.
References used: 11, 18.
6. BELL POLYNOMIALS AND RANDOM VARIABLES
Miloud Mihoubi and Yamina Saidi
Abstract. In this paper, We present the application of Bell polynomials in a
combinatoric and probabilistic study. We propose two identieies and a
combinatoric interpretation for these polynomials. We apply the Central
limit Theorem to deduce some approximations.
7. THE m-ASSOCIATED R-STIRLING NUMBERS OF THE SECOND KIND
(suite)
Miloud Mihoubi and Meriem Tiachachat
Abstract. This work is motivated by many applications of the r-Stirling
numbers of the second kind and their restictions. We use the r- Stirling
numbers and the r-Stirling numbers of the second kind to introduce and
study a new class (restriction) of
r-Stirling numbers, named, the m-
associated r-Stirling numbers of the second kind. These counts the number
of partitions of a n-set into k blocks such that the r first elements are in
different blocks with the condition that the blocks are of cardinality ≥ m. In
25
this paper, we give the exponential generating function of these numbers and
we present some of their properties such recurrence relations, congruences.
8. THE (r1 ,…, rp)-STIRLING NUMBERS OF THE SECOND KIND
Miloud Mihoubi and Mohammed Said Maamra
The Second International Symposium on Operational Research, ISOR'11
Algiers, Algeria : May 30th - June 02nd, 2011
Abstract. Let R and S be two subsets of the set {1, . . . , n} with R ∩ S = Ø,
|R| = r and |S| = s. The (r; s)-Stirling numbers of the second kind count the
number of partitions of this set such that the elements of R and the elements
of S are in different subsets. The combinatorial and algebraic properties of
these numbers are explored. We study also the multivariate case.
References used: 12, 13, 14, 17, 20, 22, 23, 30, 32, 40, 41, 50, 58.
9. ON THE RECIPROCALS AND COMPOSITIONAL INVERSES OF POWER
SERIES
Miloud Mihoubi and Rachida Mahdid
The Second International Symposium on Operational Research, ISOR'11
Algiers, Algeria : May 30th - June 02nd, 2011
Abstract. Among the different applications of the partial Bell polynomials,
we give in this paper the explicit compositional inverses and/or the
reciprocals of some power series. We illustrate the obtained results by some
examples on Stirling numbers.
26
References used: 2, 3, 4, 11, 14, 21, 22, 26, 27, 28,39, 44, 45, 46, 47, 50, 58,
59, 60, 68.
10. APPROCHE MULTI-AGENTS POUR L’ORDONNANCEMENT
DYNAMIQUE D’ATELIER DE PRODUCTION
A. Kouider, S. Ourari, B. Bouzouia and M. Mihoubi
9e Conférence Internationale de Modélisation, Optimisation et SIMulation MOSIM’12
06 au 08 Juin 2012 - Bordeaux – France « Performance, interopérabilité et
sécurité pour le développement durable »
Résumé. Ce papier s’intéresse à un problème d’ordonnancement de type
job shop où les travaux sont supposés définis au fur et à mesure que les
commandes apparaissent. L’approche multi-agents de résolution proposée
considère que les entités décisionnelles sont décentralisées au sein d’une
organisation distribuée. Le problème d’ordonnancement global est
décomposé en sous-problèmes d’ordonnancement alloués à des entités
décisionnelles locales de résolution, chacune correspondant à une ressource
de l’atelier et gérant son propre ordonnancement local. L’architecture du
système multi-agents proposée comporte m agents machines (décideurs) qui
coopèrent et s’échangent des messages pour produire, au niveau local, un
ordonnancement réalisable et ce dans le but de converger au mieux vers une
solution globale qui minimise le makespan. Chaque agent définit un plan de
réalisation en utilisant des règles de priorité tout en minimisant les espaces
d’inoccupation de sa machine. En considérant à la fois deux cas de
problèmes d’ordonnancement, statique et dynamique, des expérimentations
27
ont été effectuées et ont permis de prouver l’efficacité de l’approche
proposée.
Reférences utilisées: 24, 25, 33, 34, 35, 36, 37, 38, 54, 55, 56, 60, 64, 66.
11. THE (EXPONENTIAL) MULTIPARTITIONAL POLYNOMIALS:
INTERPRETATIONS AND RECURSSIVNESS
5th Saudi science conference, Umm Al Qura University, Makkah, Arabie
Saoudite, Avril 16-18, 2012.
Abstract. The bipartitional polynomials are introduced in a book of
Charalambos A. Charalambides, with some important properties and
recently studies by Mihoubi and Belbachir. These polynomials generalize
the Bell polynomials introduced by Bell and studies by Comtet and several
authors. Christopher and Nadarajah gave an introduction and some
applications of the multipartitional polynomials. In this paper, we establish
some results on multipartitional polynomials and their combinatorial and
probabilistic interpretations.
References used: 10, 14, 22, 44, 101.
12. THE COMPOSITIONAL INVERSES OF REMARQUABLE POWER SERIES
International conference on discrete mathematics and computer science,
DIMACOS’11, University Hassan II, Mohammedia, Morocco,
2011.
5-8 Mai
28
Abstract. Recently, several applications of Bell polynomials are given in
different frameworks such that in integration, inverse relations, congruences
and generalized Blissard problem. In the present paper, we show that the
partial Bell polynomials and binomial type sequences define a large family
of power series for which we can obtain easily their compositional inverses.
For the applications, we give several examples on power series whose
coefficients in their expansions in power series are related to Touchard
polynomials, Lah numbers, Stirling and generalized Stirling numbers and
others.
References used: 14, 22, 28, 45, 46, 47.
13. THE POLYNOMIAL SEQUENCES OF MULTINOMIAL TYPE AND THEIR
APPLICATIONS ON THE MULTIPARTITIONAL POLYNOMIALS
International conference on discrete mathematics and computer science,
DIMACOS’11, University Hassan II, Mohammedia, Morocco,
5-8 Mai
2011.
Abstract. In recent papers, Mihoubi studied the connection between Bell
polynomials and binomial type sequences and gave some of their
applications in congruences, inverse relations, reciprocals and compositional
inverses of power series. The bipartitional polynomials are introduced in a
book of Charalambos A. Charalambides with some important properties and
studied recently by Mihoubi and Belbachir. Christopher and Nadarajah gave
an introduction and some applications of the multipartitional polynomials. In
this paper, we introduce the multipartitional polynomials and polynomials of
29
multinomial type, study their properties and establish some connections
between them. Our results are illustrated by some comprehensive examples.
References used: 10, 17, 28, 44, 49.
14. POLYNOMIALS OF MULTIPARTITIONNAL TYPE AND INVERSE
RELATIONS
The international days of applied algebra, JIAA’2011, M’sila, Algeria , 2930 Nov. 1 Dec. 2011
Abstract. Chou, Hsu and Shiue gave some applications of Faà di Bruno.s
formula to characterize inverse relations. Our aim is to develop some inverse
relations connected to the multipartitional type polynomials involving to
binomial type sequences.
References used: 10, 19, 44, 47, 49.
15. THE MULTIPARTITIONAL POLYNOMIALS AND RANDOM VECTORS
Conférence internationale sur les mathématiques pures et appliquées,
ICPAM’2012, Université 8 Mai 1945, Guelma, Algérie, 28, 29 et 30 Mai
2012
Abstract. In previous papers, we studied the (exponential) bipartitional
polynomials and established several properties for them. In previous
contributions, we de_ned and studied an extension of these polynomials,
named the (exponential) multipartitional polynomials. These polynomials
30
appear as a natural extension of the Bell polynomials. In the present paper,
we give a probabilistic interpretation of these polynomials. We also give,
some applications on moments associated to a _nite sum of (iid) random
vectors and other applications on the probability distribution of a finite sum
of discrete (iid) random vectors.
31
NATIONALES
1. TOUCHARD POLYNOMIALS AND PARTIAL BELL POLYNOMIALS
Congrès des Mathématiciens Algériens, CMA'2012, Annaba, 7-8 Mars 2012
Abstract. Touchard generalized the Bell polynomials in order to give some
combinatorial interpretation on permutations. Chrysaphinou introduced and
studied a class of polynomials related to Touchard's generalization. In the
present paper, we establish some relations between Touchard polynomials,
Bell polynomials and the polynomials of binomial type. Several identities
and relations with Stirling numbers are obtained.
2. A GENERALIZED RECURRENCE FOR r-BELL POLYNOMIALS
Congrès des Mathématiciens Algériens, CMA'2012, Annaba, 7-8 Mars 2012
Abstract. Letting Bn(x) the n-th r-Bell polynomial. Our aim is to prove that
the sequences Bn+mr (x) , Bn,r+s (x) and Bn,r (x) admit a Binomial Recurrence
Coefficient in different basis of the Q-vectorial space formed by
polynomials of Q[X].
32
CHAPITRE 6 : MEMOIRES SOUTENUS
MAGISTER:
1. SAIDI Yamina
Titre: Polynômes de Bell et variables aléatoires.
Numéro d’ordre : 37/2012-M/MT
Encadé par Miloud MIHOUBI et mémoire soutenu en juin 2012.
MASTER:
1. KOUIDER Ahmed
Titre: Approche distribuée pour l’ordonnancement d’atelier de production.
Numéro d’ordre : 123456
Encadés par Miloud MIHOUBI et mémoire soutenu en juin 2012.
2. KHALED Abdelkrim et LEBDIRI Sofiane
Titre: Optimisation de la durée de déménagement d’un appareil de forage
Code: 03/2MIR/12
3. LAZDAM Ibtissem et SALAH SALAH Meriem
Titre : Optimisation du transport du gaz par canalisation
Code : 14/RO2MIR/13
33
CONCLUSION
Durant ces deux dernières années, le travail du groupe de ce projet de
recherche a donné ses fruits. Les séances de travail hebdomadaires et les
motivations n’ont a permis de réaliser, à notre estimation, plus de 75% des
objectifs visés au moment où nous nous sommes engagés dans ce PNR. En
effet, en premier lieu, nous avons abordés des travaux sur les fonctions
réciproques. Malgré que ce domaine est très connu de telle façon qu’il soit
difficile pour un chercheur (moyen) de réaliser un nouveau résultat, nos
connaissances sur la combinatoire des polynômes de Bell nous ont données
un bon coup et on a finalement aboutit de réaliser un bon travail dans le
domaine. Ensuite, on s’est dirigé vers une étude sur les nombres de Stirling
et les polynômes Bell. Cette étude est finie par une réalisation de deux autres
travaux très intéressant dans le domaine de la combinatoire énumérative.
Avec des contributions individuelles et des contributions collectifs avec
d’autres chercheurs en dehors de notre projet, on a aussi réussit de réaliser
huit autres travaux qui touchent : les polynômes multipartitionnels et leurs
liens avec la combinatoire énumérative et les vecteurs aléatoires, les
nombres de Stirling et les polynômes de Bernoulli d’Euler, le polynôme
chromatique et son rôle dans la caractérisation de plusieurs propriétés dans
les graphes et dans la combinatoire énumérative. Au total, on est arrivé à
réaliser plus de dix publications internationales. Disons qu’on a touché la
plus grande partie des thèmes et travaux programmés pour le projet. A
travers, les résultats mentionnés dans ce document, notre équipe est vraiment
très satisfaite des résultats réalisés et maintenant elle est en train de
continuer certains travaux qui sont en cours ou d’autres qui ne sont encore
réalisés.
34
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44
INFORMATION FINANCIERE
REPUBLIQUE ALGERIENNE DEMOGRATIQUE ET POPULAIRE
Ministère de l’Enseignement Supérieur et de la Recherche Scientifique
Université des Sciences et de la Technologie HOUARI Boumediene
BUDGET DE FONCTIONNEMENT DU FNR : (Année 2013)
Laboratoire de Recherche Opérationnelle, Combinatoire, Informatique Théorique et
Méthodes Stochastiques (dénommé RECITS)
Directeur du Laboratoire : Pr. BOUDHAR Mourad
Chap.
Art.
Par.
REMBOURSEMENT DES FRAIS
I
Crédits alloués
0.00
1
--
Frais de mission et de déplacement en Algérie et à l’étranger
0,00
2
--
Rencontres scientifiques : Frais d’organisation,
d’hébergement, de restauration et de transport
0.00
3
--
Honoraires des enquêteurs
0,00
4
--
Honoraires des guides
0,00
5
--
Honoraires des experts et consultants
0,00
6
--
Frais d'études, de travaux et de prestation réalisé pour le
compte de l’entité
0,00
1
2
---
3
4
---
1
2
---
Produits chimiques
Produits consommables
0,00
0,00
3
--
Composants électroniques, mécaniques et audio-visuels
0,00
4
5
6
7
-----
Papeterie et fournitures de bureau
Périodiques
Documentation et Ouvrages de recherche
Fournitures des besoins de laboratoires( animaux, plantes,
etc…)
CHARGES ANNEXES
II
III
IV
Libellé
MATERIELS ET MOBILIERS
Matériels et instruments scientifiques et audio-visuels
Renouvellement du matériel informatique, achat accessoires,
logiciels et consommables informatiques
Mobilier de laboratoire
Entretien et réparation
FOURNITURES
850 000,00
500 000,00
200 000,00
150 000.00
0.00
350 000,00
200 000,00
0,00
150 000,00
0,00
0,00
45
1
2
---
Impression et édition
Affranchissements postaux
0,00
0,00
3
--
Communications téléphoniques, fax, télex, télégramme,
Internet
0,00
4
--
Autres frais( Impots et taxes, droits de douane, frais
financiers, assurances, frais de stockage, et autre)
0,00
5
--
Banque de données( acquisition et abonnement)
0,00
PARC AUTOMOBILE
V
1
--
2
--
Carburant et lubrifiants
0,00
0,00
Location de véhicules pour les travaux de recherche sur
terrain
FRAIS DE VALORISATION ET DE DEVELOPPEMENT
TECHNOLOGIQUE
-Frais de formation et d'accompagnement des porteurs de
projets
Frais de propriété intellectuelle
0,00
0,00
2
Frais de formation et d'accompagnement des porteurs de
projets
Demandes de dépôt de brevet, de marque et de modele
0,00
3
4
Dépôt de logiciel
Protection des obtentions végétales, animales et autres
0,00
0,00
5
Frais des mandataires
0,00
VI
1
2
1
0,00
0,00
0,00
3
--
Frais de conception et de définition du projet à mettre en
valeur
0,00
4
--
Frais d'évaluation et de faisabilité du projet valorisable,(
Maturation=Plan d'affaire)
0,00
5
--
Frais d'expérimentation et de développement des produits à
mettre en valeur
0,00
6
7
8
----
Frais d'incubation
Frais de service à l'innovation
Frais de conception et de réalisation de prototypes,
maquettes, présérie, installations pilotes et démonstrations
0,00
0,00
0,00
VII
1
RETRIBUTION DES ACTIVITES DES CHERCHEURS
-La rétribution des activités de recherche des chercheurs
mobilisés dans le cadre des programmes nationaux de
recherche
TO T A L G E N E R A L
0,00
0,00
1 200 000,00

1. THE PARTIAL r-BELL POLYNOMIALS

Transcription

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