DAFx-06 Proceedings - Algeria, Sidi Bel Abbès

Conference-School on Discrete Mathematics and Computer ScienceAlgeria, Sidi Bel Abbès

November 15 - 19, 2015.

Published by:Recits LaboratoryFaculty of MathematicsUSTHBhttp://www.lrecits.usthb.dz/activites.htm

ISBN: X-XXXX-XXXXXX

LATEX editor: A Busing LATEX’s ‘confproc’ package, version 0.8Printed in Algiers — November 2015

Program Committee Organization Committee

A. Bouyakoub, Oran IUniv., Algeria , Chair A. Yousfate, SBA Univ., ChairM. Al Zohairi, KSU, Saudia Arabia B. Belarbi, Oran I Univ.,N. Badache, USTHB, Algeria H. Belbachir, USTHB,M. Bekkali, Fez Univ., Morocco A. Belkhir, USTHB,H. Belbachir, USTHB, Algeria M. Benchohra, SBA Univ.,M. A. Benoumhani, M’Sila Univ., Algeria S. Boukli-Hacène, SBA Univ.,I. Boudabbous, Sfax Univ., Tunisia I.-E. Bousbaa, USTHB,Y. Boudabbous, KFU, Saudia Arabia A. Bouyakoub, Oran I Univ.,M. Boudhar, USTHB, Algeria B. Chafi, SBA UnivM. E.-A. Chergui, USTHB, Algeria H. Harik, CERIST,A. Dil, Akdeniz Univ., Turkey M. Lakrib, SBA Univ.,A. El Sahili, Lebanese Univ., Lebanon E. Lamini, CDTAK. M. Faraoun, SBA Univ., Algeria M. Mechab, SBA Univ.,O. Khadir, Mohammadia Univ., Morocco F. Tebboune, Saida Univ.H. Kheddouci, Lyon I Univ., FranceE. Kiliç, TOBB Univ. TurkeyT. Komatsu, Wuhan Univ., ChinaJ.-G. Luque, Rouen Univ., FranceM. Mihoubi, USTHB, AlgeriaA. Mokrane, Paris 8 Univ., FranceM. Pouzet, Calgary Univ., CanadaS. Pirzada, Kashmir Univ., IndiaH. Si Kaddour, Lyon I Univ., FranceL. Szalay, Sopron Univ., HungaryN. Thiéry, Paris 11 Univ., FranceA. Yousfate, SBA Univ., AlgeriaD. Ziadi, Rouen Univ., France

http://www.lrecits.usthb.dz/activites.htm

Conference-School on Discrete Mathematics and Computer Science, Algeria, Sidi Bel Abbès, November 15 - 19, 2015.

CONFERENCE PROGRAM

Day 1

Plenary Conference1 Németh LÁSZLÓ

Hyperbolic Pascal triangles and pyramid

5 Tahar MOURIDStatistics of Functional Autoregressive Processes

Oral Session - Enumerative Combinatorics6 Nesrine BENYAHIA-TANI

Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon

9 Assia Fettouma TEBTOUBThe 2-successive Eulerian numbers and unimodality

12 Ali CHOURIAThe boson normal ordering problem: Combinotorial interpretation

Oral Session - Graph Theory16 Mohammed YAGOUNI

Meta-Storming, un outil pour la mise en œuvre collaborative des Mtaheuristiques

19 Aymen BEN AMIRAReconstruction of digraphs up to complementation

22 Daouya LAICHEPacking coloring of some undirected and oriented coronae graphs

Plenary Conference26 Fairouz TCHIER

Some topics in Discrete Mathematics

29 Imed BOUDABBOUSThe non −2-recognizable indecomposable tournaments

Oral Session - Arithmetics32 Amina SOUYAH

Image enciphering schemes: State of the art

36 Samia MASROURNested double binomial sums

DIMACOS-iii


39 Djamel BERKANEOn a constant related to the prime counting function

Oral Session - Graph Theory41 Mouna YAICH

Graphes indécomposables et leurs sous-graphes indécomposables à 7 ou à 8 sommets

44 Rahma SALEMHereditarily hemimorphy of −k-hemimorphic digraphs for k ≥ 8

48 Nour El Houda Asma MERABETn Out of n Audible Password Secret Sharing Scheme With Unexpanded Shares

Day 2

Plenary Conference52 Moussa BENOUMHANI

Polynomial associated with finite topologies

55 Szalay LÁSZLÓArithmetical triangles

Oral Session - Enumerative Combinatorics58 Athmane BENMEZAI

Chu-Vandermonde identity for q-analogue and p,q-analogue of bisnomial coefficients

61 Said AMROUCHETriangles arithmétiques associées aux suites de Fibonacci généralisées

64 Oussama IGUEROUFACentral Bisnomial Coefficients on Hypergrids

Oral Session - Graph Theory and Relational Structures69 Djamila OUDRAR

Monomorphic decomposition and profile

73 Brahim BENMEDJDOUBColoration dincidences et des carrés des graphes circulants et des graphes distances

77 Fatma MESSOUDIThe upper paired-domination subdivision numbers of graphs

Plenary Conference79 Omar KHADIR

Attacks on ElGamal digital signature scheme : an overview

DIMACOS-iv


Oral Session - Operational Research81 Nour El Houda TELLACHE

Scheduling the two-machine flow shop problem with unit-time operations and conflict graph

85 Mohamed BOUALEMStochastic Analysis and Bounds for the Stationary Distribution of an M/G/1 Retrial Queue with Breakdowns

88 Nadjat MEZIANIPSO and Simulated annealing for the two machines flowshop scheduling problem with coupled-operations

91 Amina HANEDScheduling with preemption and setup times

Oral Session - Graph Theory95 Maria ABI AAD

Paths in tournaments

97 Mohamed Amine BOUTICHEForwarding indices of some graph operations

101 Hakim HARIKOn bandwidth of some classes of graphs

103 Mortada MAIDOUNThe b-Chromatic Number and f -Chromatic Vertex Number of Regular Graphs

Day 4

Plenary Conference106 Hamid ABCHIR

Coloriage des noeuds

108 Shariefuddin PIRZADASum of the Laplacian eigenvalues of a graph and Brouwer’s conjecture

Oral Session - Operational Research111 Karim AMROUCHE

Résolution approchée d’un problème d’ordonnancement de type flowshop continu avec recirculation

113 Zahia BOUABBACHEOptimal control of dynamic systems with random input

116 Slimane BELLAOUARTowards a Set-String Subsequence Kernel

Oral Session - Enumerative Combinatorics120 Amine BELKHIR

Statistic on linear tilings and generalized q-Fibonacci polynomials

123 Mourad RAHMANISome results on Cauchy numbers

DIMACOS-v


124 Imène BENRABIATiling interpretation for combinatorial identity associated to order three Fibonacci sequence

Plenary Conference126 Fayçal HAMDI

Modeling the periodic conditional volatility as a mixture process

Oral Session - Stochastic Processes128 Billel ALIAT

Periodic stationarity and existence of moments of Markov-switching PARMA processes132 Rokia HEMIS

Estimation Bayesienne d’un Melange de Modéles GARCH Périodiques par la Méthode de Griddy-Gibbs134 Nadia BOUSSAHA

Periodic multivariate stochastic volatility model: structure and estimation136 Nesrine KARA TERKI

Inégalités de Grandes Déviations de l’Estimateur Sieves d’un Opérateur dun Processus AR Fonctionel

Oral Session Enumerative and Algebraic Combinatorics138 Boudekhil CHAFI

Sur les fonctions holomorphes discrèetes141 Zakaria CHEMLI

Shifted domino tableaux143 Moussa AHMIA

On the weighted sums associated to rays in Pascal’s triangle145 Fariza KRIM

Recurrences associated to rays in negative Generalized Arithmetic Triangle

Plenary Conference148 Nour-Eddine FAHSSI

The Many Aspects of Polynomial Triangles

Oral Session - Mathematical Programming152 Younes GUELLOUMA

Towards Integrals of Rational Tree Expressions156 Amel BELABBACI

Approche pour l’optimisation d’une fonction quadratique indéfinie159 Slimane OULAD-NAOUI

Frequent Pattern Mining : a Reduction to Acyclic Weighted Automata Determinisation

Oral Session - Enumerative Combinatorics164 Mohammed Said MAAMRA

Note on some restricted Stirling numbers of the second kind167 Imad Eddine BOUSBAA

The Generalized r-Lah Numbers Revisited

DIMACOS-vi


170 Yamina SAIDIAn identity on pairs of Appell-type polynomials

Day 5

Plenary Conference173 Ayhan DIL

Generalized Geometric Polynomials and Applications to Series with Zeta Values176 Abdelghani OUAHAB

Existence and Compactness Results for System of Difference Equations

Oral Session - Enumerative Combinatorics181 Assia MEDJERREDINE

Associated Stirling numbers for some families of graphs185 Asmaa RAHIM

The r-Jacobi-Stirling numbers of the second kind189 Abdelghani MEHDAOUI

The Stirling Numbers with constraint

Oral Session - Stochastic Processes192 Tayeb BLOUHI

Existence Results for Couple Stochastic Difference Equations with Delay197 Wahiba BOUABSA

Some asymptotic normality result of k-Nearest Neighbor estimator200 Ahmed BOUDAOUI

Existence of mild solutions to stochastic delay evolution equations with a fractional Brownian motion and impulses204 Meriem TIACHACHAT

Some applications of the translated Whitney numbers

Poster Session206 Mohammed BENATALLAH

b-Domatic number of the join graphs209 Abdelkader HAMTAT

On the Diophantine equation x2 +13k = yn

212 Tassadit LACHEMOT

An MX/G/1retrial queue with breakdown, repair, loss and two TYPES of customers214 Boubaker MECHAB

Asymptotic normality of high risk estimate for functional data218 Selma MERADJI

The local time under sublinear expectations and applications220 Sara STIHI

Stochastic calculus for matrix fractional brownian motion and applications

DIMACOS-vii


222 Nesrine ZIDANIApproximation of multiserver retrial queues by value extrapolation technique

224 Asma BENCHEKORProcessus semi-markoviens à temps discret. Théorie et applications

227 List of Authors

DIMACOS-viii

Conference on Discrete Mathematics and Computer Science, Sidi Bel Abbess, Algeria, November 15-19, 2015

Hyperbolic Pascal triangles and pyramid

Nemeth, Laszlo

University of West Hungary, Institute of MathematicsAdy E. ut 5., Sopron, Hungary

[email protected]

Abstract

We introduce a new generalization of Pascal’s triangle and pyramid based on the hyperbolic regular mosaicsin plane and space. Then we study certain quantitative properties.

Key words: Pascal triangle, Pascal pyramid, regular mosaics on hyperbolic plane, cube mosaic in hyperbolic space

2010 Mathematics Subject Classification: 11B99, 05A10.

1. Hyperbolic Pascal trianglesThere are several approaches to generalize the Pascal’s arithmetic triangle (see, for instance [3, 4]). A new type of variationsof it is based on the hyperbolic regular mosaics denoted by Schlafli’s symbol p, q, where (p − 2)(q − 2) > 4 ([5]). Eachregular mosaic induces a so called hyperbolic Pascal triangle (see [2]), following and generalizing the connection between theclassical Pascal’s triangle and the Euclidean regular square mosaic 4, 4. For more details see [2], but here we also collectsome necessary information.

The hyperbolic Pascal triangles based on the mosaic p, q can be figured as a digraph, where the vertices and the edgesare the vertices and the edges of a well defined part of the lattice p, q, respectively, and the vertices possess a value thatgive the number of the different shortest paths from the base vertex. Figure 1 illustrates the hyperbolic Pascal triangle whenp, q = 4, 5. Here the base vertex has two edges, the leftmost and the rightmost vertices have three, the others have fiveedges. The quadrilateral shape cells surrounded by the appropriate edges correspond to the squares in the mosaic. Apartfrom the winger elements, certain vertices (called “Type A”) have 2 ascendants and 3 descendants, while the others (“TypeB”) have 1 ascendant and 4 descendants. In the figures we denote the vertices type A by red circles and the vertices typeB by cyan diamonds, further the wingers by white diamonds. The vertices which are n-edge-long far from the base vertexare in row n. The general method of preparing the graph is the following: we go along the vertices of the jth row, accordingto the type of the elements (winger, A, B), we draw the appropriate number of edges downwards (2, 3, 4, respectively).Neighbour edges of two neighbour vertices of the jth row meet in the (j + 1)th row, constructing a new vertex type A. Theother descendants of row j have type B in row j + 1. In the sequel, ) nk ( denotes the kth element in row n, which is either thesum of the values of its two ascendants or the value of its unique ascendant. We note, that the hyperbolic Pascal triangles hasthe property of vertical symmetry.

In studying the quantitative properties of the hyperbolic Pascal triangle 4, q, first we determine the number of theelements of the nth row of the graph. Denote by an and bn the number of vertices of type A and B, respectively, further letsn = an + bn + 2, which gives the total number of the vertices of row n ≥ 1. Recall, that q ≥ 5.

Theorem 1. The three sequences an, bn and sn can be described by the same ternary homogeneous recurrencerelation

xn = (q − 1)xn−1 − (q − 1)xn−2 + xn−3 (n ≥ 4),1


Figure 1: Hyperbolic Pascal triangle linked to 4, 5 up to row 5

the initial values are

a1 = 0, a2 = 1, a3 = 2, b1 = 0, b2 = 0, b3 = q − 4, s1 = 2, s2 = 3, s3 = q.

Let an, bn and sn denote the sum of type A, type B and all elements of the nth row, respectively. We will justify thefollowing statements.

Theorem 2. The three sequences an, bn and sn can be described by the same ternary homogenous recurrence relation

xn = qxn−1 − (q + 1)xn−2 + 2xn−3 (n ≥ 4),

the initial values are

a1 = 0, a2 = 2, a3 = 6, b1 = 0, b2 = 0, b3 = 2(q − 4), s1 = 2, s2 = 4, s3 = 2q.

Let sn be the alternating sum of elements of the hyperbolic Pascal triangles (starting with positive coefficient) in row n,and we distinguish the even and odd cases.

Theorem 3. Let q be even. Then

sn =sn−1∑

i=0

(−1)i )ni

( =

0, if n = 2t+ 1, n ≥ 1,−2(5− q)t−1 + 2, if n = 2t, n ≥ 2,

hold, further s0 = 1.

Theorem 4. Let q ≥ 5 be odd. Then s0 = 1, further

sn =sn−1∑

i=0

(−1)i )ni

( =

0, if n = 3t+ 1, n ≥ 1,(−2)t(q − 5)t−1 + 2, if n = 3t− 1, n ≥ n1,

2(−2)t(q − 5)t−1 + 2, if n = 3t, n ≥ n2,

where (n1, n2) = (2, 3) and (5, 6) if n > 5 and n = 5, respectively. In the latter case s2 = 0, s3 = −2.

2


Along paths of the hyperbolic Pascal triangle 4, 5 we can find some well-known sequences for example the Fibonacciand the Pell sequence (see Figure 2). Generally, we can prove Theorem 5.

Theorem 5. Let η, and f0 < f1 denote positive integers with gcd(f0, f1) = 1. Further let fn denote a binary recurrencesequence given by

fn = ηfn−1 ± fn−2, (n ≥ 2).

The values f0 and f1 appear next to each other in a suitable row of the Pascal triangle, such that the type of f1 is A. Thenall elements of the sequence are descendants of the vertex labelled by f1, all have type A.

Figure 2: Fibonacci and Pell sequences in the hyperbolic Pascal triangle 4, 5

2. Hyperbolic Pascal pyramidThe 3-dimensional analogue of the original Pascal’s triangle is the well-known Pascal’s pyramid (or more precisely Pascal’stetrahedron) (left part in Figure 3). Its layers are triangles and the numbers along the three edges of the nth layer are thenumbers of the nth line of Pascal’s triangle. Each number inside in any layers is the sum of the three adjacent numbers in thelayer above.

We can defined a hyperbolic Pascal pyramid (right part in Figure 3) in the hyperbolic space based on the hyperbolic reg-ular cube mosaic (cubic honeycomb) with Schlafli’s symbol 4, 3, 5 generalised to hyperbolic Pascal triangles and classicalPascal’s pyramid which is based on the Euclidean regular cube mosaic 4, 3, 4.

We denote the sums of the vertices A, B, C, D and E in level i by ai, bi, ci, di and ei, respectively.

Theorem 6. The growing of the numbers of the different types of the vertices are described by the system of linear inhomo-

3


geneous recurrence sequences (n ≥ 1)

an+1 = an + bn + 3,bn+1 = an + 2bn,

cn+1 =13an + cn +

23dn,

dn+1 =12bn +

32cn + 2dn +

52en,

en+1 = 3cn + 4dn + 6en,

with zero initial values.

1

4 4

4

6

12

12

4

6

4

641

12

1

3 3

3

33

3

1

11

6

2 2

2

1

11

1

1

1 1

1

1

2 23 34 45 5

22

3

3

4

5

5

4

22

3

34

4

5

5

12

8 6

66

8

8

12

12

1

2

2

23 3

3

33

3

1

11

6

2 2

2

1

11

1

1 1

1

Figure 3: Euclidean and hyperbolic Pascal pyramid

References[1] Ahmia, M. – Szalay, L., On the weighted sums associated to rays in generalized Pascal triangle, submitted.

[2] Belbachir, H., Nemeth, L., Szalay, L., Hyperbolic Pascal triangles, submitted, (arXiv:1503.02569).

[3] Belbachir, H. – Szalay, L., On the arithmetic triangles, Siauliai Math. Sem., 9 (17) (2014), 15-26.

[4] Bondarenko, B. A., Generalized Pascal triangles and pyramids, their fractals, graphs, and applications. Translatedfrom the Russian by Bollinger, R. C. (English) Santa Clara, CA: The Fibonacci Association, vii, 253 p. (1993).www.fq.math.ca/pascal.html

[5] Coxeter, H. S. M., Regular honeycombs in hyperbolic space, Proc. Int. Congress Math., Amsterdam, Vol. III. (1954),155-169.

[6] Harris, J, M., Hirst, J. L., Mossinghoff, M. J, Combinatorics and Graph Theory, Springer, (2008).

4


Statistics of Functional Autoregressive Processes

Tahar Mourid

Laboratoire de Statistiques et Modelisations AleatoiresUniversity of Abou Bekr Belkaid. Tlemcen 13000. Algeria

Abstract

We give an over review on some statistical problems related to the class of Functional Autoregressive Pro-cesses. We especially emphasis on some aspects of statistical estimation of parameter operator based on sampleof AR functional process and discuss the prediction problem in this setting. An extension to Functional AR Pro-cesses with random coefficients is also considered. We finish by the notion of closed sub space of Fortet givinga new insight in the prediction problem in functional statistic framework and its application in AR functionalprediction.

Key words: Functional autoregressive process- Limit theorems- Parameter operator estimation- Prediction- Covarianceoperator- Exponential Bounds- Rate of convergence Closed sub space of Fortet.

2010 Mathematics Subject Classification:

5


Ordered and non-ordered non-congruent convex quadrilateralsinscribed in a regular n-gon

Nesrine BENYAHIA-TANI1∗, Sadek BOUROUBI2, Zahra YAHI31 Algiers 3 University, L’IFORCE Laboratory

2 Ahmed Waked Street, Dely Brahim, Algiers, [email protected]

2 USTHB, Faculty of Mathematics, L’IFORCE LaboratoryP.B. 32 El-Alia, 16111, Bab Ezzouar, Algiers, Algeria.

[email protected] Bejaia Abderahmane Mira University, L’IFORCE Laboratory

Targa Ouzemour Road, 06000, Bejaia, [email protected]

AbstractUsing several arguments, some authors showed that the number of noncongruent triangles inscribed in a

regular n-gon equals n2/12, where x is the nearest integer to x. In this paper, we revisit the same problem,but study the number of ordered and non-ordered non-congruent convex quadrilaterals, for which we givesimple closed formulas using Partition Theory. The paper is complemented by a study of two further kinds ofquadrilaterals called proper and improper non-congruent convex quadrilaterals, which allows to give a formulathat connects the number of triangles and ordered quadrilaterals. This formula can be considered as a newcombinatorial interpretation of a certain identity in Partition Theory.

Key words: Congruent triangles; Congruent quadrilaterals; Ordered quadrilaterals; proper quadrilaterals; Integer partitions.

2010 Mathematics Subject Classification: 05A17.

1. IntroductionIn the 1938’s, Norman Anning from university of Michigan proposed the following problem [6]:”From the vertices of aregular n-gon three are chosen to be the vertices of a triangle. How many essentially different possible triangles are there?”.For any given positive integer n ≥ 3, let ∆ (n) denotes the number of such triangles.

Using a geometric argument, Frame showed that ∆ (n) = n2/12, where x is the nearest integer to x. After that,other solutions were proposed by some authors, such as Auluck [2].In 1978, Reis posed the following natural general problem: From the vertices of a regular n-gon k are chosen to be thevertices of a k − gon. How many incongruent convex k-gons are there ?Let us first specify that two k-gons are called congruent if one k-gon can be moved to the other by rotation or reflection.The aim of this paper is to enumerate two kinds of non-congruent convex quadrilaterals, inscribed in a regular n-gon, theordered ones which have the sequence of their sidess sizes ordered, denoted by RO(n, 4) and those which are non-ordereddenoted by RO(n, 4), using the Partition Theory.

∗Speaker6


2. Notations and preliminariesWe denote by Gn a regular n-gon and by N the set of nonnegative integers. The partition of n ∈ N into k parts is a tupleπ = (π1, . . . , πk) ∈ Nk, k ∈ N, such that

n = π1 + · · ·+ πk, 1 ≤ π1 ≤ · · · ≤ πk,

where the nonnegative integers πi are called parts. We denote the number of partitions of n into k parts by p(n, k), thenumber of partitions of n into parts less than or equal to k by P (n, k) and by q(n, k) we denote the number of partitions of ninto k distinct parts. We sometimes write a partition of n into k parts π = (πf1

1 , . . . , πfss ), where

∑si=1 fi = k, the value of

fi is termed as frequency of the part πi. Let m ∈ N,m ≤ k, we denote cm(n, k) the number of partitions of n into k partsπ = (πf1

1 , . . . , πfss ) for which 1 ≤ fi ≤ m and fj = m for at least one j ∈ 1, . . . , s. For example c2(12, 4) = 10, the such

partitions are 1128, 1137, 1146, 1155, 1227, 1335, 1344, 2235, 2244, 2334. Let δ(n) ≡ n (mod 2), so δ(n) = 1 or 0,bxc the integer part of x and finally x the nearest integer to x.

3. Main resultsIn this section, we give the explicit formulas of RO(n, 4) and RO(n, 4).

Theorem 1. For n ≥ 1,

RO(n, 4) =n3

144+n2

48− δ(n) · n

32

,

where δ(n) equals 0 if n is even and 2 otherwise.

To give an explicit formula for RO(n, 4), we need the following lemma.

Lemma 2. For n ≥ 4,

c2(n, 4) = p(n, 4)− q(n, 4)−⌊n− 1

3

⌋.

Now we can derive the following theorem.

Theorem 3. For n ≥ 4,

RO (n, 4) =n3

144+n2

48− δ(n) · n

16

+

(n− 6)3

144+

(n− 6)n2

48− (n− 6)δ(n)

16

−⌊n− 1

3

⌋.

4. Connecting formula between ∆ (n) and RO(n, 4)

There are two further kinds of quadrilaterals inscribed in Gn, the proper ones, those which do not use the sides of Gn and theimproper ones, those using them.Let denote by RP

O(n, 4) and RPO(n, 4) respectively, the number of these two kinds of quadrilaterals. The goal of this section

is to prove the following theorem.

Theorem 4. For n ≥ 4,∆ (n) = RO(n+ 1, 4)−RO(n− 3, 4)·

Remark 1. The well-known recurrence relation [4, p. 373],

p(n, k) = p(n+ 1, k + 1)− p(n− k, k + 1), (1)

implies by setting k = 3,p(n, 3) = p(n+ 1, 4)− p(n− 3, 4)· (2)

Thus, as we can see, the formula of Theorem 4 can be considered as a combinatorial interpretation of identity (2).

7


For k ≤ n, we have the following generalization, using the same arguments to prove Theorem 4.

Theorem 5. For n ≥ k,RO(n, k) = RO(n+ 1, k + 1)−RO(n− k, k + 1)·

The formula of Theorem 5 can be considered as a combinatorial interpretation of the recurrence formula (1).

References[1] Andrews G. E., Eriksson, K., Integer Partitions, Cambridge University Press, Cambridge, United Kingdom, 2004.

[2] Anning A, Frame J. S., Auluck F. C., Problems for Solution: 3874-3899, The American Mathematical Monthly, Vol. 47,No. 9, (Nov., 1940), 664-666.

[3] Charalambos Charalambides A., Enumerative Combinatorics, Chapman & Hall /CRC, 2002.

[4] Comtet L., Advanced Combinatorics, D. Reidel Publishing Company, Dordrecht-Holland, Boston, 1974, 133–175.

[5] Gupta, H., Enumeration of incongruent cyclic k-gons, Indian J. Pure Appl. Math. 10, 1979, No. 8, 964 999.

[6] Musselman, J. R., Rutledge, G., Anning, N., Leighton, W., Thebault V.:Problems for Solution: 3891 3895, The Ameri-can Mathematical Monthly, Vol. 45, No. 9, Nov1938, 631 632.

[7] OEIS: The online Encyclopedia of Integer sequences. Published electronically at http://oeis.org, 2008.

[8] Shevelev, V. S., Necklaces and convex k-gons, Indian J. Pure Appl. Math. 35, no. 5,(2004), 629 638.

8


The 2-successive Eulerian numbers and unimodality

Assia Fettouma TEBTOUB1∗, Hacene BELBACHIR2, Zakaria CHEMLI3

1 USTHB University, RECITS LaboratoryPo. Box 32, El Alia, 16111, Algiers, Algeria

[email protected] USTHB University, RECITS Laboratory

Po. Box 32, El Alia, 16111, Algiers, [email protected]

3 Paris-Est Marne la Vallee University, LIGM Laboratory5 bd Descartes, Champs sur Marnes, Marne la Vallee, Paris, France

[email protected]

Abstract

Using a combinatorial approach, we introduce the 2-successive Eulerian numbers, we give the recurrencerelation. We establish the unimodality of sequences lying over diagonal rays of Eulerian triangle. Some combi-natorial identities are given.

Key words: Eulerian numbers; Recurrence relations; Log-concavity; Unimodality.

2010 Mathematics Subject Classification: 05A19, 11B37, 11B39, 11B73, 05A15.

1. IntroductionThe Eulerian numbers, denoted by, A(n, k), count the number of n-permutations with k − 1 descents. They satisfy thefollowing recurrence relation:

A(n, k) = (k + 1)A(n− 1, k + 1) + (n− k)A(n− 1, k).

For other properties we can see for instance and reference therein [5].It is important in combinatorics to know if the sequences lying over an arithmetical triangle are unimodal or not, specially incomputer science, the mode constitute the maximal value to consider in programming.A finite sequence (an)

nk=0 is unimodal if it increases to a maximum and then decreases. That is, there exists k such that

a1 ≤ a2 ≤ · · · ≤ ak ≥ ak+1 ≥ · · · ≥ an.

The sequence (an)nk=0 is log-concave (LC for short) if for k = 2, . . . , n− 1,

a2k ≥ ak+1ak−1, (1)

it is strictly log-concave (SLC for short) when (1) holds with the strict inequality.It is known that log-concavity implies unimodality [11].

∗Speaker9


The first result dealing with unimodality of Pascal’s triangle elements is due to Tanny and Zuker [9], who showed that thesequence of terms

(n−kk

)(k = 0, . . . , bn/2c) is unimodal. They also investigated for α the unimodality of the sequence(

n−αkk

)in [8, 10]. Belbachir and al, also treat in [1] some cases of binomial sequence

(n−αkβk

). In [2], Belbachir and Szalay

proved that any ray crossing Pascal’s triangle provide Unimodal sequence.In second kind’s Stirling triangle, Harper in [7], showed that

∑k

nk

xk has only real roots, Canfield [6] showed that the

sequence(nk

)k

is unimodal for a fixed n with at most two consecutive modes. In [3] Belbachir and Tebtoub proved that

the sequence(

n−kk

)k

is strictly log-concave and unimodal with at most two consecutive modes, they also investigated for

α the unimodality of the sequence(

n−αkk

)k

in [4].By analogy to these works, our main interest is to examine combinatorial sequences connected to arithmetical triangles as theEulerien triangle.

2. Main resultsThe 2-successive Eulerian numbers denoted, A2(n, k), satisfy the following recurrence relation.

Theorem 1. For n ≥ 2k, we have

A2(n, k) = (k + 1)A2(n− 1, k) + (n− 2k)A2(n− 2, k − 1),

where A2(n, 0) = 1, n ≥ 0.

The 2-successive Eulerian numbers are linked to the Eulerian’s triangle elements as follows

Theorem 2. For n ≥ 2k, we haveA2(n, k) = A(n− k, k).

Corollary 2.1. For n ≥ 2k, we have

A2(n, k) =k∑

i=0

(−1)i(n− k + 1

i

)(k − i+ 1)(n−k).

Corollary 2.2. For n ≥ 2k, we have

A2(n, n− k − 1) =k∑

i=0

(−1)k−i(k

i

)ik.

We will also, give a combinatorial interpretation of the A2(n, k), and will establish the unimodality of the sequence(A2(n, k))k.

References[1] H. Belbachir, F. Bencherif, L. Szalay, Unimodality of certain sequences connected with binomial coefficients, J. Integer

Seq. Vol. 10, Art. 07.2.3 (2007).

[2] H. Belbachir, L. Szalay, Unimodal rays in the regular and generalized Pascal triangles, J. of Integer Seq. 11 (2008),Article. 08.2.4.

[3] H. Belbachir, A. F. Tebtoub, Les nombres de Stirling associes avec succession d’ordre 2, nombres de Fibonacci-Stirlinget unimodalite, C. R. Acad. Sci. Paris, Ser. I (2015).

[4] H. Belbachir, A. F. Tebtoub, The t-successive associated Stirling numbers, t-Fibonacci Stirling numbers and Unimodal-ity, submitted.

10


[5] M. Bona, Combinatorics of Permutations, Chapman & Hall/CRC, 2004.

[6] E. R. Canfield, Location of the maximum Stirling number(s) of second kind, Studies in App Math, 59, (1978), 83–93.

[7] L. H. Harper. Stirling behaviour is asymptotically normal. Ann. Math. Stat.38 (1967), p. 410–414.

[8] S. Tanny, M. Zuker, Analytic methods applied to a sequence of binomial coefficients, Discrete Math. 24 (1978) 299-310.

[9] S. Tanny, M. Zuker, On unimodality sequence of binomial coefficients, Discrete Math. 9 (1974) 79–89.

[10] S. Tanny, M. Zuker, On unimodal sequence of binomial coefficients II, Combin. Inform. System Sci. 1 (1976) 81–91.

[11] H. S. Wilf. Generatingfunctionology. Academic Press, 1994.

11


The boson normal ordering problem:Combinatorial interpretation

Ali Chouria1∗, Imad Eddine Bousbaa2

1 Universite de Rouen, Laboratoire LITIS - EA 4108Av de l’Universite, BP 876801 Saint-Etienne-du-Rouvray Cedex, France

2 USTHB, Faculty of Mathematics, RECITS Laboratory, DG-RSDTBP 32, El Alia, 16111, Bab Ezzouar, Algiers, Algeria

[email protected] [email protected]

Abstract

In this work we are concerned with the one mode boson creation a+ and annihilation a operators satisfyingthe relation [a, a+] = 1. We are interested in the combinatorial interpretation of a+ and a in the boson normalordering problem using new object coled B-Diagram. In the aim to explain the computations, Blasiak et al.constructed a combinatorial Hopf algebra on these structures [1, 2]. We give a combinatorial interpretationof the B-diagrams using set partitions and set partitions into lists, . . . to find as a result Stirling, r-Stirling,rp-Stirling numbers and other series of numbers.

Key words: Boson operators, Normal ordering problem, Stirling and Bell numbers.

2010 Mathematics Subject Classification: 11B73, 05A19.

1. IntroductionThe creation and annihilation operators, denoted respectively by a† and a are non-Hermitian operators acting on the Fockspace by

a†∣n⟩ = √n + 1∣n + 1⟩ and a∣n⟩ = √

n∣n − 1⟩. (1)

These operators generate the Heisenberg algebra abstractly defined as the free algebra generated by the elements a, a† andquotiented by the relation [a, a†] = 1. (2)

Since J. Katriel pioneered the study of the combinatorial aspects of the normal ordering problem [4], several papers proposedcombinatorial and algebraic interpretations of these computations. A first example is ordering of the power of the numberoperator:

(a+a)n = n∑k=1S(n, k)(a+)kak (3)

where S(n, k) are the Stirling numbers of the second kind.set of n elements into k nonempty subsets.In this paper, we first describe the combinatorial structure of B-diagrams. In the second section we give new interpretation

in terms of Stirling, r-Stirling and rp-Stirling numbers . . . using these combinatorial objects.

∗Speaker12


2. Main results

2.1. B-diagrams: definition and first examples

Let us first introduce notation. Let #E denote the cardinal of the set E, Ja, bK ∶= a, a + 1, . . . , b − 1, b for any pairs ofintegers a ≤ b, E = E′ ⊎E′′ when E = E′ ∪E′′ and E′ ∩E′′ = ∅.

Definition 1. A B-diagram is a 5-tuple G = (n,λ,E↑,E↓,E) such that n is an integer, λ = [λ1, . . . , λn] with λi ∈ N ∖ 0for each i, E↑,E↓ ⊂ J1, λ1 + ⋯ + λnK, E ⊂ (a, b) ∶ a ∈ E↑, b ∈ E↓, v(a) < v(b) where v ∶ J1, λ1 + ⋯ + λnK Ð→ J1, nK isdefined by v(k) = i if k ∈ Jλ1 + ⋯ + λi−1 + 1, λ1 + ⋯ + λiK, finally for each a ∈ E↑ and b ∈ E↓, the sets (a, c) ∶ (a, c) ∈ Eand (c, b) ∶ (c, b) ∈ E contain at most one element.

Graphically, a B-diagram can be represented as a graph with n vertices. The vertex i has exactly λi inner (resp. outer)half-edges labeled by Jλ1 +⋯+ λi−1 + 1, λ1 +⋯+ λiK. The inner (resp. outer) half edges which does not belong to E↓ (resp.E↑) are denoted by ×. An element of E is represented by an edge relying an outer half edge a of a vertex i to an inner halfedge b of a vertex j with i < j.Example 1. For instance, the B-diagram G = (3, [3,1,2], J1,5K, J1,6K,(1,6), (2,4), (4,5)) is represented in Figure 1

11

1

2

2 3

3

24

4

35

5 6×

6

Figure 1: The B-diagram (3, [3,1,2],1,2,3,4,5,1,2,3,4,5,6,(1,6), (2,4), (4,5))If G = (n,λ,E↑,E↓,E) we define some tools in the aim to manipulate more easily the B-diagrams

1. the number of vertices is ∣G∣ = n, of half-edges ω(G) = λ1 +⋯ + λn and of edges is τ(G) = #E,

2. the number of non used outer (resp. inner) half edges, h↑(G) = ω(G) − #e↑(E) (resp. h↓(G) = ω(G) − #e↓(E))where e↑(a, b) = a (resp. e↓(a, b) = b),

3. the set of non used outer (resp. inner) non cut half edges, H↑f(G) = E↑ ∖ e↑(E) (resp. H↓f(G) = E↓ ∖ e↓(E)) ,

4. the number of non used outer (resp. inner) non cut half edges, h↑f(G) = #H↑f(G) (resp. h↓f(G) = #H↓f(G)), thenumber of non used cut half edges, hc(G) = h↑(G) − h↑f(G) + h↓(G) − h↓f(G),

5. the set of half edges associated to a vertex i, H(i) = Jλ1 +⋯ + λi−1 + 1, λ1 +⋯ + λiK,

6. the set of outer (resp. inner) non cut half edges associated to a vertex i, H↑f(i) = E↑ ∩ Jλ1 +⋯+λi−1 + 1, λ1 +⋯+λiK(resp. H↓f(i) = E↓ ∩ Jλ1 +⋯ + λi−1 + 1, λ1 +⋯ + λiK),

7. we will also use the map v of Definition 1; this map will be denoted vG in case of ambiguity.

Example 2. Consider the B-diagram G = (n,λ,E↑,E↓,E) represented in Figure 1. We have ∣G∣ = 3, ω(G) = 6, andτ(G) = 3. We also have h↑(G) = 2, h↓(G) = 3, h↑f(G) = 2, h↓(G) = h↓f(G) = 3, and hc(G) = 1 since H↑f(G) = 3,5 and

H↓f(G) = 1,2,3. Furthermore H(1) = H↑f(1) = H↓f(1) = 1,2,3, H(2) = H↑f(2) = H↓f(2) = 4, H(3) = H↓f(3) =5,6, and H↓f(3) = 5. Finally, v(1) = v(2) = v(3) = 1, v(4) = 2, and v(5) = v(6) = 3.

13


2.2. Combinatorial interpretation

Let dp,q denote the number of B-diagram G such that ω(G) = p and h↑f(G) = q.We find the induction

dp,q = p∑i=1

i∑j=0

i∑k=0

j∑=0 `!(j

`)(q − k + `

`)(ij)( ik)dp−i,q−k+j , (4)

with the special cases d0,0 = 1 and dp,q = 0 if p, q ≤ 0 and (p, q) ≠ (0,0). Indeed, we obtain a diagram with p half edges andq non used outer non cut half edges by branching ` inner half edges of an elementary B-diagram with i half edges, j ≤ i nonused inner non cut half edges, and k ≤ i non used outer non cut half edges to a B-diagram with p − i half edges and q − k + jnon used outer non cut half edges. The number of ways to do that is `!(j

`)(q−k+`

`)(i

j)(i

k). Indeed, the factor `! is the number

of permutation of the inner half edges of the elementary B-diagram, the factor (j`) corresponds to the choice of ` half edges

in the set of the j possible non cut half edges, the coefficients (q−k+``

) is the number of ways to choose ` outer half edges inthe second B-diagram, and the factors (i

j)(i

k) is the number of ways to select j inner half edges and k outer half edges in i

half edges. The number αp of B-diagrams having exactly p half edges is given by αp = ∑pq=0 dp,q.

The first values are 1,4,36,372,4372,57396,828020,12962164,218098356, . . .

Now, we give a combinatorial interpretation of the B-diagrams using set partitions and set partitions into lists. Let G =(n, [λ1, . . . , λn],E↑,E↓,E) be a B-diagram, we split each vertex i into λi vertex such that each one has one inner and oneouter (see Figure 2), we denote by Ri(G) the set of those elements (#Ri = λi). The configuration of the new vertices canbe regrouped into four subsets, denoted A, M+, M− and S with A ∩M+ ∩M− ∩ S = ∅ and V (G) = A ∪M+ ∪M− ∪ S =⋃n

i=1Ri(G), such that

• A = E↑ ∩E↓, which corresponds to the elements having valid inner and outer.

• M+ = a ∈ E↑ ∧ a ∉ E↓, which corresponds to the elements having valid outer and stump inner.

• M− = a ∈ E↓ ∧ a ∉ E↑, which corresponds to the elements having valid inner and stump outer.

• S = a ∉ E↑ ∧ a ∉ E↓. which corresponds to the elements having stump inner and outer.

Definition 2. Let S(G,m) be the number of B-diagram G having exactly m non used inner half-edges and B(G) =∑k S(G,k) is the number of B-diagram G.

Definition 3. Let D be a diagram and k an integer, we define The G = V (G)k

as the number of partitions of the set V (G)(or V ) into k parts such that :

• the elements of each subset Ri(G) (1 ≤ i ≤ n) are in distinct parts,

• the elements of the subset S are singletons,

• the elements of the subset M+ are maximal in there parts,

• the elements of the subset M− are minimal in there parts.

We define B(G) as B(G) ∶= ∑k V (G)k

.Proposition 1.

S(G,m) = V (G)m +#(S ∪M−) (5)

Example 3. We consider the B-diagram defined by G = 4, [1,3,2,1],1,3,4,6,1,3,6,7,(1,6), (3,7), correspond-ing to a set V (G) = 1, . . . ,7 with A = 1,3,6, M+ = 7, M− = 4, S = 2,5, R1 = 1, R2 = 2,3,4, R3 = 5,6and R4 = 7. G can be interpreted as the set partition: 1,623,745.

14


r-Stirling and (r1, . . . , rp)-Stirling numbers of the second kind

Mihoubi and Maamra [6] defined the (r1, . . . , rp)-Stirling numbers of the second kind, denoted nk

r1,...,rp, as the number

of partitions of n elements into k parts such that the elements of each subset Ri are in distinct parts. The (r1, . . . , rp)-Bellnumbers, denoted Br1,...,rp(n), count the number of partition of n elements under the same condition.

Br1,...,rp =∑k

nk

r1,...,rp

. (6)

Let us consider the B-diagram Br1,...,rn = (n, [r1, . . . , rn],1, . . . , r1 +⋯+ rn,1, . . . , r1 +⋯+ rn,E) correspondingto the set representation V = 1, . . . , r1 +⋯ + rn with A = V and M+ =M− = S = ∅.

Proposition 2. We have V (Br1,...,rn)k

= r1+⋯+rn

k

r1,...,rnand B(V ) = Br1,...,rn(r1 +⋯ + rn).

Example 4. We consider the B3,2,3,2,1 corresponding to the set partition 1,623,104,957811

11

1

2

2 3

3

24

4 5

5

36

6

7

7 8

8

49

9 10

10 511

11

11

1

22

2

33

344

4

55

5

66

6

77

7

88

8 99

9

1010

10

1111

11

Figure 2: The B-diagram (5, [3,2,3,2,1], J1,10K, J1,10K,(1,5), (4,6))We denoted by n

k

r, the r-Stirling numbers of the second kind [3]. Let us consider the B-diagramBr1,...,rn = (n, [r1,1, . . . ,1],1, . . . , r + n − 1,1, . . . , r + n − 1,E) corresponding to the set representation V = 1, . . . , r + n − 1 with A = V ,

M+ =M− = S = ∅, R1 = 1, . . . , r and Rj = r + j − 1 (2 ≤ j ≤ n).

Proposition 3. We have V (Br1,...,rn)k

= n+r−1k

r. and B(V ) = Bn+r−1,r.

Example 5. We consider theR = (5, [3,1,1,1,1], J1,7K, J1,7K,(1,5), (4,6)) corresponding to the set partition 1,5234,67.

References[1] Blasiak Pawel, Combinatorics of boson normal ordering and some applications, arXiv preprint quant-ph/0507206, 2005.

[2] Blasiak Pawel and others, Combinatorial algebra for second-quantized Quantum Theory,Advances in Theoretical andMathematical Physics, 14(4),1209–1243, 2010.

[3] A. Z Broder, The r-Stirling numbers,Discrete Mathematics 49(3), 241-259, 1984.

[4] J. Katriel, Combinatorial aspects of Boson algebra, Lettere al Nuovo Cimanto., 1974, 10(13), 565–567

[5] I. Mezo, The r-Bell numbers,Journal of Integer Sequences, 14(1), 2011.

[6] M. Mihoubi and M. Maamra, The (r1, . . . , rp)-Stirling numbers of the second kind,Integers, 12,1047–1059, 2012.

15


Le Meta-Storming, un outil pour la mise en œuvrecollaborative des Métaheuristiques

Mohammed Yagouni1∗, Haoi An Le Thi2,

1 LaROMAD, USTHBBP 32 El Alia - Bab Ezzouar, Alger 16111, Algérie

[email protected] Université de Lorraine, LITA

Ile du Saulcy 57445 Cedex, Metz, [email protected]

RésuméL’essentiel de cette contribution réside dans la conception et la construction d’un système basé

sur la collaboration et la coopération des métaheuristiques pour la résolution des problèmes difficilesde l’optimisation et particulièrement ceux de l’optimisation combinatoire. Nous avons conçu et misen oeuvre une approche coopérative et collaborative en vue essentiellement de fédérer et mettre àprofit la complémentarité des propriétés désirables, telle que l’intensification et la diversification, enles regroupant dans une approche de résolution collective pour la résolution d’une même instance d’unproblème donné de l’optimisation combinatoire (POC).

Key words : Métaheuristiques, Brainstorming, Collaboration, Programmation Parallèle, Programmation Distri-buée.

1. IntroductionL’idée de concevoir cette approche d’optimisation est inspirée de la célèbre technique du Brainstorming. Unetechnique efficace de résolution collective des problèmes de sociétés, inventée en 1935 par Alex F. Osborn [1], leBrainstorming est, en effet, une technique efficace utilisée à ce jour dans le management opérationnel des Entrepriseset la recherche participative de solutions à un large éventail de problèmes.L’apport de l’approche proposée réside dans la transposition de la technique du Brainstorming à la prise encharge de la problématique de l’optimisation combinatoire, à travers l’adaptation de ses règles de déroulementet la mise en oeuvre de ses principes fondateurs tels qu’ils ont été imposés par son inventeur. En effet, en vuede résoudre un problème d’optimisation donné, le système obtenu, «Brainstorming des Métaheuristiques ou leMeta-Storming»[2], dans son mode opératoire, utilise plusieurs processus indépendants et parallèles, implémentantchacun une méthode ou une variante d’une méthode donnée, collaborent et coopèrent entre-eux, par des échangesasynchrones d’informations (solutions courantes) pour trouver la meilleure solution d’une même instance. Le Méta-Storming obtenu est un système où, un véritable équilibre entre la diversification de la recherche et l’intensificationdans les zones susceptibles de contenir une solution optimale sont assurées, voire renforcées dans le processus de∗Speaker 16


recherche. En effet, la diversification recherchée est prise en charge par exploration d’une manière concurrente desdivers sous-espaces de l’espace de recherche tout en assurant une intensification effective réalisée au niveau dechaque processeur. Une accélération du temps d’exécution du processus d’optimisation et de recherche globale estrendue possible par une implémentation parallèle et distribuée du système.

2. Le BrainstormingLe mot "Brainstorming" et la technique à laquelle il fait référence, ont été inventés en 1935 par Alex Faickney

Osborn. A l’origine, la méthode Brainstorming, est mise en œuvre en tant que mode efficace pour la conduitedes réunions de groupes au sein des Entreprises, notamment, pour trouver collectivement des solutions à desproblèmes complexes. La flexibilité de son adaptation et la simplicité de sa mise en œuvre sont autant d’ingrédientsqui ont contribués à son large succès, pour se voir aujourd’hui, décliné en un éventail de méthodes adaptéesdans divers domaines, allant des compagnies publicitaires, à la gouvernance participative dans des grands projetsd’aménagement du territoire et de protection de l’environnement, en passant par son utilisation dans la recherchede politiques d’investissement et autres projets budgétivores et de décisions stratégiques.

2.1. Le Méta-Storming ou le Brainstorming des métaheuristiquesLes notions et les concepts de base liés au Brainstorming ; les principes et les règles de la définition et de la

composition d’une équipe ainsi que les étapes complémentaires du déroulement d’une séance de Brainstormingsont détaillés dans [1,2]. Nous nous contentons dans ce travail de résumer essentiellement le schéma suivi dansla transposition de ce mode de réunion efficace au domaine des POCs, et sa mise en oeuvre pour la résolutiond’un POC, à l’exemple de l’archétype de cette classe de problèmes, en l’occurrence le Problème du Voyageur deCommerce (PVC). Le tableau 1 illustre la forte analogie entre le brainstorming et le processus d’optimisation.

Brainstorming OptimisationProblème à résoudre par Brainstorming Instance d’un POCIdée dans une réunion Solution (réalisable, partielle, complète)Evaluation d’une idée Evaluation d’une solution (fonction objectif)Modérateur Algorithme de TriParticipants(Assistants) Méthodes de résolutionDispositif de mémorisation Mémoire AdaptativeCombinaison et association d’idées Combinaison de solutions

Tab. 1 – Analogie entre le Brainstorming et le Processus d’Optimisation

Le schéma d’implémentation suivi pour la mise en oeuvre du Meta-Storming est celui présenté à la figure 1 suivante.

2.2. Mise en oeuvre de l’approche proposée pour le PVCL’approche proposée a été appliquée pour la résolution des instances du PVC. Les méthodes participantes

implémentées pour le PVC sont le Recuit Simulé, les Algorithmes Génétiques, la Recherche à Voisinages Variables.Leurs solutions initiales ont été obtenues à l’aide des heuristiques de construction de solutions (Plus Proche Voisin,Regret Maximum, les algorithmes d’insertions...). La mise en oeuvre s’est faite en suivant le schéma d’implémen-tation présenté à la figure 2 suivante.

3. Conclusions et PerspectivesNous avons présenté une approche distribuée collaborative pour résoudre des POCs. L’idée de collaboration

entre les métaheuristiques est inspirée de la technique du brainstorming, nous l’avons mise en oeuvre pour la

17


résolution du PVC et les résultats probants des premiers tests réalisés sur des benchmarks de la TSPLIB, et d’autresen cours, nous laissent escompter que ce schéma générique puisse être adapté à d’autres domaines d’optimisationtelle que l’optimisation non convexe et l’optimisation multiobjectif.

Références[1] M. Yagouni, An Hoai Le Thi, A Collaborative Metaheuristic Optimization Scheme : Methodological Issues, Ad-

vances in Intelligent Systems and Computing, Springer International Publishing Switzerland 282 : 3-14,.2014.[2] M. Yagouni, An Hoai Le Thi, Et si la collaboration était la solution ?, Roadef 2015.[3] A.F, Osborn, Your Creative Power, Lightning Source Incorporated,1948.

18


Reconstruction of digraphs up to complementation

Aymen Ben Amira1∗, Bechir Chaari2, Jamel Dammak3, Hamza Si Kaddour4

1 Sfax University, Laboratory: Dynamical Systems and CombinatoryFaculty of Sciences of Sfax BP 1171, 3000, Sfax, Tunisia

[email protected] Sfax University, Laboratory: Dynamical Systems and Combinatory

Faculty of Sciences of Sfax BP 1171, 3000, Sfax, [email protected]


[email protected] ICJ, Department of Mathematics, University of Lyon, University Claude-Bernard Lyon1

43 Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, [email protected]

Abstract

Two digraphs G = (V,E) and G′ = (V,E′) are isomorphic up to complementation if G′ is isomorphicto G or to the complement G of G. Let k be a nonnegative integer, G and G′ are k-hypomorphic up tocomplementation if for every k-element subset X of V , the induced sub-digraphs GX and G′X are isomorphicup to complementation. The digraphs G and G′ are (≤ k)-hypomorphic up to complementation if they aret-hypomorphic up to complementation for all integer t ≤ k. We conjecture that for k large enough, if G andG′ are (≤ k)-hypomorphic up to complementation then G′ is isomorphic up to complementation to G or tothe dual G∗ of G. Here we prove that this conjecture is true whenever the boolean sum U := G ⊕ G′ and thecomplement U are both connected, in this case the smallest value of k is 5.

Key words: Digraph, isomorphism, k-hypomorphy up to complementation, boolean sum, symmetric graph, tournament.

2010 Mathematics Subject Classification: 05C50; 05C60.

1. IntroductionThis work is about isomorphy and reconstruction of digraphs up to complementation. The case of symmetric digraphs wasstudied by J.Dammak, G.Lopez, M.Pouzet and H.Si Kaddour [2, 3].

A directed graph or simply digraph G consists of a finite and nonempty set V of vertices together with a prescribedcollection E of ordered pairs of distinct vertices, called the set of the arcs of G. Such a digraph is denoted by (V (G), E(G))or simply (V,E). Given a digraph G = (V,E), to each nonempty subset X of V associate the subdigraph (X,E∩ (X×X))of G induced by X denoted by GX . Given a proper subset X of V , GV−X is also denoted by G − X , and by G − vwhenever X = v. With each digraph G = (V,E) associate its dual G∗ = (V,E∗) and its complement G = (V,E) defined

∗Aymen Ben Amira19


as follows. Given x 6= y ∈ V, (x, y) ∈ E∗ if (y, x) ∈ E, and (x, y) ∈ E if (x, y) 6∈ E.Let G = (V,E) be a digraph, for x 6= y ∈ V , x −→

Gy or y ←−

Gx means (x, y) ∈ E and (y, x) /∈ E; x

Gy means

(x, y) ∈ E and (y, x) ∈ E; x . . .Gy means (x, y) /∈ E and (y, x) /∈ E.

Two interesting types of digraphs are symmetric graphs and tournaments. A digraph G = (V,E) is a symmetric graph orgraph (resp. tournament) whenever for x 6= y ∈ V , x

Gy or x . . .

Gy (resp. x −→

Gy or y −→

Gx). If G = (V,E) is a

symmetric graph, each arc (x, y) of G is identified with the pair x, y and is called an edge of G.Given two digraphsG = (V,E) andG′ = (V ′, E′), a bijection f from V onto V ′ is an isomorphism fromG ontoG′ providedthat for any x, y ∈ V , (x, y) ∈ E if and only if (f(x), f(y)) ∈ E′. The digraphs G and G′ are isomorphic, which is denotedby G ' G′, if there exists an isomorphism from one onto the other, otherwise G 6' G′.Given two digraphs G and G′ on the same vertex set V . They are equal up to complementation if G′ = G or G′ = G. Let kbe an integer with 0 < k < |V |, the digraphs G and G′ are k-hypomorphic if for every k-element subset X of V , the inducedsubdigraphs GX and G′X are isomorphic. The digraphs G and G′ are (≤ k)-hypomorphic if they are t-hypomorphic forall integer t ≤ k. The digraphs G and G′ are isomorphic up to complementation if G′ is isomorphic to G or G (resp. G orG∗). Let k be a positive integer, the digraphs G and G′ are k-hypomorphic up to complementation if for every k-elementsubset X of V , the induced subdigraphs GX and G′X are isomorphic up to complementation. The digraphs G and G′ are(≤ k)-hypomorphic up to complementation if they are t-hypomorphic up to complementation for all integer t ≤ k.

The symmetric graph Pn is defined in the following manner. For i 6= j ∈ 0, 1, . . . , n − 1, vi, vj is an edge of Pn

when |i− j| = 1. Thus Pn := v0 v1 . . . vn−2 vn−1.A path is a symmetric graph isomorphic to Pn. A cycle is a symmetric graph isomorphic to Cn := (V (Pn), E(Pn) ∪v0, vn−1) = vn−1 v0 v1 . . . vn−2 vn−1.We define the digraph

−→Pn by, for i 6= j ∈ 0, 1, . . . , n−1, vi −→−→

Pnvj when j = i+1. Thus

−→Pn := v0 −→ v1 −→ . . . −→

vn−2 −→ vn−1. We denote by directed path or oriented path a digraph isomorphic to−→Pn and by directed cycle or oriented

cycle a digraph isomorphic to−→Cn := (V (

−→Pn), E(

−→Pn) ∪ (vn−1, v0))=vn−1 −→ v0 −→ v1 −→ . . . −→ vn−2 −→ vn−1,

and−→P f

n (resp.−→Cf

n) is obtained from−→Pn (resp.

−→Cn) by switching the void pairs by the full pairs. Thus

−→P f

n = (−→Pn)∗ and−→

Cfn = (

−→Cn)∗.

Dammak, Lopez, Pouzet and Si Kaddour [2, 3] proved that, for symmetric digraphs of size n, k-hypomorphy up tocomplementation implies isomorphy up to complementation if 4 ≤ k ≤ n− 3, in particular if 4 ≤ k ≤ n− 4 they obtainedthe equality up to complementation. The case k = n− 3, needs the following result of Pouzet, Si Kaddour and Trotignon [4]on the boolean sum of two symmetric graphs which are 3-hypomorphic up to complementation.

Theorem 1. [4] If G and G′ are two symmetric graphs, 3-hypomorphic up to complementation and |V (G)| ≥ 10, then theconnected components of U := G⊕G′, or of its complement U , are cycles of even length or paths.

2. Main resultsOur main result is the following.

Theorem 2. Let G and G′ be two digraphs on the same set V of n ≥ 4 vertices such that G and G′ are (≤ 5)-hypomorphicup to complementation. Let U := G⊕G′. If U and U are connected, then G′ is isomorphic up to complementation to G orG∗. Moreover if G′ and G are not chains then G′ is equal up to complementation to G or G∗

To prove this, we establish first the following result.

Theorem 3. Let G and G′ be two digraphs on the same set V of n ≥ 4 vertices such that G and G′ are (≤ 5)-hypomorphicup to complementation. Let U := G⊕G′. If U and U are connected, then one of the following holds:

1. G and G′ are two chains.

2. G ' −→Pn or G ' −→Cn, and G′ = G∗.

3. G ' −→Pn or G ' −→Cn, and G′ = G∗.

4. G '−→P f

n or G '−→Cf

n , and G′ = G∗.

5. G '−→P f

n or G '−→Cf

n , and G′ = G∗.

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From Theorem, 3 we deduce trivially the following result for digraphs which is similar to Theorem 1.

Theorem 4. Let G and G′ be two digraphs on the same set V of n ≥ 4 vertices such that G and G′ are (≤ 5)-hypomorphicup to complementation and U := G ⊕G′. If U and U are connected and G is not a total order, then U or U is a cycle or apath.

References[1] J. Dammak, La (-5)-demi-reconstructibilite des relations binaires connexes finies, Proyecciones. 22 (2003), no. 3, 181–

199.

[2] J. Dammak, G. Lopez, M. Pouzet and H. Si Kaddour, Hypomorphy of graphs up to complementation. JCTB, Series B99 (2009), no. 1, 84–96.

[3] J. Dammak, G. Lopez, M. Pouzet and H. Si Kaddour, Boolean sum of graphs and reconstruction up to complementation.Advances in Pure and Applied Mathematics 4 (2013), 315–349.

[4] M. Pouzet, H. Si Kaddour and N. Trotignon, Claw freeness, 3-homogeneous subsets of a graph and a reconstructionproblem, Contributions Discrete Mathematics, 6 (2011), 92–103.

21


Packing coloring of some undirected and oriented coronae graphs

Daouya LAICHE1∗, Isma BOUCHEMAKH2 Eric SOPENA3

1 USTHB, Faculty of Mathematics, L’IFORCE LaboratoryB.P. 32 El-Alia, Bab-Ezzouar, Algiers, Algeria

[email protected] USTHB, Faculty of Mathematics, L’IFORCE Laboratory

B.P. 32 El-Alia, Bab-Ezzouar, Algiers, Algeriaisma [email protected]

3 Univ. Bordeaux, LaBRIUMR5800, F-33400 Talence, France

[email protected]

Abstract

The packing chromatic number χρ(G) of a graph G is the smallest integer k such that its set of verticesV (G) can be partitioned into k disjoint subsets V1, ..., Vk, in such a way that every two distinct vertices in Viare at distance greater than i in G for every i, 1 ≤ i ≤ k. For a given integer p ≥ 1, the generalized coronaG pK1 of a graph G is the graph obtained from G by adding p degree-one neighbors to every vertex of G. Inthis presentation, we determine the packing chromatic number of generalized coronae of paths and cycles.Moreover, by considering digraphs and the (weak) directed distance between vertices, we get a natural extensionof the notion of packing coloring to digraphs. We then determine the packing chromatic number of orientationsof generalized coronae of paths and cycles.

Key words: Packing coloring; Packing chromatic number; Corona graph; Path; Cycle.

2010 Mathematics Subject Classification: 05C15,05C70.

1. IntroductionAll the graphs we considered are simple and loopless. For an undirected graph G, we denote by V (G) its set of vertices andby E(G) its set of edges. The distance dG(u, v), or simply d(u, v), between vertices u and v in G is the length (number ofedges) of a shortest path joining u and v. The diameter of G is the maximum distance between two vertices of G. We denoteby Pn the path of order n and by Cn, n ≥ 3, the cycle of order n.The corona GK1 of a graph G is the graph obtained fromG by adding a degree-one neighbor to every vertex of G. We call such a degree-one neighbor a pendant vertex or a pendantneighbor. More generally, for a given integer p ≥ 1, the generalized corona G pK1 of a graph G is the graph obtainedfrom G by adding p pendant neighbors to every vertex of G.Packing coloring has been introduced by Goddard, Hedetniemi, Hedetniemi, Harris and Rall [1] under the name broadcastcoloring and has been studied by several authors in recent years.

∗Speaker

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2. Main results

2.1. Packing coloring of some undirected coronae graphs

We study in this section coronae of paths and cycles. We first determine the packing chromatic number of coronae of paths.Note that any corona Pn K1 is also a caterpillar of length n.

Theorem 1. The packing chromatic number of the corona graph Pn K1 is given by:

χρ(Pn K1) =

2 if n = 1,3 if n ∈ 2, 3,4 if 4 ≤ n ≤ 9,5 if n ≥ 10.

In [4], William, Roy and Rajasingh proved that χρ(Cn K1) ≤ 5 for every even n ≥ 6. We complete their result asfollows:

Theorem 2. The packing chromatic number of the corona graph Cn K1 is given by:

χρ(Cn K1) =

4 if n ∈ 3, 4,5 if n ≥ 5.

When considering generalized coronae of paths or cycles, the following proposition is useful:

Proposition 1. Let Pn = x1 . . . xn, n ≥ 2, be a path and Pn pK1, p ≥ 1, be a generalized corona of Pn. Any packingcoloring π of Pn pK1 with π(xi) = 1 for some vertex xi must use at least p+ 3 colors if 2 ≤ i ≤ n− 1, or at least p+ 2colors if i ∈ 1, n.

Similarly, if Cn pK1, p ≥ 3, is a generalized corona of Cn = y1 . . . yn, then any packing coloring π′ of Cn pK1

with π′(yi) = 1 for some vertex yi must use at least p+ 3 colors.

The value of the packing chromatic number of generalized coronae of paths Pn pK1 with p ≥ 4 is given by thefollowing theorem:

Theorem 3. Let Pn pK1, p ≥ 4, be a generalized corona of the path Pn. Then we have:

χρ(Pn pK1) =

2 if n = 1,3 if n = 2,4 if n ∈ 3, 4,5 if 5 ≤ n ≤ 8,6 if 9 ≤ n ≤ 34,7 otherwise.

The value of the packing chromatic number of generalized coronae of paths Pn pK1, when p ∈ 2, 3, is given by thenext two results. We will see that the maximum value of the packing chromatic number of such graphs is 6, slightly betterthan the bound given in Theorem 3. This is due to the fact that the number of pendant vertices is now bounded by 3, whichallows us to use color 1 for coloring the vertices of the path Pn.

Theorem 4. Let Pn 2K1 be a generalized corona of the path Pn. Then we have:

χρ(Pn 2K1) =

2 if n = 1,3 if n = 2,4 if n ∈ 3, 4,5 if 5 ≤ n ≤ 11,6 otherwise.

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Theorem 5. Let Pn 3K1 be a generalized corona of the path Pn. Then we have:

χρ(Pn 3K1) =

2 if n = 1,3 if n = 2,4 if n ∈ 3, 4,5 if 5 ≤ n ≤ 8,6 otherwise.

We now turn to generalized coronae of cycles Cn pK1. When p ≥ 4, we have the following (note the particular casewhen n = 11):

Theorem 6. Let Cn pK1, p ≥ 4, be a generalized corona of the cycle Cn. Then we have:

χρ(Cn pK1) =

4 if n = 3,5 if n = 4,6 if n ∈ 5, 6,8 if n = 11,7 otherwise.

Finally, for p = 3, we have the following:

Theorem 7. Let Cn 3K1 be a generalized corona of the cycle Cn. Then we have:

χρ(Cn 3K1) =

4 if n = 3,5 if n = 4,7 if n ∈ 7, . . . , 13, 15, . . . , 22, 24, . . . , 27, 30, . . . , 36, 39, 40, 41

∪ 45, 47, . . . , 50, 53, 54, 55, 59, 62, 63, 64, 68, 77, 78, 91,6 otherwise.

2.2. Packing coloring of some oriented coronae graphs

Proposition 2. For every orientation−→G of an undirected graph G, χρ(

−→G) ≤ χρ(G).

Note also that Proposition is still valid for oriented graphs:

Proposition 3. If−→H is a subgraph of

−→G , then χρ(

−→H ) ≤ χρ(

−→G).

The characterization of oriented graphs with packing chromatic number 2 is given by the following result:

Proposition 4. For every orientation−→G of an undirected graph G, χρ(

−→G) = 2 if and only if (i) G is bipartite and (ii) one

part of the bipartition of G contains only sources or sinks in−→G .

We now determine the packing chromatic number of orientations of paths, cycles, and coronae of paths and cycles.For oriented paths, we have the following:

Theorem 8. Let−→Pn be any orientation of the path Pn = x1 . . . xn. Then, for every n ≥ 2, 2 ≤ χρ(

−→Pn) ≤ 3. Moreover,

χρ(−→Pn) = 2 if and only if one part of the bipartition of Pn contains only sources or sinks in

−→Pn.

For oriented cycles, we have the following:

Theorem 9. Let−→Cn be any orientation of the cycle Cn = x0 . . . xn−1x0. Then, for every n ≥ 3, 2 ≤ χρ(

−→Cn) ≤ 4. Moreover,

(1) χρ(−→Cn) = 2 if and only if Cn is bipartite (that is, n is even) and one part of the bipartition contains only sources or

sinks in−→Cn.

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(2) χρ(−→Cn) = 4 if and only if

−→Cn is a directed cycle (all arcs have the same direction), n ≥ 5 and n 6≡ 0 (mod 4).

For orientations of generalized coronae of paths, we have the following:

Theorem 10. Let−→G be any orientation of a generalized corona Pn pK1, with p ≥ 1 and Pn = x1 . . . xn. Then, for every

n ≥ 1, 2 ≤ χρ(−→G) ≤ 3. Moreover, χρ(

−→G) = 2 if and only if one part of the bipartition of Pn pK1 contains only sources

or sinks in−→G .

Finally, for orientations of generalized coronae of cycles, we have the following:

Theorem 11. Let−→G be any orientation of a generalized corona Cn pK1, with p ≥ 1 and Cn = x0 . . . xn−1. Then, for

every n ≥ 3, 2 ≤ χρ(−→G) ≤ 4. Moreover,

(1) χρ(−→G) = 2 if and only if CnpK1 is bipartite (that is, n is even) and one part of the bipartition contains only sources

or sinks in−→G .

(2) χρ(−→G) = 4 if and only if either:

(2.1)−→Cn is a directed cycle, n ≥ 5 and n 6≡ 0 (mod 4), or

(2.2)−→G contains the oriented graph depicted in Figure ?? as a subgraph, or

(2.3) n ≡ 0 (mod 4) and there exists a vertex xi, 0 ≤ i ≤ n − 1, such that the paths xixi+1xi+2xi+3 andxi+4 . . . xi−1(indices are taken modulo n) are both directed paths, but in opposite direction.

Proposition 5. Let−→Pn be any orientation of the path Pn = x1 . . . xn of odd length n− 1 and S be a set of sources or sinks

in−→Pn with odd indices not containing x1. Consider the coloring π of

−→Pn produced by SCP with (c, c′) = (1, α) for some

α ∈ 2, 3 and S. Then we have:

(i) π(xn) = α if |S| is even (resp. odd) and n ≡ 2 (mod 4) (resp. n ≡ 0 (mod 4)),

(ii) π(xn) = 5− α otherwise.

3. ConclusionIn this presentation, we have determined the packing chromatic number of coronae and generalized coronae of paths andcycles. We also extended to digraphs the notion of packing coloring and determined the packing chromatic number oforientations of such graphs.

References[1] W. Goddard, S.M. Hedetniemi, S.T. Hedetniemi, J.M. Harris and D.F. Rall. Broadcast chromatic numbers of graphs.

Ars Combinatoria 86 (2008), 33-49.

[2] D. Lache. Sur les nombres broadcast chromatiques. Magister thesis, University of Sciences and Technology HouariBoumediene, Algeria, 2010 (in french).

[3] C. Sloper. An eccentric coloring of trees. Australasian Journal of Combinatorics 29 (2004), 309-321.

[4] A. William, I. Rajasingh and S. Roy. Packing chromatic number of enhanced hypercubes. International Journal ofMathematics and its Applications 2 (2014), no. 3, 1-6.

[5] A. William, S. Roy and I. Rajasingh. Packing chromatic number of cycle related graphs. International Journal of Math-ematics and Soft Computing 4 (2014), no.1, 27-33.

25


Some topics in Discrete Mathematics

Fairouz Tchier∗

Mathematics department,King Saud University, Riyadh 11495, Saudi Arabia

[email protected]

Abstract

The importance of relations is almost self-evident. Science is, in a sense, the discovery of relations betweenobservables. Zadeh has shown the study of relations to be equivalent to the general study of systems (a systemis a relation between an input space and an output space). Goguen, J.A. [6]

Key words: Relational algebra, nondeterministic semantics, demonic calculus, fuzzy calculus, demonic calculus.

2010 Mathematics Subject Classification: 18C10 ,18C50, 68Q55 ,68Q65,03B70,06B35

1. Introduction

The calculus of relations has been an important component of the development of logic and discrete mathematics since themiddle of the nineteenth century. George Boole, in his ”Mathematical Analysis of Logic” [2], initiated the treatment of logicas part of mathematics, specifically as part of algebra. Quite the opposite conviction was put forward early this century byBertrand Russell and Alfred North Whitehead in their Principia Mathematica [?]: that mathematics was essentially groundedin logic. The logic is developed in two streams. On the one hand algebraic logic, in which the calculus of relations playeda particularly prominent part, was taken up from Boole by Charles Sanders Peirce, who wished to do for the ”calculus ofrelatives” what Boole had done for the calculus of sets, Peirce’s work was in turn taken up by Schroder in ”Algebra undLogik der Relative” [10]. Schroder’s work, however, lay dormant for more than 40 years, until revived by Alfred Tarski in hisseminal paper ”On the calculus of binary relations” [11]. Tarski’s paper is still often referred to as the best introduction to thecalculus of relations. It gave rise to a whole field of study, that of relation algebras, and more generally Boolean algebras withoperators. This important work defined much of the subsequent development of logic in the 20th century, completely eclipsingfor some time the development of algebraic logic. In this stream of development, relational calculus and relational methodsappear with the development of universal algebra in the 1930’s, and again with model theory from the 1950’s onwards. Inso far as these disciplines in turn overlapped with the development of category theory, relational methods sometimes appearin this context as well. It is fair to say that the role of the calculus of relations in the interaction between algebra and logicis by now well understood and appreciated, and that relational methods are part of the toolbox of the mathematician and thelogician.

The main advantages of the relational formalization are uniformity and modularity. Actually, once problems in thesefields are formalized in terms of relational calculus, these problems can be considered by using formulae of relations, that is,we need only calculus of relations in order to solve the problems. This makes investigation on these fields easier.

∗Speaker

26


Demonic relational calculus In the context of software development, one important approach is that of developing pro-grams from specifications by stepwise refinement, see, e.g. [1, 13, 15].

One point of view is that a specification is a relation constraining the input-output (respectively, argument-result) behaviorof programs. Inspired by interpretation of relations as non-deterministic programs, demonic, angelic and erratic variants ofrelational operations have been studied. The demonic interpretation of non-deterministic turns out to closely correspond tothe concept of under-specification, and therefore the demonic operations are used in most refinement calculus.

Fuzzy Calculus Fuzzy ”As the complexity of a system increases, our ability to make precise and yet significant statementsabout its behavior diminishes until a threshold is reached beyond which precision and significance (or relevance) becomealmost mutually exclusive characteristics.” [?]

Let us consider characteristic features of real-world systems again: real situations are very often uncertain or vague ina number of ways. Due to lack of information the future state of the system might not be known completely. This type ofuncertainty has long been handled appropriately by probability theory and statistics. This Kolmogoroff-type probability isessentially frequentistic and bases on set-theoretic considerations. Koopman’s probability refers to the truth of statement andtherefore bases on logic. On both types of probabilistic approaches it is assumed, however, that the events (elements of sets)or the statements, respectively, are well defined. We shall call this type of uncertainty or vagueness stochastic uncertaintyby contrast to the vagueness concerning the description of the semantic meaning of the events, phenomena or statementsthemselves, which we shall call fuzziness. Fuzziness can be found in many areas of daily life, such as in engineering, inmedicine, in meteorology, in manufacturing [8]; and others. The first publications in fuzzy set theory by Zadeh [17] andGoguen [6, 7] show the intention of the authors to generalize the classical notion of a set. Zadeh writes:”The notion of afuzzy set provides a convenient point of departure for the construction of a conceptual framework which parallels in manyrespects the framework used in the case of ordinary sets, but is more general than the latter and, potentially, may prove to havea much wider scope of applicability, particularly in the fields of mathematics and computer science (pattern classification andinformation processing).

Fuzzy set theory provides a strict mathematical framework ( there is nothing fuzzy about fuzzy set theory) in whichvague conceptual phenomena can be precisely and rigorously studied. It can also be considered as a modeling language wellsuited for situations in which fuzzy relations, criteria, and phenomena exist. It will mean different things, depending on theapplication area and the way it is measured. In the meantime, numerous authors have contributed to this theory. In 1984as many as 4000 publications may already exist. The specialization of those publications conceivably increases, making itmore and more difficult for new comers to this area. Roughly speaking, fuzzy set is a formal theory which, when maturing,became more sophisticated and specified and was enlarged by original ideas and concepts as well as by ”embracing” classicalmathematical areas such as algebra, graph theory, topology, and so on by generalizing (fuzzifying) them. As a very powerfulmodeling language, that can cope with a large fraction of uncertainties of real-life situations. There are countless applicationsfor fuzzy logic. In fact, some claim that fuzzy logic is the encompassing theory over all types of logic. The items in this listare more common applications that one may encounter in everyday life, for example Temperature control (heating/cooling),Medical diagnoses, Predicting travel time, auto-Focus on a camera, predicting genetic traits and bus Time Tables.

References[1] ok Back, R. J. R. : On correct refinement of programs . J. Comput. System Sci., 23(1):49–68, 1981.

[2] Boole, G..: The Mathematical Analysis of Logic, Being an Essay Toward a Calculus of Deductive Reasoning. Macmil-lan, Cambridge, 1847.

[3] Desharnais, J. B. Moller, and F. Tchier. Kleene under a demonic star. 8th International Conference on AlgebraicMethodology And Software Technology (AMAST 2000), May 2000, Iowa City, Iowa, USA, Lecture Notes in ComputerScience, Vol. 1816, pages 355–370, Springer-Verlag, 2000.

[4] Desharnais, J., Belkhiter, N., Ben Mohamed Sghaier, S., Tchier, F., Jaoua, A., Mili, A. and Zaguia, N.: Embedding aDemonic Semilattice in a Relation Algebra. Theoretical Computer Science, 149(2):333–360, 1995.

[5] Dijkstra, E. W. and Feijn, W. : Predicate calculus and program semantics. Addison-Wesley, 1988.

[6] Goguen, J. A. . : L-fuzzy sets. JMAA 18, 145-174.

[7] Goguen, J. A. . : The logic of inexact concepts. Synthese 19, 325-373.

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[8] Mamdani, E. H. : ”Advances in the Linguistic Syntesis of Fuzzy Controllers In Fuzzy Reasoning and Its Applications”,Academic Press, London (1981).

[9] Schmidt, G. and Strohlein, T.: Relations and Graphs. EATCS Monographs in Computer Science. Springer-Verlag,Berlin, 1993.

[10] Strohlein, E.: Vorlesungen uber die Algebra der Logik(exacte Logik). Teubner, Leipzig, 1895.Vol.3, Algebra und Logikder Relative, part I,2ndedition published by Chelsea, 1966.

[11] Tarski, A.: On the calculus of relations. J. Symb. Log. 6, 3, 1941, 73–89.

[12] Tchier, F.: Semantiques relationnelles demoniaques et verification de boucles non deterministes. Theses of doctorat,Departement de Mathematiques et de statistique, Universite Laval, Canada, 1996.

[13] Tchier, F.: Demonic semantics by monotypes. International Arab conference on Information Technology(Acit2002),University of Qatar, Qatar, 16-19 December 2002.

[14] Tchier, F.: Demonic relational semantics of compound diagrams. In: Jules Desharnais, Marc Frappier and WendyMacCaull, editors. Relational Methods in computer Science: The Quebec seminar, pages 117-140, Methods Publishers2002.

[15] Tchier, F.: While loop demonic relational semantics monotype/residual style. 2003 International Conference onSoftware Engineering Research and Practice (SERP03), Las Vegas, Nevada, USA, 23-26, June 2003.

[16] Tchier, F..: Demonic Semantics: using monotypes and residuals. IJMMS 2004:3 (2004) 135-160. (International Journalof Mathematics and Mathematical Sciences)

[17] Zadeh, L. A. .: Fuzzy Sets. Inform and Control 8 1965, 338–353.

[18] Zimmermann, H. J. .: Fuzzy Set Theory And its Applications, Second edition. Kluwer Academic Publishers, Boston,Dordrecht, London, 1990.

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The non −2-recognizable indecomposable tournaments

Nadia Amri1, Rim Ben Hamadou2, Imed Boudabbous3, ∗

1 Universite de Sfax,Faculte des Sciences de Sfax, Tunisie

[email protected] Universite de Gabes

Institut Suprieur des Sciences Appliquees et Technologie de Gabes, [email protected]

3 Universite de SfaxInstitut Preparatoire aux Etudes d’Ingenieurs de Sfax, Tunisie

[email protected]

Abstract

Given a tournament T = (V,A), a subset X of V is an interval of T provided that for any a, b ∈ Xand x ∈ V \ X , (a, x) ∈ A if and only if (b, x) ∈ A. For example, ∅, the single-vertex sets and V areintervals of T , called trivial intervals. A tournament whose intervals are trivial is indecomposable; otherwise,it is decomposable. Two tournaments are said to be equivalent if both are indecomposable or not. Given twotournaments T = (V,A) and T ′ = (V,A′), consider an integer k such that 0 < k <| V |. We say that T and T ′

are −k-equivalent if T −X and T ′−X are equivalent for every subset X of V with | X |= k. A tournamentT is said to be −k-recognizable if any tournament −k-equivalent to T is equivalent to it. We describe thenon −2-recognizable indecomposable tournaments.

Key words: Indecomposability graph, Interval, Indecomposable tournament.

2010 Mathematics Subject Classification: 05C50, 05C60.

1. IntroductionA digraph G is defined by a finite and nonempty vertex set V (G) and an arc set A(G), where an arc is an ordered pair of dis-tinct vertices. Such a digraph is denoted by (V (G), A(G)) or simply (V,A). The cardinality of G, denoted by | V | or v(G),is that of V (G). The digraph dual G? = (V,A?) associated to G is defined on V such that for any x 6= y ∈ V , (x, y) ∈ A?

if and only if (y, x) ∈ A. With each nonempty subset X of V we associate the subdigraph G[X] = (X,A ∩ (X ×X)) of Ginduced by X . Given a proper subset X of V , G[V \X] is also denoted by G−X , and by G− x whenever X = x.A digraph G is a tournament if | A(G) ∩ (v, w), (w, v) |= 1 for any v 6= w ∈ V (G). Let T = (V,A) be a tourna-ment. For any (distinct) vertices x, y of V , the notation x −→ y signifies that (x, y) ∈ A. For each subset X of V , setN−T (X) = y ∈ V : y −→ X and N+

T (X) = y ∈ V : X −→ y. For convenience, given x ∈ V , N−T (x) is denoted byN−T (x), and N+

T (x) is denoted by N+T (x).

∗Speaker

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Given a tournament T = (V,A), a subset X of V is an interval of T provided that for any a, b ∈ X and x ∈ V \ X ,(a, x) ∈ A if and only if (b, x) ∈ A. For example, ∅, the single-vertex sets and V are intervals of T , called trivial intervals.A tournament whose intervals are trivial is indecomposable[4],[5],[1]; otherwise, it is decomposable.

Let T = (V,A) be a tournament. Given a proper subsetX of V such that |X| ≥ 3 and T [X] is indecomposable, considerthe following subsets of V \X .

• Ext(X) is the set of v ∈ V \X such that T [X ∪ v] is indecomposable;

• 〈X〉 is the set of v ∈ V \X such that X is an interval of T [X ∪ v], that is,

〈X〉 = N−T (X) ∪N+T (X)

• for each u ∈ X , X(u) is the set of v ∈ V \X such that u, v is an interval of T [X ∪v]. Besides, for u ∈ X , X(u)is divided into X−(u) and X+(u) as follows

– X−(u) is the set of the elements x of X(u) such that x −→ u;

– X+(u) is the set of the elements x of X(u) such that u −→ x.

The family constituted by Ext(X), 〈X〉 and X(u), where u ∈ X , is denoted by pX .

2. Main resultsTwo digraphs are said to be equivalent under indecomposability or simply equivalent [2] if both are indecomposable or not.Given two digraphs G = (V,A) and G′ = (V,A′), we consider an integer k such that 0 < k <| V |. We say that G and G′

are −k-equivalent if G−X and G′ −X are equivalent for every subset X of V with | X |= k. A digraph G is said to be−k-recognizable if any digraph −k-equivalent to G is equivalent to it.In [2], the authors studied the −2-recognition of digraphs. They gave a counter example to −2-recognizable graph.Thus, the following problem arises:

Problem[2] Given k > 1, are the digraphs with at least 2k + 1 vertices are −2k-recognizable?To characterize the non −2-recognizable indecomposable tournaments we have to introduce the following tournaments

[3].

• For n ≥ 3, the tournament Pn = (Nn, An) is defined as follows.For x 6= y ∈ Nn, (x, y) ∈ An if

y = x+ 1ory ≤ x− 2.

• For n ≥ 5, the tournament Qn is defined on Nn as follows.

Q[Nn−2] = Pn−2, Nn−2 → n, Nn−3 → n− 1 and n→ n− 1→ n− 2.

This theorem to describe the non −2-recognizable indecomposable tournaments by the following theorem.

Theorem 1. Let T be an indecomposable tournament with at least 14 vertices. T is not −2-recognizable if and only ifT or T ∗ is isomorphic to a tournament T ′ where T ′ is defined on Nn (n ≥ 14) and satisfies one of the following assertions.

1. T ′ − n = Qn−1 and :

N+T ′(n) =

Nn−3 ∪ n− 1orNn−5 ∪ n− 1, n− 2orNn−5 ∪ n− 1, n− 2, n− 3.

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2. T ′[X] = Qn−3 where X = Nn−3 and satisfying one of the following assertions.

H1 : n− 2 ∈ N+T ′(X), n− 1 ∈ X−(n− 4), and

1. n ∈ X+(n− 4) andn− 2 −→ n− 1, nor2. n ∈ X(n− 4), n− 2 −→ n− 1 andn −→ n− 1, n− 2.

H2 : n− 2 ∈ X+(n− 3), n− 1 ∈ N−T ′(X), n ∈ N+T ′(X) and n −→ n− 2 −→ n− 1.

H3 : n− 2 ∈ X+(n− 3), n− 1 ∈ N−T ′(X), and

1. n ∈ N+T ′(X), n− 2 −→ n− 1, n andn −→ n− 1

or2. n ∈ N−T ′(X) andn −→ n− 2 −→ n− 1 −→ nor3. n ∈ X+(n− 3), n− 2 −→ n− 1, n andn− 1 −→ n.

H4 : n− 2 ∈ X+(n− 3), n− 1 ∈ N+T ′(X), and

1. n ∈ N+T ′(X) andn −→ n− 1 −→ n− 2 −→ n

or2. n ∈ N−T ′(X), n− 1 −→ n, n− 2 andn −→ n− 2or3. n ∈ X+(n− 3) andn −→ n− 1 −→ n− 2 −→ n.

H5 : n− 2 ∈ X+(n− 3), n− 1 ∈ X(n− 4), and

1. n ∈ N−T ′(X) andn −→ n− 1 −→ n− 2 −→ nor2. n ∈ N+

T ′(X), n− 1 −→ n, n− 2 andn −→ n− 2or3. n ∈ N+

T ′(X), n− 2 −→ n− 1 andn −→ n− 1, n− 2or4. n ∈ N+

T ′(X), n− 1 −→ n− 2, n− 4 andn −→ n− 1, n− 2.

References[1] H. Belkhechine and I. Boudabbous, Indecomposable tournaments and their indecomposable subtournaments on 5 and 7

vertices, to appear in Ars Combinatoria.

[2] A. Boussaıri, A. Chaıchaa and P. Ille, Indecomposability graph and indecomposability recognition, European Journalof Combinatorics (2014) 37 : 32-42.

[3] A. Cournier and P. Ille, Minimal indecomposable graphs, Discrete Math 183 (1998) 61-80.

[4] A. Ehrenfeucht and G. Rozenberg, Primitivity is hereditary for 2-structures, Theoret. Comput. Sci. 70 (1990), pp. 343-358.

[5] R. Fraısse, L’intervalle en theorie des relations, ses generalisations, filtre intervallaire et cloture d’une relation, in:Orders, Description and Roles, M. Pouzet et D. Richard ed. North-Holland. (1984), pp. 313-342.

31


Image enciphering schemes: State of the art

Souyah Amina1 ∗

1 Djilalli LiabesSidi Bel Abbes, Algeria

[email protected]

Abstract

Several works for secure image encryption schemes are proposed in the literature based on different strate-gies to deal with the challenge of fast and highly secure image cryptosystems suited in using for real-timeInternet image encryption and transmission applications. The intend of this paper is to highlight the most recentand prevalent works based on chaos strategy toward image encryption schemes.

Key words: Cryptography, image encryption, chaotic systems, ergodicity, diffusion.

1. IntroductionRecently, and with the ever-increasing advancement and development of both computer science and Internet, the wideneduse of multimedia data such as digital images take place in several applications as military, broadcasting, video-conferencing,Telemedicines etc.. In the other hand, the demand of security, confidentiality and integrity toward images come quite naturalespecially with the existing wired or wireless insecure transmission mediums/networks [10]. One given solution to theaforementioned problems of image security is cryptography, the art of science that is actually considered as a branch of bothcomputer science and mathematics [11]. Due to the intrinsic features of digital images such as: large date size, bulk datacapacity, high redundancy and strong correlation between adjacent pixels that trace their difference from textual informationin encryption, several image encryption schemes are proposed in the literature based on different strategies among whichchaos based methods owing to the interesting relationship between chaos and cryptography in which many chaotic propertiesas ergodicity and sensitivity to initial conditions and control parameters have their corresponding cryptographic properties ofconfusion and diffusion respectively [12]. In this paper, the most recent and prevalent approaches based on chaos that adoptFridrich architecture[9] are analyzed, characterized and highlighted in section 2, conclusion is given in section 3.

2. Different strategies used in image encryption schemesImage encryption is mainly divided into two groups according to the encryption scheme strategy used in designing the cryp-tosystem. Firstly, the classical approaches which are based on the traditional encryption algorithms after their modificationsto cover the above-mentioned requirements of image encryption as [17], secondly, the non classical approaches based ondiscrete dynamical systems and mathematical models as chaos strategy which is the focus of this paper.

∗Speaker

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2.1. The chaotic strategy toward image encryption schemes

Since 1990s, chaos as a kind of discrete dynamical system and a mathematical model has been widely investigated as a newstrategy for designing image encryption schemes [8]. The dependency to initial conditions, control parameters and ergodicityof its temporal evolution permit its direct connectivity to cryptographic properties of confusion and diffusion [7] and its use asa base of new image cryptosystems. Fridrich in Ref [9], proposed that chaos based image encryption scheme should consistof two stages: permutation and diffusion in which pixels are relocated using two dimensional chaotic map in the first stage,then their values are changed sequentially using any discretized or continuous chaotic map in the second stage. 1 shows theclassical permutation-diffusion architecture of chaos-based image scheme. Based on this architecture which has an origin totraditional confusion-diffusion architecture of Shannon[16], various works are proposed in the literature in which we put afocus on the most recent/prevalent ones.

Figure 1: Classical architecture of chaos-based image encryption cryptosystems

Rather than the commonly use of permutation and diffusion as two separate stages in several existing image encryptionschemes, the proposition of an architecture combining these two stages by considering them as one phase is proposed byYong et al.[1], the main motivation of the authors is that the encryption speed and the efficiency will be improved since theimage-scanning for obtaining pixels values is done just once. At first, the plain-image is divided into 8× 8 pixels blocks andafter a spatiotemporal chaos which has attracted the interest of several researchers in the fields of mathematics, physics andengineering is used to generate the pseudorandom sequence used in shuffling/diffusing the blocks simultaneously to obtainthe matching ciphered-image. Both theoretical analysis and experimental tests have been performed which reflect the highsecurity and time efficiency of the proposed scheme.Despite of the commonly used permutation in chaos based approaches architecture at a pixel level by its consideration asthe smallest unit for image’s shuffling step , a bit level permutation is proposed by Zhi-liang et al.[2], by dividing the pixelinto a number of bits , on which bit-level operations are performed. The notable effect of this permutation is that notonly the locations of pixels are exchanged, but also the value of each pixel is modified making the achievement of bothconfusion/diffusion possible in the permutation phase. It was shown that the higher 4 bits of a pixel carry 94.125 % of thetotal information of the image whereas the lower 4 bits carry less than 6 %, by taking this result into account the authorsproposed in the permutation phase that the higher 4 bits were permuted individually while the remaining lower 4 ones arerelocated as a whole using cat chaotic map, in the diffusion phase, each pixel value is modified sequentially at the pixel levelusing the output of the logistic map. The superior security and time efficiency of the work is shown comparing it with otherexisting works.An extension of the aforementioned work is suggested with the same authors[3], it is applied to an RGB image for oneround confusion-diffusion encryption. The work aimed to overcome the limitations of bit-level permutation in which the bitdistribution information in each bit plane still unchanged for that an expand-and-shrink strategy is employed to significantlyreduce the high correlation among the higher bit planes and to further smooth the changeable distributions in each bit planeafter the modification of each element of the expanded permuted plane is applied under the diffusion phase.A novel image encryption scheme is proposed by Yushu et al.[4], it is based on confusion-diffusion architecture in where bit-level permutation with random rotation matrix that relies on plain image are used in the confusion phase, a diffusion phase ischaracterized by the adoption of a block diffusion to spread the influence of each bit in the plain image all over the cipheredone, the scheme can be extended easily to be employed in parallel as there is no sequential mode between the encrypted

33


blocks like the conventional diffusion operation mode. The scheme is also shown to have a satisfactory security performancebesides to its suitability for executing in parallel mode and noisy environment.The main focus of the proposed work by Jun-xin et al.[5] is to improve the classical diffusion phase to further achieve thesame level of security under few numbers of rounds since this phase is featured by its highest cost in runtime over all thecryptosystem. In this context, an improved diffusion scheme named continuous diffusion strategy is applied which containstwo relevant diffusion rounds in one overall round encryption, the scheme composed of a supplementary diffusion procedureafter the conventional one and the control parameters are modified by the cipher image after the first diffusion procedure, forenhancing the confusion effect of the diffusion, a cyclic shift to cipher pixels is performed using stretched generated chaotickey-stream. A high level of security and fast encryption speed were well clarified by the authors.A combination of two existing one-dimensional (1D) chaotic maps is proposed by Yicong et al.[6] to overcome the securityweaknesses of many broken works under the non-resistance to chosen plaintext attacks besides to the known limitations of1D chaotic maps. The scheme is able to produce many 1D chaotic maps with excellent chaotic features, by mean of thelarge chaotic range and uniform distributed density function. Moreover, using the same set of keys with the same originalplain image in each time, the algorithm is able to generate a completely different, random, non-repeated and unpredictableencrypted images, making the resistance to chosen-plaintext, data loss and noisy attacks possible in addition to the excellentdiffusion and confusion properties.A novel symmetrical image encryption scheme is proposed by Guamin et al.[13]. It is based on the use of both skew tent mapand new chaos based line map that overcomes the limitation of image’s width is as equal as its height in the confusion phase,the diffusion phase is characterized by the simple use of xor operation of both rows and columns using a generated sequencein the previous step of skew tent map. The authors clarified the high security and speed performance of the proposed schemevia the numerical simulations, the scheme is able to be implemented in parallel since no complex sub-block operation exists.An RGB image encryption scheme based on the total plain image features is proposed by Murillo et al.[15]. One-roundpermutation-diffusion is required to achieve a sufficient level of security, in this work an optimized pseudorandom sequencefrom 1D logistic map is used based on the previous work of the authors[14], to overcome the drawbacks of one-dimensionalchaotic maps of discontinuous range, non-uniform data distribution etc., the generated sequence is based totally on the plainimage characteristics and the used external secret key to withstand the commonly used chosen/known plain-image attacks forbreaking cryptosystems. Numerical simulations were performed to demonstrate the effectiveness of the proposed scheme tobe used in practical applications.

3. ConclusionChaotic cryptography still be a source of new image encryption schemes due to some dynamical properties of chaotic systemsthat can be utilized in achieving the cryptographic properties of good ciphers. The intend of the current paper is to put thefocus on the most important and prevalent existing works in the literature based on chaos strategy toward image encryptionschemes.In the other hand, and more recently DNA cryptography is one of the new attractive and promising direction ofresearch in cryptography that has come to light with the development of DNA computing, its combination with chaos theoryfor designing cryptographic systems that benefit from both the goodness of each field have attracted the intention of manyresearchers recently.

References[1] Wang, Y., Wong, K. W., Liao, X., Chen, G., A new chaos-based fast image encryption algorithm, Applied soft comput-

ing., 2011,11(1), 514-522

[2] Zhu, Z. L., Zhang, W., Wong, K. W., Yu, H., A chaos-based symmetric image encryption scheme using a bit-levelpermutation, Information Sciences., 2011, 181(6), 1171-1186

[3] Zhang, W., Wong, K. W., Yu, H., Zhu, Z. L., A symmetric color image encryption algorithm using the intrinsic featuresof bit distributions, Communications in Nonlinear Science and Numerical Simulation., 2013, 18(3), 584-600

[4] Zhang, Y., Xiao, D., An image encryption scheme based on rotation matrix bit-level permutation and block diffusion,Communications in Nonlinear Science and Numerical Simulation., 2014, 19(1), 74-82

34


[5] Chen, J. X., Zhu, Z. L., Yu, H., A fast chaos-based symmetric image cryptosystem with an improved diffusion scheme,Optik-International Journal for Light and Electron Optics., 2014, 125(11), 2472-2478

[6] Zhou, Y., Bao, L., Chen, C. P., A new 1D chaotic system for image encryption, Signal processing., 2014, 97, 172-182

[7] Pellicer-Lostao, C., Lopez-Ruiz, R., Notions of Chaotic Cryptography: Sketch of a Chaos based Cryptosystem, it arXivpreprint arXiv., 2012.

[8] Arroyo Guardeo, D. Framework for the analysis and design of encryption strategies based on discrete-time chaoticdynamical systems , Polytechnic University of Madrid Higher Technical School of Agricultural Engineers , Spain,2009.

[9] Fridrich, J., Symmetric ciphers based on two-dimensional chaotic maps, International Journal of Bifurcation and chaos.,1998, 8(06), 1259-1284

[10] Ahmad, J., Hwang, S. O., Ali, A., An Experimental Comparison of Chaotic and Non-chaotic Image Encryption Schemes, Wireless Personal Communications., 2015, 1-18

[11] Acharya, B., Patra, S. K., Panda, G., Image encryption by novel cryptosystem using matrix transformation, In EmergingTrends in Engineering and Technology, 2008. ICETET’08 . IEEE., 2008, 77-8

[12] Alvarez, G., Li, S., Some basic cryptographic requirements for chaos-based cryptosystems, International Journal ofBifurcation and Chaos., 2006, 16(08), 2129-2151

[13] Zhou, G., Zhang, D., Liu, Y., Yuan, Y., Liu, Q., A novel image encryption algorithm based on chaos and Line map,Neurocomputing., 2015

[14] Murillo-Escobar, M. A., Abundiz-Prez, F., Cruz-Hernndez, C., Lpez-Gutirrez, R. M., A novel symmetric text encryptionalgorithm based on logistic map, A novel symmetric text encryption algorithm based on logistic map. In InternationalConference on Communications, Signal Processing and Computers., 2014, 49–53

[15] Murillo-Escobar, M. A., Cruz-Hernndez, C., Abundiz-Prez, F., Lpez-Gutirrez, R. M., Del Campo, O. A., A RGB imageencryption algorithm based on total plain image characteristics and chaos, Signal Processing., 2015, 56(2), 243–256

[16] El Assad, S., Farajallah, M., Vladeanu, C., Chaos-based block ciphers: An overview, In Communications (COMM),2014 10th International Conference on IEEE., 2014, 1-4

[17] Nithin, N., Bongale, A. M., Hegde, G. P., Image Encryption based on FEAL algorithm., International Journal ofAdvances in Computer Science and Technology., 2013,2(3),14-20

35


Nested double binomial sums

Samia Masrour1,2∗, Abdelkader Bouyakoub2

1 Fac. of Sc., Algiers 1 Univ.GEAN Laboratory, DG-RSDT

Algiers, [email protected]

2 Depart. of Math., Fac. of Sc., Oran Univ.GEAN Laboratory, DG-RSDT

Po. Box 1524, ELM Naouer, 31000, Oran, [email protected]

Abstract

In their paper, Prodinger and Kilic [1] give some binomial sums with complex coefficients. Kilic andBelbachir [2] extend their work and establish some generalized nested binomial sums. To do they compute thegenerating functions of some sequences satisfying recurrence relations. In this work, we propose an unifiedapproach to merge some double sums given in [2]. We also obtain new results which regroup and generalizesome of them.

Key words: Nested sums, binomial coefficient, generating functions.

2010 Mathematics Subject Classification: 05A10, 05A15, 11B65.

1. IntroductionIt is established in [2] the following results

Theorem 1. Let n and s be positive integers, and t and u any complex numbers. The generating function of the sequenceA

(s)n

defined by

A(s)n (t, u) :=

∑

i,j

(n− isj

)(n− sji

)tiuj ,

is

A (z) =∑

n

A(s)n zn =

(1− (1 + t) z)s−1

(1− (1 + t) z)s − uzs (1− tz)s .

Theorem 2. Let n and s be positive integers, and t and u any complex numbers. The generating function of the sequenceB

(s)n

defined by

B(s)n (t, u) :=

∑

i,j

(n− isj

)(n− ji

)tiuj ,

∗Speaker36

is

B (z) =∑

n

B(s)n zn =

(1− (1 + t) z)s−1

(1− (1 + t) z)s − uzs (1− tz) .

Theorem 3. Let n and s be positive integer, and t and u any complex numbers. The generating function of the sequenceC

(s)n

defined by

C(s)n (t, u) :=

∑

i,j

(n− isj

)(n− ji− j

)tiuj ,

is

C (z) =∑

n

C(s)n zn =

(1− (1 + t) z)s−1

(1− (1 + u) z)s − tuzs+1.

Theorem 4. Let n and s be positive integers, and t and u any complex numbers. The generating function of the sequenceD

(s)n

defined by

D(s)n (t, u) :=

∑

i,j

(n

i+ j

)(i

sj

)tiuj ,

is

D (z) =∑

n

D(s)n zn =

(1− (1 + t) z)s−1

(1− (1 + t) z)s − uts (1− z)s−1zs+1

.

2. Main resultWe propose to present an unified theorem for the cited ones in the introduction, and present some other corollaries on themain theorem.

Theorem 5. Let n, s, α and β be positive integers, and t and u any complex numbers. The generating function of thesequence

V

(s)n

defined by

V (s)n (t, u) :=

∑

i,j

(n− isj

)(n− αji− βj

)tiuj ,

is

V (z) =∑

n

V (s)n zn =

(1− (1 + t) z)s−1

(1− (1 + t) z)s − uzs (1− tz)α−β (tz)β.

Remark 1. Here are given some specializations

1. α = s and β = 0 give Theorem 1;

2. α = 1 and β = 0 give Theorem 2;

3. α = 1 and β = 1 give Theorem 3;

4. α = 0 and β = 1 permit to obtain Theorem 4 with an adequate change of variables.

Corollary 5.1. Let n, α and s be positive integers, and t and u any complex numbers. The generating function of thesequence

Ω(s)n

defined by

Ω(s)n (t, u) :=

∑

i,j

(n− isj

)(n− αji

)tiuj ,

is

Ω (z) =∑

n

Ω(s)n zn =

(1− (1 + t) z)s−1

(1− (1 + t) z)s − uzs (1− tz)α .

37

Corollary 5.2. Let n and s be positive integers, and t and u any complex numbers. The generating function of the sequenceΓ(s)n

defined by

Γ(s)n (t, u) :=

∑

i,j

(n− isj

)(n− sji− sj

)tiuj ,

is

Γ (z) =∑

n

Γ(s)n zn =

(1− (1 + t) z)s−1

(1− (1 + t) z)s − uz2sts.

Corollary 5.3. Let n, r and s be positive integers, and t and u any complex numbers. The generating function of the sequenceΛ(s)n

defined by

Λ(s)n (t, u) :=

∑

i,j

(n− isj

)(n− rji− rj

)tiuj ,

is

Λ (z) =∑

n

Λ(s)n zn =

(1− (1 + t) z)s−1

(1− (1 + t) z)s − uzs (tz)r.

References[1] E. Kılıc and H. Prodinger, Some Double binomial sums related with the Fibonacci, Pell and generalized order-k Fi-

bonacci numbers, Rocky Mount. J. Math., 43 (3) (2013), 975–987.

[2] H. Belbachir and E. Kilic, Generalized double binomial sums families by generating functions. Utilitas Math., to appear.Available at http://ekilic.etu.edu.tr/list/DoubleSums2.pdf

38


On a constant related to the prime counting function

Djamel Berkane1∗, Pierre Dusart2

1 USTHB, Faculty of Mathematics, RESITS laboratoryPo. Box 32 EL ALIA 16111 BAB EZZOUAR ALGIERS.

[email protected]

2 Universite de LimogesPo. Box 123, avenue Albert Thomas 87060 Limoges Cedex

[email protected]

Abstract

Let π (x) be the number of primes not exceeding x. We produce new explicit bounds for π(x) and we usethem to obtain a fine frame for the remainder term in the asymptotic formula of the sum

∑2≤n≤x 1/π(n).

Key words: Explicit estimates, Prime numbers.

2010 Mathematics Subject Classification:11A41, 11Y60, 26D15, 40A25.

1. IntroductionThe problem of finding asymptotic expansion for sums involving quotients of arithmetical functions has been investigated inseveral papers, we quote here the first papers in this axe of De Koninck, Ivic and Erdos in [2-3] and [5], and recently the workof De Koninck and Luca [4], which obtains an estimate for the sum of the reciprocals of the middle prime factor of a giveninteger. In particular, the reciprocal of the prime counting function follows some steps and generate more auxiliary works.The reader will find in [1] a good survey.Let, as usual, π (x) be the number of primes not exceeding x and let us denote by SN (for N ≥ 2) a sequence involving thereciprocal of the prime counting function defined by

SN =∑

2≤n≤N

1π (n)

− 12

log2N + logN + log logN.

At first, in 1980, De Koninck and Ivic in [3] showed that

∑

2≤n≤N

1π (n)

=12

log2N +O (logN) .

Shortly after, Ivic [7] expanded the above formula by proving the following result:

∗Speaker

39


Theorem 1 (A. Ivic). For any fixed integer m ≥ 2, we have∑

2≤n≤N

1π (n)

=12

log2N − logN − log logN + C+

k2

logN+ · · ·+ km

(m− 1) logm−1N+O

(1

logmN

),

where C is an absolute constant and knn is the sequence of integers given by the recurrence relation:

k1 = 1, kn + 1!kn−1 + 2!kn−2 + · · ·+ (n− 1)!k1 = n · n! .

2. Main resultsIn this work, our interest is to determine a fairly accurate value of the constant C which appears in the Ivic’s result (seeTheorem1) and therefore of the limit of SN as N → ∞. Our idea is to study an explicit viewpoint the O-term of thesummatory function of the 1/π (n)-function i.e, to study

S(x) =∑

2≤n≤x

1π (n)

− 12

log2 x+ log x+ log log x.

A first approach is given by Hassani and Moshtagh in [6]. They obtain

α(x) ≤∑

2≤n≤x

1π (n)

− 12

log2 x+ log x ≤ β(x), ∀x ≥ 2,

with α(x) = −1.51 log log x+ 0.8994, β(x) = −0.79 log log x+ 6.4888, and conjecture based on numerical evidence thatC ≈ 6.9. Here, we propose to improve this last result. Our main theorem is the following

Theorem 2. For all x ≥ 150721071, we have

Clw +6

10√

log11 x≤ S(x) ≤ 2

log x+ Cup,

with Clw = 6.68400420 and Cup = 6.78291066.

The upper bound also holds for x ≥ 25555987. Using this result, we readily deduce a good estimate for the limit ofour sequence and the computations we ran lead us to believe that C ≈ 6.7. With this technique, obtaining more decimals isclosely linked with progress on the zero-free region of the Riemann zeta-function.

References[1] H. Belbachir and D. Berkane: Asymptotic expansion for the sum of inverses of arithmetical functions involving iterated

logarithms. Appl. Anal. Discrete Math. 5 (2011), 80–86.

[2] J.-M. De Koninck: Sums of quotients of additive functions. Proc. Amer. Math. Soc. 44 (1974), 35–38.

[3] J.-M. De Koninck and A. Ivic: Topics in Arithmetical Functions. Notas de Matematica North-Holland, Amsterdam. 43(1980).

[4] J.-M. De Koninck and F. Luca: On the middle prime factor of an integer. Journal of Integer Sequences. 16 (2013), 1–10.

[5] P. Erdos and A. Ivic: On sums involving reciprocals of certain arithmetical functions. C. R. Math. Acad. Sci. Soc. R.Can. 32(46) (1982), 49–56.

[6] M. Hassani and H. Moshtagh: A remark on a sum involving the prime counting function. Appl. Anal. Discrete Math. (2)(2008), 51-55.

[7] A. Ivic: On a sum involving the prime counting function. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 13 (2002),85–88.

40


Graphes indécomposables et leurs sous-graphesindécomposables à 7 ou à 8 sommets

Imed Boudabbous 1, Jamel Dammak 2, Mouna Yaich 3∗

1 Institut Préparatoire aux Études d’Ingénieurs de SfaxUniversité de Sfax, Tunisie.

[email protected] Faculté des Sciences de Sfax

Université de Sfax, [email protected]

3 Faculté des Sciences de SfaxUniversité de Sfax, Tunisie

[email protected]

RésuméÉtant donné un graphe G = (S,A), une partie X de S est un intervalle de G lorsque pour tous a,

b ∈ X et x ∈ S \X , a, x ∈ A si et seulement si b, x ∈ A. Par exemple, ∅, x(x ∈ S) et S sontdes intervalles deG, appelés intervalles triviaux. Un graphe dont tous ses intervalles sont triviaux, estdit indécomposable ; sinon, il est décomposable. On dit qu’un graphe G abrite un graphe G′ si G′ estisomorphe à un sous-graphe de G. Nous classifions les graphes indécomposables à partir de graphesindécomposables à 7 ou à 8 sommets qu’ils abritent.Mots clés : Graphes, indécomposable, intervalle, critique, abritement.

Key words : Graphes, indécomposable, intervalle, critique, abritement.

2010 Mathematics Subject Classification :

1. Introduction1.1. Définitions

Un graphe (ou graphe simple) G = (S,A) est constitué d’un ensemble fini S (noté aussi S(G)) de sommetsde G et d’un ensemble A de paires de sommets distincts, appelées les arêtes de G (noté aussi A(G)). Pour deux

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sommets distincts x et y d’un grapheG la notion x y (resp. x . . . y), signifie que x, y ∈ A (resp. x, y /∈ A).Pour toute partie X de S, on appelle sous-graphe de G induit par X , le sous-graphe G[X] = (X,A ∩ P2(X)).Étant donné deux graphes G = (S,A) et G′ = (S′, A′), un isomorphisme de G sur G′ est une bijection f deS sur S′ vérifiant : pour tous x 6= y ∈ S, x, y ∈ A si et seulement si f(x), f(y) ∈ A′. Lorsqu’un telisomorphisme existe, on dit que G et G′ sont isomorphes et on note G ' G′. On dit que G abrite G′ si G′ estisomorphe à un sous-graphe de G. Le graphe complémentaire d’un graphe G = (S,A) est le graphe G = (S,A)où A = x, y ∈ P2(S); x, y /∈ A.

1.2. Graphes indécomposables

Étant donné un graphe G = (S,A), une partie X de S est un intervalle [6, 7, 10] de G lorsque pour tous a,b ∈ X et x ∈ S \X , a, x ∈ A si et seulement si b, x ∈ A. Par exemple, ∅, x(x ∈ S) et S sont des intervallesdeG, appelés intervalles triviaux. Un graphe dont tous ses intervalles sont triviaux, est dit indécomposable [8, 10] ;sinon, il est décomposable. Un sommet x d’un graphe indécomposable G est dit critique si le sous-graphe G[S \x] est décomposable. Un graphe indécomposable G est dit critique si tous ses sommets sont critiques. Ongénéralise cette définition, en disant qu’un graphe G est (−k)-critique lorsqu’il admet exactement k sommets noncritiques (où k ≥ 1 ). Un graphe est dit (−1)-critique en a si a est l’unique sommet non critique de G.

Afin de rappeler la caractérisation des graphes critiques, nous introduisons pour n ≥ 1, le graphe G2n définiesur (0, . . . , 2n − 1) comme suit : pour tous i 6= j ∈ 0, . . . , 2n − 1, i, j ∈ A(G2n) s’il existe k ≤ l avecl ∈ 0, . . . , n− 1 tel que i, j = 2k, 2l + 1.

Théorème 1.1. (Schmerl and Trotter[10])Soit G = (V,E) un graphe indécomposable. G est critique si et seulement si G est isomorphe à G2n ou à G2n telque n ≥ 2.

Suite à cette étude, Houmem. Belkhechine, Imed. Boudabbous et Mohamed. Baka Elayech [1] ont caractérisé,en 2010 les graphes (−1)-critiques. Il est à noter que la morphologie des graphes (−1)-critiques est difficile àdécrire.

2. Présentation des résultatsLes dernières années, le concept d’indécomposabilité est devenu fondamental dans l’étude des structures finies.

Ce concept et d’autres notions voisines ont fait l’objet de plusieurs articles, par exemple, voir ( [3], [4], [5], [9]).En 2012, M. Chudnovsky et P. Seymour [4] ont présenté une méthode de construction des graphes indécomposablesà partir des sous-graphes indécomposables qu’ils abritent.Dans cette étude, nous présentons une autre caractérisation des graphes critiques et des graphes (−1)-critiqueset nous classifions les graphes indécomposables à partir des graphes indécomposables à 7 ou à 8 sommets qu’ilsabritent. Il est à noter qu’une étude analogue dans le cas des "tournois" a été faite, en 2015, par Imed. Boudabbous[2].

Théorème 2.1.Soit G = (S,A) un graphe indécomposable d’au moins 8 sommets.

1. G est critique si et seulement si sur chaque ensemble de sommets à 5 éléments le graphe induit est décom-posable.

2. Si G n’est pas critique. Posons k = 7, si |S| est pair ;8, si |S| est impair.

2.1. G est (−1)-critique en a si et seulement si sur chaque ensemble de sommets à k éléments contenant a,

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le graphe induit est décomposable.

2.2. G n’est pas (−1)-critique si et seulement pour tout sommet x il existe un ensemble de sommets à kéléments contenant x sur lequel le graphe induit est indécomposable.

Nous montrons en construisant des exemples que la valeur k dans le théorème 2.1 est optimale.

Références[1] H. Belkhechine, Imed. Boudabbous et M. B Elayech, Les graphes (−1)-critiques (to apear in Arrs Combina-

toria).

[2] I. Boudabbous, C. R. Acad. Sci. Paris, Ser. I 353 (2015) 671-675

[3] Y. Boudabbous, J. Dammak, P. Ille, Indecomposablity and duality of tournaments, Discrete Math. 223 (2000)55-82.

[4] M. Chudnovsky, P. Seymour, Growing without cloning, SIAM J. Discrete Math. 26 (2012) 860-880.

[5] E. Clarou, G. Lopez, Configurations inévitables dans un tournoi indécomposable, C. R. Acad. Sci. Paris, Ser.I 324 (1997) 1201-1204.

[6] A. Cournier and P. Ille, Minimal indecomposable graphs, Discrete Math 183 (1998) 61-80.

[7] R. Fraïssé, L’intervalle en théorie des relations, ses généralisations, filtre intervallaire et clô ture d’une re-lation, in : M. Pouzet and D. Richard eds, Order, Description and Roles, North-Holland, Amsterdam.(1984)313-342.

[8] P. Ille, Indecomposable graphs, Discrete Math. 173 (1997) 71-78.

[9] M. Pouzet, I. Zaguia, On minimal prime graphs and posets, Order 26(4)(2009) 357-375.

[10] J.H. Schmerl, W.T. Trotter, Critically indecomposable partially ordered sets, graphs, tournaments and otherbinary relational structures, Discrete Math. 113 (1993) 191-205.

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Hereditarily hemimorphy of −k-hemimorphic digraphs for k ≥ 8

Houcine Bouchaala1, Jamel Dammak2, Rahma Salem2∗

1 Faculty of Sciences, Taif University, Turaba branch, Department of Mathematics21995, Turaba, Taif, Kingdom of Saudi Arabia

[email protected] Faculty of Sciences, Sfax University, Department of Mathematics

B.P. 1171, 3000 Sfax, [email protected]

[email protected]

Abstract

Given a digraph G = (V,E), with each nonempty subset X of V , we associate the subdigraph G[X] =(X,E ∩ (X × X)) of G induced by X . Two digraphs G = (V,E) and G′ = (V,E′) are hemimorphic ifG′ is isomorphic to G or to G? = (V, (x, y) : (y, x) ∈ E). The digraphs are −k-hemimorphic if for all(|V | − k)-element subset K of V , G[K] and G′[K] are hemimorphic. The digraphs G and G′ are hereditarilyhemimorphic if for all subset X of V , G[X] and G′[X] are hemimorphic. In this work we prove: two −k-hemimorphic digraphs, on at least k+12 vertices, are hereditarily hemimorphic for every k ≥ 8. The particularcase of tournaments was obtained by M. Bouaziz, Y. Boudabbous and N. El Amri in 2011.

Key words: Digraph, tournament, hereditarily hemimorphy, reconstruction.

2010 Mathematics Subject Classification: 05C50;05C60.

1. IntroductionA directed graph or simply digraphG consists of a finite nonempty set V (G) and an arc setE(G), where an arc is an orderedpair of distinct vertices. Such a digraph is denoted by (V (G), E(G)) or simply by (V,E). Given a digraph G = (V,E), witheach nonempty subset X of V , we associate the subdigraph G[X] = (X,E ∩ (X ×X)) of G induced by X . Given a propersubset X of V , G[V −X] is also denoted by G−X and G− x whenever X = x.

Given a digraph G = (V,E), for x 6= y ∈ V , x −→G y or y ←−G x means (x, y) ∈ E and (y, x) /∈ E, xGy means

(x, y) ∈ E, (y, x) ∈ E and x · · ·G y means (x, y) /∈ E, (y, x) /∈ E. Say x and y of G form a directed pair or an orientedpair if either x −→G y or y −→G x; otherwise, they form a neutral pair; x, y is full if x

Gy, and void if x . . .G y. For

x ∈ V and for Y ⊆ V , x −→G Y signifies that for every y ∈ Y , x −→G y. For X,Y ⊆ V , X −→G Y signifies that forevery x ∈ X , x −→G Y . For x ∈ V and for X,Y ⊆ V , x←−G Y , x

GY , x · · ·G Y , X

GY , X · · ·G Y are defined in

the same way.A digraph T = (V,E) is a tournament if for all x 6= y ∈ V , either x −→T y or y −→T x. A tournament T is a total

order or a chain provided that for x, y, z ∈ V (T ), if x −→T y and y −→T z, then x −→T z. Given a total orderO = (V,E),x < y means x −→O y for x, y ∈ V .

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Given a digraph G = (V,E), a subset I of V is an interval of G if for any x ∈ V − I , x −→G I or x ←−G I orx

GI or x . . .G I . For instance, the empty set, the singletons of V and the set V are intervals of G, called trivial intervals.

A digraph is indecomposable if all its intervals are trivial, otherwise it is decomposable.Given two digraphs G = (V,E) and G′ = (V ′, E′), a bijection f from V onto V ′ is an isomorphism from G onto G′

provided that for any x, y ∈ V , (x, y) ∈ E if and only if (f(x), f(y)) ∈ E′. The digraphs G and G′ are then isomorphic,which is denoted by G ' G′, if there exists an isomorphism from G onto G′. If G and G′ are not isomorphic, we writeG 6' G′. For instance, a digraph H embeds into a digraph G if H is isomorphic to a subdigraph of G. With each digraphG = (V,E) associate its dual G? = (V,E?) defined as follows. Given x 6= y ∈ V , (x, y) ∈ E? if (y, x) ∈ E. Two digraphsG and G′ are hemimorphic, if G′ is isomorphic to G or to G?. A digraph is said to be self dual if it is isomorphic to its dual.

Given two digraphs G and G′ on the same vertex set V , consider an integer k such that 0 ≤ k ≤ |V | = v. The digraphsG′ and G are k-hypomorphic (resp. k-hemimorphic) whenever for every K ⊆ V with |K| = k, the subdigraphs G′[K]and G[K] are isomorphic (resp. hemimorphic). G′ and G are −k-hypomorphic (resp. −k-hemimorphic) wheneverG′ and G are v − k-hypomorphic (resp. v − k-hemimorphic). Notice that G and G′ are trivially 0-hypomorphic,however G and G′ are −0-hypomorphic if and only if they are isomorphic. A digraph is k-self dual (resp. −k-selfdual) if it is k-hypomorphic (resp. −k-hypomorphic) to its dual. Given two integers p and q such that 0 < |p|, |q| ≤ v,G and G′ are p, q-hypomorphic (resp. p, q-hemimorphic) if they are p-hypomorphic and q-hypomorphic (resp.p-hemimorphic and q-hemimorphic). The notation is extended to (≤ k)-hypomorphic (resp. (≤ k)-hemimorphic) ifthey are p-hypomorphic (resp. p-hemimorphic) for p ∈ 1, 2, ..., k. A digraph is p, q-self dual (resp. (≤ k)-selfdual) if it is p, q-hypomorphic (resp. (≤ k)-hypomorphic) to its dual. A digraph G is k-reconstructible (resp. k-half-reconstructible) if any digraph k-hypomorphic (resp. k-hemimorphic) to G is isomorphic (resp. hemimorphic)to it. Similarly, we denote the −k-reconstructible, p, q-reconstructible, (≤ k)-reconstructible, likewise for the half-reconstruction. Finally,G andG′ are hereditarily isomorphic (resp. hereditarily hemimorphic) if they are (≤ v)-hypomorphic(resp. (≤ v)-hemimorphic), i.e. if G[X] is hypomorphic (resp. hemimorphic) to G′[X] for all X ⊆ V . A digraph ishereditarily self dual if it is hereditarily isomorphic to its dual.

The problem of (≤ k)-reconstruction was introduced by R. Fraısse in 1970 [7]. In 1972, G. Lopez [9, 10] proved that:

Theorem 1. [9, 10] The digraphs on at least 6 vertices are (≤ 6)-reconstructible (i. e. if G and G′ are (≤ 6)-hypomorphicdigraphs then G′ and G are isomorphic).

In 2015, using Theorem 1, J. Dammak and R. Salem proved that:

Corollary 1.1. [6] Let G and G′ be two digraphs on at least 6 vertices.

1. If G and G′ are (≤ 6)-hypomorphic, then G′ and G are hereditarily isomorphic.

2. If for every C ∈ DG,G′ satisfying C is an interval of G and G′, G′[C] and G[C] are hereditarily isomorphic, then Gand G′ are hereditarily isomorphic.

Given a digraph G with a non-self dual subdigraph, Cdual(G) denotes the smallest cardinality of a non-self dual subdi-graph of G. Applying Theorem 1 to G′ = G?, the digraph G embeds a non-self dual subdigraph with at most 6 vertices.So, as all digraphs on at most 2 vertices are self dual, 3 ≤ Cdual(G) ≤ 6. In the case where G has no non-self dual finitesubdigraph, we convene Cdual(G) = ∞. G. Lopez and C. Rauzy [12] proved that: the digraphs on at least 7 vertices are4,−1-reconstructible. So, we deduce:

Corollary 1.2. [6] Given an integer k ≥ 1, the digraphs on at least k + 6 vertices are 4,−k-reconstructible.

The following two notions play an important role in this paper. Given two (≤ 2)-hypomorphic digraphs G = (V,E) andG′ = (V,E′), denote DG,G′ the equivalence relation on V defined as follows. For every x ∈ V , xDG,G′x and for everyx 6= y ∈ V , xDG,G′y if there are vertices x0 = x, ..., xn = y such that (xi, xi+1) ∈ E if and only if (xi, xi+1) /∈ E′, forevery i ∈ 0, ..., n − 1. The relation DG,G′ is called the difference relation of G and G′, its classes are called differenceclasses. The set of difference classes of G and G′ is denoted by DG,G′ .

Given a digraph G = (V,E), we define an equivalence relation C on V in the following way. For every x ∈ V , xCxand for every x 6= y ∈ V , xCy if there are vertices x0 = x, ..., xn = y such that xi −→G xi+1 or xi ←−G xi+1, for everyi ∈ 0, ..., n− 1. The equivalence classes of C are called arc-connected components of G. A digraph is arc-connected if Vis the unique arc-connected component. The set of the arc-connected components of G is denote by ρ(G).

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2. Main resultsIn this work, first, we prove:

Theorem 2. Given an integer k ≥ 7, consider two (≤ k)-hemimorphic digraphs G and G′ on the same vertex set V . Let I0be a subset of V such that |I0| = Cdual(G) and G[I0] is non-self dual.

1. Let C ∈ DG,G′ . If C /∈ ρ(G), then C is an interval of G and G′, and G′[C] and G[C] are hereditarily isomorphic.

2. If G′[I0] ' G[I0] and, (k ≥ 8) or (k = 7 and Cdual(G) 6= 4), we have:

(a) For every C ∈ DG,G′ , C is an interval of G and G′, and G′[C] and G[C] are (≤ 4)-hypomorphic.

(b) If for every C ∈ DG,G′ and for every A ⊂ C with |A| ≤ 3 we have G′[C − A] and G[C − A] are isomorphic,then G′ and G are hereditarily isomorphic.

In 2011, M. Bouaziz, Y. Boudabbous and N. El Amri [1] proved that two −k-hemimorphic tournaments, on at leastk+ 7 vertices, are hereditarily hemimorphic for every k ≥ 5. We give a generalization of the previous work [1] for digraphs,for k ≥ 8. Our main result is:

Theorem 3. Given an integer k ≥ 7, consider two −k-hemimorphic digraphs G and G′ on at least k + 12 vertices. Ifk ≥ 8 or (k = 7 and Cdual(G) 6= 4), then G′ and G are hereditarily hemimorphic.

Finally, we set the following conjecture.

Conjecture 1. two −7-hemimorphic digraphs on at least 19 vertices, are hereditarily hemimorphic.

References[1] Bouaziz M., Boudabbous Y., El Amri N., Hereditary hemimorphy of −k-hemimorphic tournaments for k ≥ 5, J.

Korean Math. Soc.., 2011, 48(3), 599–626

[2] Bouchaala H., Boudabbous Y., Lopez G., −1-self-dual finite prechains and applications, C. R. Acad. Sci. Paris, Ser.I., 2013, 351, 859–864

[3] Boudabbous Y., Dammak J., Sur la −k-demi-reconstructibilite des tournois finis, C. R. Acad. Sci. Paris, Ser. I., 1998,326, 1037–1040

[4] Dammak J., La dualite dans la demi-reconstruction des relations binaires finies, C. R. Acad. Sci. Paris, Ser. I., 1998,327, 861–864

[5] Dammak J., La −5-demi-reconstructibilte des relations binaires connexes finies, Proycciones., 2003, 22(3), 181–199

[6] Dammak J., Salem R., Hereditary hemimorphy of 5,−4-hemimorphic arc-connected digraphs, accepted in Ars Com-binatoria.

[7] Fraısse R., Abritement entre relations et specialement entre chaınes, Symposi. Math. Instituto Nazionale di Alta Mat.,1970, 5, 203–251

[8] Kendalland M. G., Smith B., On the method of paired comparaison, Biometrika., 1940, 31, 324–345

[9] Lopez G., Deux resultats concernant la determination d’une relation par les types d’isomorphie de ses restrictions, C.R. Acad. Sci. Paris, Ser. A., 1972, 274, 1525–1528

[10] Lopez G., Sur la determination d’une relation par les types d’isomorphie de ses restrictions, C. R. Acad. Sci. Paris, Ser.A., 1972, 275, 951–953

[11] Lopez G., Rauzy C., Reconstruction of binary relations from their restrictions of cardinality 2,3,4 and (n-1), I, Z. Math.Logik Grundlag. Math., 1992, 38(1), 27–37

[12] Lopez G., Rauzy C., Reconstruction of binary relations from their restrictions of cardinality 2, 3, 4 and (n − 1), II, Z.Math. Logik Grundlag. Math., 1992, 38(2), 157–168

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[13] Moon J. W., Tournaments whose subtournaments are irreducible or transitive, Canad. Math. Bull.., 1979, 21(1), 75–79

[14] Pouzet M., Application d’une propriete combinatoire des parties d’un ensemble aux groupes et aux relations, Math.Zeirtschr.., 1976, 150(2), 117–134

[15] Pouzet M., Relations non reconstructibles par les restrictions, J. Combin. Theory, Ser. B., 1979, 26(1), 22–34

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n Out of n Audible Password Secret Sharing Scheme WithUnexpanded Shares

Nour El Houda Asma MERABET1∗, Redha BENZID2,

1 University of BatnaBatna, Algeria

[email protected] University of Batna

Batna, [email protected]

...

Abstract

Audio secret sharing is a cryptography technique whose decryption can be done by human ears. Audiosecret sharing suffers from the problem of expanded shares just like visual secret sharing. In this paper weprofit from the nice features that offers Wang et al.s image secret sharing scheme, one of the most cited paperthat overcomes the main drawback of traditional visual cryptography by using simple XOR operation in thedecryption process, and apply its (n,n) threshold scheme for sharing audible secret password.

Key words: audio secret sharing; expanded shares; XOR Boolean operation.

1. IntroductionThe number of multimedia data exchanged over the Internet is increasing day after day, that’s way it becomes very importantto protect sensitive data from illegitimate person. Traditional techniques of cryptography and secret sharing use computationaldevice to decrypt secret data, this last is not very practical in the case where computer is not available, thats way two importantcryptographic techniques that can decode secret data using human senses attract attention of searchers in the recent years,which are: visual secret sharing (VSS) and audio secret sharing (ASS). In 1994, Naor and Shamir introduced an innovativetype of cryptography which called Visual cryptography or VC [1]. VC purposes to protect secret image and sharing it at thesame time. To encrypt secret image the dealer divide it into n random images called shares or shadows, the secret can berevealed when k,2 ≤ k ≤ n shares are stacked together, if k-1 number of shares are stacked together, no information canbe given about the secret image. this is known as the threshold mechanism. The main advantage of VSS is situated in itsdecryption process that can be done using the human visual system without using any computational tools. Another type ofcryptography technique called audio secret sharing (ASS) introduced by Desmedt et al in 1998 [2], in which the encryptionprocess is done in the same way as the conventional VC, but the secret data and/or shares are sound waves and the decryptionprocess can be done by human ears unlike VC. One of the main parameter that would be respected in both of VSS and ASS toget an ideal scheme is the unexpanded shares (fixed in term of size), but unfortunately the problem of expanded shares exists

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in these two type of cryptography cited above: Traditional VC suffers from the problem of pixel expansion ( the number ofencoded pixels in each share m) that affect both of the size of shares and the reconstructed secret image in term of size, morethe number of encoded pixels increases, more the quality of the reconstructed image decreases, also the size of recoveredimage will be m times big than the original secret image which influence on cost and memory storage. To overcome thiscritical issue many works have been developed schemes searching to enhance the quality of reconstructed image by reducingthe number of pixel expansion to one m = 1, using the concept of probability [3-4], random grid [5-6] and XOR-based[7-8]. In the other hand: the issue of expanded shares exists also in ASS. unlike VSS where the dealer expands the numberof pixels in each share, in ASS the dealer expands the number of beeps in each share which affects the size of shares andthe reconstructed secret audio. In order to overcome the problem of expanded shares and get an ideal scheme: Lin et al. [9],Ehdaie et al. [10] and Patil et al. [11] proposed audio secret schemes with unexpanded shares. In this paper we choose one ofthe most cited article in VSS that solves the problem of pixel expansion, just by using simple XOR Boolean operation in thedecryption process and apply its (n,n) threshold scheme to share secret audible password between n number of participants.knowing that XOR operation is widely used in many cryptosystem because of its simplicity in term of computation andefficacy in term of encryption. The rest of this paper is organized as follow: in section two we present the modified Wang etal.s scheme for ASS [12], then in section three we show our experimental results, finally we give a conclusion.

2. Application of Wang et al scheme for secret audio schemeWang et al.s secret image sharing scheme [12] is based on XOR Boolean algebra in the decryption process. Before applyingWang et al.s scheme for sharing secret audio, we have to preprocess the secret audio (.wav), firstly we quantize the secretwave file in order to divide it into equal intervals of eight bits using this formula:

Q = SecretAudio−Min(SecretAudio)/(Max(SecretAudio)−Max(SecretAudio)). (1)

Where: Min and Max represent the minimum and maximum value of secret audio respectively, then we round the quantizedsecret audio to the nearest integer, then instead of using random images to construct shares we use audio data to create shares,the size of secret is the same as the shares.

Wang et al.s Algorithm for secret images:

Input: secret password S, and an integer n2,Output: n share shadows T1, . . . , Tn.Phase one: quantization of the secret signal using the well-known linear quantizer: Quantized secret password= (Q-Min(Q) /(Max(Q)-Min(Q)))

Phase two: Audio share construction:1)generate n 1 random matrices B1, . . . , Bn − 1,

2) compute n audio shadows:T1=B1,T2 = B1 ⊕B2,Tn − 1 = Bn − 2⊕Bn − 1,Tn = Bn − 1⊕APhase three: reconstruction of the secret audio S:S = T1 ⊕ T2 ⊕ ...⊕ T2

3. Main resultsThe experimental results of the use of Wang et al.s algorithm are shown in figure 1, 2, 3 and 4: in figure 1 the secret audiois presented, however, in figure two and three the audio shares are generated by the above mentioned algorithm, and finallythe revealed secret audio is shown in figure 4 when all shares are played simultaneously, these results have been implementedusing Matlab2013. Thanks to Peak Signal to Noise Ratio (PSNR) we measure the similarity between the secret audio andthe reconstructed secret audio, high value of PSNR indicates that the reconstructed signal has a good quality, giving that:PSNR = 20∗ log10(MAX2/MSE) (1) Where: MSE is the mean squared error between the secret audio and reconstructedaudio, however MAX is the maximum value of signal, note that MAX value equals to 255 in our case. By using equation

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(1) we find that: PSNR=Infinity, which interprets that the revealed secret audio has reconstructed with high quality withoutany distortion. As for statistical similarities: we calculate the correlation between the two signals, thus we find its value 1which means that secret audio has a strong correlation with the reconstructed secret audio, hence no information losses inthe reconstructed secret audio. The able below gives a comparison between Wang et al. for ASS scheme and others ASSschemes.

Year Authors Schemes Shares Decryption Secret1998 Desmedt et al. [2] (2,n) Expanded HAS Bit string2005 Ehdaie et al. [8] (k,n) Unexpanded HAS Audio2012 Yoshida et al. [13] (n,n) Expanded HAS Audio2014 Abukari et al.[14] (k,n) Expanded Computer Audio2015 Pati et al. [9] (k,n) Reduced Computer Audio2015 Wang et al. [12] for ASS (n,n) Enexpanded Computer Audio

Figure 1: Secret audio Figure 2: the first share

Figure 3: Lgende Figure 4: Reconstructed secret audio

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4. ConclusionIn some cases we need to share not only visual information like text, image, but also audible secret, hence the developmentof audio secret sharing schemes, however sharing audio secret between participants is very important in some applicationslike in military, banks In this paper we have chosen Wang and al.s algorithm which offers good properties mention: 1) gettingrid of the weakness of Naor and Shamirs scheme by reducing the number of pixel expansion to one, hence we can applyits scheme to solve the same problem in ASS. 2) the use of Boolean simple XOR operation in the decryption process, theexperimental results show that this algorithm is perfectly applicable for audio data, the reconstructed secret audio is perfectlyrecovered with high quality, so we conclude that Wang et al.s scheme is audiovisual secret sharing which means that theirscheme is available to share both of secret images and secret audio.

References[1] Naor, A. Shamir, Visual cryptography, Advances in cryptology-EUROCRYPT94, Lecture Notes in Computer.

[2] Y. Desmedt and Y. Frankel Threshold cryptosystems Proc of CRYPTO89 volume 435 of LNCS, paper 307-315 SpringerVerlag 1990.

[3] P. Tuyls, H.D.L. Hollmann, J.H.van Lint, L. Tolhuizen, Xor-based visual cryptography Schemes, Designs Codes andCryptography, vol.37, 2005, pp.169186.

[4] C.-N. Yang. New visual secret sharing schemes using probabilistic method. Pattern Recognition Letters, 481-494, 2004.

[5] O. Kafri and E. Keren, Encryption of pictures and shapes by random grids, Optics Letters, Vol. 12, No. 6, pp. 377-379,1987.

[6] Shyong Jian Shyu, Image encryption by random grids, Science Direct, Pattern Recognition, 2006.

[7] D. Wang, L. Zhang, N. Ma, and X. Li, Two Secret Sharing Schemes Based on Boolean Operations, Pattern Recognition,Vol. 40, pp. 2776- 2785, 2007.

[8] Lin Dong, Daoshun Wang, Min Ku, Yiqi Dai, (2, n) secret image sharing scheme with ideal contrast, InternationalConference on Computational Intelligence and Security, pp 421-423, 2010.

[9] Lin, C. C., Laih, C. S., Yang, C. N., New Audio Secret Sharing Schemes With Time Division Technique, J. of Informa-tion Science and Engineering, 19, pp. 605-614, 2003.

[10] M. Ehdaie, T. Eghlidos, M. Reza Aref, A Novel Secret Sharinf from Audio Perspective, Internatioal Symposium onTelecommunications, 2008.

[11] S. Pati, P. R. Deshmukh, T. Chavan, V. Shastri., Reduced Size Share Audio Secret Sharing, International Conference onPervasive Computing (ICPC), 2015.

[12] D. Wang, L. Zhang, N. Ma, and X. Li, Two Secret Sharing Schemes Based on Boolean Operations, Pattern Recognition,Vol. 40, pp. 2776- 2785, 2007.

[13] Yoshida, K., Watanabe, Y. Security of audio secret sharing scheme encrypting audio secrets. International Conferencefor Internet Technology and Secured Transactions, pp. 294295. IEEE, London 2012.

[14] Abukari Yakubu, Namunu C. Maddage, Pradeep K. Atrey Audio Secret Management Scheme Using Shamirs SecretSharing, (LNCS) vol. 8935, pp. 396407, Springer 2015.

51


Polynomial associated with finite topologies

Moussa Benoumhani∗

Department of Mathematics, University of SharjahSharjah. UAE

[email protected]

Abstract

Let τ be a topology on the finite set Xn. We consider the open-set polynomial associated with the topologyτ . Its coefficients are the cardinalities of sets Oj = Oj(τ) of open sets of size j = 0, . . . , n.J. Brown asked when this polynomial has only real zeros. We prove that this polynomial has only real zerosonly in the trivial case where τ is the discrete topology. Then, we weaken Brown’s question: for which topologyis this polynomial log–concave, or at least unimodal? A partial answer is given. More specifically, we provethat if the topology has a large number of open sets, its open polynomial is unimodal.

Key words: open-set polynomial, log–concave sequence, polynomial with real zeros,topology, unimodal sequence.

2010 Mathematics Subject Classification: Primary 11B39; Secondary 11B75.

1. IntroductionA topology τ ⊆ P(X) on a set X is a family of subsets (called open sets) containing X, φ, and is closed under union andfinite intersection. A base or a basis for τ is subset of τ , such that each open set is a union of members of the basis. Let T (n)be the total number of topologies one can define on a finite set X = Xn of cardinality n. Little is known about this sequence;in fact, T (n) is known just for n ≤ 18.

Finite topologies are associated with digraphs [6, 10]. Another important topic linked to finite topologies is that ofpreorders (or quasiorders) which are reflexive and transitive relations. The one-to-one correspondence between topologiesand quasiorders was already observed in the 30’s of the last century by Alexandroff [1] . In this correspondence, the T0

topologies correspond to the orders (that is, antisymmetric quasiorders). Recall that a topology is T0 if, for every x 6= y, thereis an open set containing exactly one of them.

Recall also that, as a topology can be viewed as a graph [6], polynomials are powerful tools to study graphs. The oldestone related to graphs is certainly the chromatic polynomial investigate the four-colour problem.

We say that a polynomial has a certain property if its coefficients do have this property. For example, a polynomial ofdegree n is said to be symmetric or palindromic if ak = an−k holds for every 1 ≤ k ≤

⌊n2

⌋.

The independence polynomial of a graph, I(G, x), was introduced by Gutman and Harary [7], its coefficients ak arethe numbers of stable sets (sets of pairwise nonadjacent vertices) having cardinality k. A famous conjecture stated that theindependence polynomial of any tree is unimodal (the sequence of its coefficients first is increasing, then decreasing). Forchemical purposes, Hosoya [8] defined a polynomial H(G, x), now known as Hosoya polynomial as follows: the coefficientak of the Hosoya polynomial is the number of pairs of vertices whose distance is k, 1 ≤ k ≤ d, where d is the diameter of thegraph. The value H ′(G, 1) = W (G), is the famous Weiner index, an important chemical invariant. All of these polynomialsas well as their roots supply a lot of information about the graph itself.

∗Speaker52


In this paper, we consider the open-set polynomial P (τ, x) of a topology τ defined on a finite set X . Its coefficientsare the number of open sets of cardinality k, 0 ≤ k ≤ n, or, which is the same, the number of ideals of the correspondingquasiorder of cardinality k. The polynomials I(G, x) and P (τ, x) are related by I(G, 1) = P (τ, 1).This relationship betweenI(G, x) and P (τ, x) shows that the open-set polynomial is yet another one of these graph-counting polynomials.

In this talk, we will answer Brown’s question: which open-set polynomials have only real zeros? We then consider aweaker question than Brown’s, namely, which open-set polynomials are log-concave or at least unimodal? A partial answeris given to this question by proving that unimodality is guaranteed in case the topology has a large number of open sets. Weend the paper by some open questions concerning the open-set polynomials.

2. PreliminariesIn this section, we recall some definitions and results concerning the partitions, finite topologies, as well as the notions ofunimodality and log concavity of real sequences.

Definition 1. A partition topology τ on X , is a topology that is induced by a partition of X .

For example the topology τ = φ, a, b, c, d, e, a, b , c, d, a, b , e, c, d , e, X on X =a, b , c, d, e is induced by the partition π = (a, b, c, d, e).Let τ be a topology on Xn. Let us designate by uj the number of open sets in τ having cardinality j, and consider the

polynomial P (x) = P (τ, x) =n∑

j=0

ujxj , called the open–set polynomial of τ . J. Brown [4] raised the following question:

When are all of the roots of an open-set polynomial real?In the next section, we answer this question. Before doing this, we recall some facts about polynomials and log–concavesequences. A real positive sequence (aj)nj=0 is said to be unimodal , if there exist integers k0, k1, k0 ≤ k1,( integersk0 ≤ j ≤ k1 are the modes of the sequence) such that

a0 ≤ a1 ≤ . . . < ak0 = ak0+1 = . . . = ak1 > ak1+1 ≥ . . . ≥ an.

It is log-concave if a2j ≥ aj−1aj+1, for 1 ≤ j ≤ n − 1. A real sequence (aj)nj=0 is said to be with no internal zeros (NIZ),

if i < j, ai 6= 0, aj 6= 0 then al 6= 0 for every l, i ≤ l ≤ j. A (NIZ) log-concave sequence is obviously unimodal, butthe converse is not true. The sequence 1, 1, 4, 5, 4, 2,1 is unimodal but not log-concave. Note the importance of (NIZ):the sequence 0, 1, 0, 0, 2, 1 is log-concave but not unimodal. A real polynomial is unimodal (log-concave, symmetric,respectively) provided that the sequence of its coefficients is unimodal (log-concave, symmetric, respectively).If inequalities in the log-concavity definition are strict, then the sequence is called strictly log-concave (SLC for short), andin this case, it has at most two consecutive modes. The following result may be helpful in proving unimodality:

Theorem 2. If the polynomialn∑

j=0

ajxj associated with the sequence (aj)nj=0 has only real zeros then

a2j ≥

j + 1j

n− j + 1n− j aj−1aj+1, for 1 ≤ j ≤ n− 1 (1)

3. The roots of the open-set polynomial are not realThe main result

Theorem 3. The open-set polynomialn∑

j=0

ujxj has only real zeros if and only if τ is the discrete topology on Xn.

The natural question after this last result is a weak version of Brown’s question:

Which topologies have their open-set polynomials log-concave or at least unimodal ?

53


4. A Class of unimodal open polynomialsIntuitively, the open-set polynomial will be unimodal, if the topology has a large number of open sets (so, a large number ofopen sets which are singletons, this means that the topology looks like the discrete one). In this section, we will determineexplicitly the open-set polynomials of a class of topologies having a large number of open sets (τ is such that |τ | ≥ 6 · 2n−4

and another case where |τ | ≥ 5 · 2n−4). The unimodality of these polynomials is also proved.The symbol Xk will designate any finite set of cardinality k. If we consider a subset of Xn of cardinality k ≤ n − 1,the elements of Xn \ Xk will be designated by a, b, c, ..., or y1, y2, ..., if the set is large, those of Xk are denoted byx1, x2, x3, ...Consider the set τ(n, k) of the topologies having k open sets. In order to compute its cardinality, Kolli [9] divided it into twodisjoint subsets :

τ1(n, k) = τ ∈ τ(n, k) :⋂

U∈τU 6= φ, U nonempty set

and τ2(n, k) = τ(n, k)− τ1(n, k).

This means that τ1(n, k) is the set of topologies with k open sets having one minimal open set, while τ2(n, k) is the set oftopologies with k open sets having at least two minimal open sets. Before starting the computations, let us make the followingconvention. If τ1 is a topology on a Xj ⊆ Xn , by τ = τ1 ] A, B, C, ..., , we mean the topology τ on Xn obtained byadding or adjoining the sets A, B, C, ..., as well as all their intersections and unions with each other and with the elementsof τ1.

For τ ∈ τ1(n, k) and k ≥ 5.2n−4, all open-set polynomials are given in the following theorem.

Theorem 4. Let τ ∈ τ1(n, k) with k ≥ 5 · 2n−4 open sets. Then the open polynomial of τ is one of the three

P1(x) = x(x+ 1)n−1 + 1P2(x) = x2(x+ 1)n−2 + x(x+ 1)n−3 + 1P3(x) = x3(x+ 1)n−3 + 2x2(1 + x)n−4 + x(1 + x)n−4 + 1.

The remaining of the paper is devoted to the determination of the open-set polynomials of the topologies in the setτ2(n, k) and k ≥ 6 · 2n−4. We prove

Theorem 5. If τ is a topology on the set Xn, such that |τ | ≥ 6 · 2n−4, then its open-set polynomial is unimodal.

References[1] P. Alexandroff, Diskrete Raume. Mat. Sb. (N.S.) 2 (1937), 501–518.

[2] M. Benoumhani, The number of topologies on a finite set, J. Integer Seq. 9 (2006), Article 06.2.6.

[3] M. Benoumhani, M. Kolli, Partitions and topologies, J. Integer Seq. Vol. 13 (2010), Article 10.3.5.

[4] J. Brown, On the zeros of independence and open polynomials, Seminar. Isaac Newton Institute. Jan 2008.

[5] J. Brown, C. Hickman, H. Thomas, D. Wagner, Bounding the Roots of Ideal and open-set polynomials, Ann. Comb. ,vol. 9, no. 3, pp. 259-268 (2005).

[6] Evans, Harary, Lynn, On the computer enumeration of finite topologies, Comm. of A.C. M., 1967.

[7] I. Gutman and F. Harary, Generalizations of the matching polynomial, Utilitas Math. 24 (1983), 97–106.

[8] H. Hosoya, On some counting polynomials. Discrete Appl. Math. 1988, 19, 239–257.

[9] M. Kolli, Direct and elementary approach to enumerate topologies on finite set, J. Integer Seq. 10 (2007), Article 07.3.1.

[10] R.E. Merrifield and H.E. Simmons, Topological Methods in Chemistry, Wiley, New York, 1989.

[11] R. C. Read, An introduction to chromatic polynomials, J. Combin. Theory, 4 (1968), pp. 52–71.

54


Arithmetical triangles

Szalay, Laszlo1∗

1 University of West Hungary, Institute of MathematicsAdy E. ut 5., Sopron, Hungary

[email protected]

Abstract

We analyze some generalization directions of Pascal triangle.

Key words: Pascal triangle, generalization, Fibonacci sequence

2010 Mathematics Subject Classification: 05A10

1. IntroductionThe elements

(nk

)of the Pascal triangle represent the coefficients of xn−kyk (0 ≤ k ≤ n) in the expansion of the polynomial

(x+ y)n, therefore they are called binomial coefficients. It is known that(n

k

)=

n!k! · (n− k)!

, (0 ≤ k ≤ n) (1)

gives the number of ways of choosing k (or n − k) objects out of n distinguishable objects. Formula (1) or its combina-torial interpretation provides generalization possibilities of the Pascal triangle. Note that Pascal himself called the objectarithmetical triangle.

2. Main resultsNow we build up a sort of Generalized Arithmetical Triangle (in short GAT, see Figure 1) which is structurally identical withthe regular Pascal triangle. It resembles Ensley’s GAT [3], but here we allow a0 6= b0 in the generator sequences, further wealso vary the rule of addition.

Let an∞n=0 ∈ R∞ and bn∞n=0 ∈ R∞ be two real sequences called generator sequences, further let A, B ∈ R. This

GAT contains also rows numbered by 0, 1, 2, . . . such that the nth row possesses the elements⟨nk

⟩in the positions (say

columns) k = 0, 1, . . . , n as follows.

Let⟨

00

⟩be arbitrary denoted by Ω, for positive integer n put

⟨n0

⟩:= Anan and

⟨nn

⟩:= Bnbn, further for

n ≥ 2 and 1 ≤ k ≤ n− 1 let ⟨nk

⟩:= B

⟨n− 1k − 1

⟩+A

⟨n− 1k

⟩. (2)

∗Speaker55


Figure 1: GAT

Clearly, a GAT depends only on the generator sequences an, bn and on the real numbers A, B.

Since the definition of GAT is recursive, the aim is arising naturally to give a direct formula for the elements⟨nk

⟩by

the terms of the sequences an, bn and by the values A, B. Theorem 1 admits such a formula, where an extensions ofbinomial coefficients (1) to the whole lattice of integer vectors is provided by the Gamma function as follows. Put

(n

k

)

Γ

= limn1 → nk1 → k

Γ(n1 + 1)Γ(k1 + 1) · Γ(n1 − k1 + 1)

.

Theorem 1. Given an, bn, A, B. Then for any positive integer n and for any non-negative integer k ≤ n we have

⟨nk

⟩= An−kBk

n−k−1∑

i=0

(n− 2− ik − 1

)

Γ

ai+1 +k−1∑

j=0

(n− 2− jk − 1− j

)

Γ

bj+1

. (3)

The main difficulty in the evaluation of the formula is the two sums in Theorem 1, since no general treatment for arbitrarysequences an, bn. Hence one must consider specific cases. The next frame collects some relevant Pascal type triangles(or arrays) have been already studied. Apart from the last case we assume A = B = 1.

an bn Reference

1 1 Pascal Triangle (PT)

2 1 Hosoya [4] (Asymmetrical PT)

arbitrary arbitrary Ensley [3] (GAT)

Fn+1 Fn+1 Ensley [3] (shifted Fibonacci Triangle)

0 1n Dil – Mezo [1] (Hyperharmonic numbers)

0 Fn Dil – Mezo [1] (Hyper-Fibonacci numbers)

F2n−1 Fn−1 Dil – Mezo [1]

a b Belbachir – Szalay [2]

Here we grab out one example, namely the Fibonacci triangle (see Figure 2), which was essentially introduced by Ensley[3] (he took an = bn = Fn+1).

The triangle now is generated by an = bn = Fn, and the result is the following.

56


Figure 2: Fibonacci triangle

Theorem 2. ⟨nk

⟩

Fn

= Fn+k − qk(n), (4)

where

qk(x) = 2bk/2c∑

j=0

(x

k − 2j

)F2j (5)

is a rational polynomial of degree k − 2 if k ≥ 2, and q0(x) = q1(x) = 0.

In the next table we give the first few polynomials qk(x) explicitly.

q0(x) = 0,q1(x) = 0,q2(x) = 2,q3(x) = 2x,q4(x) = x2 − x+ 6,

q5(x) =13

(x3 − 3x2 + 20x),

q6(x) =112

(x4 − 6x3 + 47x2 − 42x+ 192).

References[1] Dil A., Mezo I., A symmetric algorithm for hyperharmonic and Fibonacci numbers, Appl. Math. Comp., 2008, 206,

942–951.

[2] Belbachir H., Szalay L., On the arithmetic triangles, Siauliai Math. Sem., 2014, 9(17), 15–26.

[3] Ensley D., Fibonacci’s triangle and other abominations, in: The Edge of the Universe: Celebrating Ten Years of MathsHorizons, MAA, 287–307, 2006.

[4] Hosoya H., Pascal’s triangle, non-adjacent numbers, and D-dimensional atomic orbitals, J. Math. Chemistry, 1998, 23,169–178.

57


Chu-Vandermonde identity for q-analogue and p, q-analogue ofbisnomial coefficients

A. Benmezai123∗, A. Bouyakoub2, H. Belbachir3

1 Univ. Algiers-3, RECITS LaboratoryDely Brahim, Fac. of Eco. Manag. Sc., Algiers, Algeria

[email protected] Univ. Oran-1, GEAN Laboratory

Po. Box 1524, ELM Naouer, 31000, Oran, [email protected]

3 USTHB, RECITS Laboratory DG-RSDTBP 32, El Alia 16111 Bab Ezzouar, Algiers, Algeria

[email protected]

Abstract

We extend the q-analogue of the Chu-Vandermonde identity to general Chu-Vandermonde identity associ-ated to the bisnomial coefficients. We also give the corresponding p, q-analogue.

Key words: Gaussian polynomials, recursions, generating function.

2010 Mathematics Subject Classification: 11B39, 05A15, 05A19.

1. IntroductionOne of the most important relations satisfied by the binomial coefficients is the Chu-Vandermonde identity

(n+m

k

)=

k∑

j=0

(n

j

)(m

k − j

).

This identity has a q-analogue satisfying by the q-binomial coefficients introduced by Gauss, and has a p, q-analogueversion, see [2].

Our aim is to extend the Chu-Vandermonde identity to a q-deformation of the generalized Chu-Vandermonde identityaccording to the q-bisnomial coefficients given in [1]. We also establish the corresponding p, q-analogue.

For n ≥ 0, the bisnomial coefficients(nk

)s

(s ≥ 1) , are defined by the expansion

(1 + x+ x2 + · · ·+ xs

)n=∑

k≥0

(n

k

)

s

xk, (1)

∗Speaker

58


We established in [1], for a = ei 2πs+1 with i2 = −1, that

(n

k

)

s

=∑

j1+j2+···+js=k

(n

j1

)(n

j2

)· · ·(n

js

)(−1)k

a−∑sr=1 rjr , (2)

and defined the q-analogue of bisnomial coefficients by

[n

k

](s)

q

:=∑

j1+j2+···+js=k

[n

j1

]

q

[n

j2

]

q

· · ·[n

js

]

q

q∑sr=1 (jr2 ) (−1)k

a−∑sr=1 rjr , (3)

where [n

k

]

q

=[n]q!

[k]q![n− k]q!, [n]q! = [1]q[2]q · · · [n]q , [n]q = 1 + q + · · ·+ qn−1.

Now, we propose the p, q-analogue of the bisnomial coefficient defined by

[n

k

](s)

p,q

:=∑

j1+j2+···+js=k

[n

j1

]

p,q

[n

j2

]

p,q

· · ·[n

js

]

p,q

p∑sr=1 (n−jr2 )q

∑sr=1 (jr2 ) (−1)k

a−∑sr=1 rjr ,

where [n

k

]

p,q

=[n]p,q!

[k]p,q![n− k]p,q!, [n]p,q! = [1]p,q[2]p,q · · · [n]p,q , [n]p,q = pn−1 + pn−2q + · · ·+ qn−1.

With this proposition we get the following identity

n−1∏

j=0

(s∑

t=0

(pjx)t (

qjy)s−t

)=

ns∑

k=0

[n

k

](s)

p,q

xkyns−k.

2. Main resultsLet[nk

](1)q

:= q(k2)[nk

]q. The q-analogue of the Chu-Vandermonde identity, (see for instance [2]), is given by

[n+m

k

]

q

=k∑

j=0

q(n−j)(k−j)

[n

j

]

q

[m

k − j

]

q

,

or equivalently [n+m

k

](1)

q

=k∑

j=0

qn(k−j)

[n

j

](1)

q

[n

k − j

](1)

q

.

The q-Chu-Vandermonde identity can be extended to the q-bisnomial coefficients as following.

Theorem 1. The q-analogue of bisnomial coefficients[nk

](s)q

satisfy the general Chu-Vandermonde identity associated to theq-bisnomial coefficients [

n+m

k

](s)

q

=k∑

j=0

qn(k−j)

[n

j

](s)

q

[n

k − j

](s)

q

.

And it’s extended to the p, q-bisnomial coefficients as following.

Theorem 2. The p, q-analogue of the generalized Chu-Vandermonde identity according to the p, q-bisnomial coefficients isgiven by [

n+m

k

](s)

p,q

=k∑

j=0

pn(k−j)qn(k−j)

[n

j

](s)

p,q

[m

k − j

](s)

p,q

.

59


References[1] Belbachir H., Benmezai A., A q-analogue for bisnomial coefficients and generalized Fibonacci sequences, Appl. Math.

Comput., 2014, 226, 691–698.

[2] Teseleanu G., New proof for the P,Q-analogue of Chu-Vandermonde’s identity, Integers, 2013, A51, 11.

[3] Cigler J., A new class of q-Fibonacci polynomials. Electron. J. Combin., 2003, 10 :Ressearch Paper 19, 15pp.

60


Triangles arithmetiques associees aux suites de Fibonaccigeneralises

Said Amrouche1∗, Hacene Belbachir1

1 USTHB University, RECITS laboratory,Faculty of MathematicsBP 32 Bab-Ezzouar, El-Alia 16111, Algiers, Algeria

said [email protected]@gmail.com

Resume

Soit s un entier non nul, on se propose de construire des triangles arithmetiques associes a la suite deFibonacci generalisee (s-bonacci) , ensuite une etude d’unimodalite et de log-concavite sur les suites recurrenteslineaires situees sur les transversales finies de ces triangles est etablie.

Key words : triangles arithmitiques, suite s-bonacci, unimodalite, log-concavite.

2010 Mathematics Subject Classification :05A10 .

1. IntroductionOn definit la suite s-bonacci comme suit :

Tn = Tn−1 + Tn−2 + · · ·+ Tn−s−1 tel que, T0 = 1, T−1 = 0, · · ·T−s = 0.

Il est bien connu que les termes de la suite de Fibonacci (Fn)n peuvent etre recuperer en sommant les elements diagonauxdu triangle de Pascal. Ainsi pour s = 1, Fn = Fn−1 + Fn−2 et Fn+1 =

∑k

(n−k

k

).

Alladi et Hoggat [1] ont propose pour, s = 2, un triangle arithemetique associe a la suite de tribonnaci, les elements de cetriangle sont obtenus par la recurrence suivante :

b(n, i) = b(n− 1, i) + b(n− 1, i− 1) + b(n− 2, i− 1),

avec b(n, 0) = b(n, n) = 1.

Barry [2] a donne une relation entre les elements du triangle associe a la suite de tribonacci et les coefficients binomiauxcomme suit :

b(n, i) =i∑

j=0

(i

j

)(n− ji

).

On ce propose d’etendre ces resultat a s quelconque

∗Speaker

61


n/i

0

1

2

3

4

5

6

0 1 2 3 4 5 6 ...

1

1

1

1

1

1

1

1

3

5

7

9

11

1

5

13

25

41

1

7

25

63

1

9

41

1

11 1...

FIG. 1 – triangle associe a la suite tribonacci.

2. un triangle arithmetique pour la suite s-bonacci

On note par(ni

)(s)les elements du triangle associe a la suite s-bonacci donne par la recurrence lineaire

(n

i

)(s)

=(n− 1i

)(s)

+(n− 2i− 1

)(s)

+ · · ·+(n− s+ 1i− 1

)(s)

.

avec (n0 )(s) = (n

n)(s) = 1.

Le resultat suivant exprime la relation entre les elements du triangle de Pascal et les coefficients (ni )(s) .

Theoreme 1. Les elements du triangle arithmetique associe a la suite s-bonacci :

(n

i

)(s)

=∑

0≤is−1≤···≤i2≤i1≤i

(i

i1

)(i1i2

)· · ·(n− i1 − i2 − · · · − is−1

i

). (1)

Si on somme les elements diagonaux du triangle associe a la suite s-bonacci on aura :

Theoreme 2. La somme des elements diagonaux associe a la suite s-bonacci satisfait la recurrence suivante :

Tn = Tn−1 + Tn−2 + · · ·+ Tn−s−1,

avec T0 = 1, T−1 = 0, · · ·T−s = 0.La forme explicite est donnee par

Tn =bn/2c∑

i=0

(n− i− 1

i

)(s)

.

3. Unimodalite et log-concaviteDans cette section on etablit l’unimodalite et la log-concavite des suites situees sur les transverasles finies du triangle

arithmetique associe a la suite s-bonacci.

62


Rappelons qu’une suite de nombres reels a0, a1, · · · , an est unimodale si pour un certain 0 ≤ j ≤ n on a a0 ≤ a1 ≤· · · · · · ≤ aj ≥ aj + 1 ≥ · · · ≥ an, et elle est log-concave si a2

i ≥ ai−1ai+1 pour tout 1 ≤ i ≤ n− 1.Notons que Tanny et Zuker [5] ont prouve que les suites situees sur la diagonale principale du triangle de Pascal sont unimo-dale. Belbachir et Szalay [3] ont etabli l’unimodalite de n’importe quelle suite situee sur les transverasles finies du trianglede Pascal.Le resultat suivant etablit la log-concavite de n’importe quelle suite situee sur les transversales finies du triangle arithmetiqueassocie a la suite s-bonacci.

Theoreme 3. La suite ((n−qip+ri

)(s))i est log-concave.

Dans [4], on trouve que si une suite positive ui est log-concave sans zero entrelaces , alors cette suite est unimodal, ainsi :

Corollaire 3.1. La suite ((n−qip+ri )

(s))i est unimodal.

References[1] Alladi K, Hoggat V.E, Jr,On tribonacci numbers and related functions, The Fibonacci Quarterly 1977,15, 1, 42-45.

[2] Barry P, On integer-sequence-based constructions of generalized Pascal triangles, Journal of Integer Sequences,2006,9, 06-2.4.

[3] Belbachir H, Szalay L. Unimodalirays in the regular and generalized Pascal triangles, Journal of Integer Sequences,2008, 11 , 08.2.4.7

[4] Brenti.F,unimodal,log-concave and polya frequency sequences in combinatorics,(Amer.Math.Soc.no), 1989,413

[5] Tanny S, Zuker M, On a unimodality sequence of binomial coefficients, Discrete Math 1974, 9 , 79-89 .

63


Central Bisnomial Coefficients on Hypergrids

Igueroufa Oussama1∗, Belbachir Hacene21 USTHB, RECITS laboratory

BP 32, El Alia, 16111, Bab Ezzouar, Algiers, [email protected]

2 USTHB, RECITS laboratoryBP 32, El Alia, 16111, Bab Ezzouar, Algiers, Algeria

[email protected]

Abstract

We study paths moving on an hypergrid, we reach some results concerning the stability by symmetry of theset of these paths under some conditions related to dimension and length. This permits to give the interpretationof the central bisnomial coefficients.

Key words: bisnomial coefficients, generalized Pascal triangle, hypergrid, suitable paths.

We need first to introduce some definitions and concepts.

Definition 1. [1] Let s ≥ 1 and L ≥ 0 be integers and k ∈ 0, 1, . . . , sL. The bisnomial coefficient denoted(Lk

)s, is the

kth element of the development

(1 + x+ x2 + · · ·+ xs)L =∑

k≥0

(L

k

)

s

xk.

As an example, see Table 1 for s = 3, we give the corresponding triangle. As central representation see Table 2. For the

L\k 0 1 2 3 4 5 6 7 8 9 10 11 120 11 1 1 1 12 1 2 3 4 3 2 13 1 3 6 10 12 12 10 6 3 14 1 4 10 20 31 40 44 40 31 20 10 4 1

Table 1: Bi3nomial coefficients(s = 3) :(Lk

)3.

properties of the bisnomial coefficients, we refer to [1, 2]. As the central binomial coefficient have the well known expressionBn =

(2nn

), n ∈ Z+, we define the central bisnomial coefficients.

∗Speaker

64


11 1 1 1

1 2 3 4 3 2 16 10 12 12 10 6

· · · 20 31 40 44 40 31 20 · · ·101 135 155 155 135 101

336 456 546 580 546 456 3361554 1918 2128 2128 1918 1554

5328 6728 7728 8092 7728 6728 5328...

......

...(2n

3n−3

)3

(2n

3n−2

)3

(2n

3n−1

)3

B3n

Table 2: Central bi3nomial coefficients in generalized Pascal triangle, for s = 3

Definition 2. Let n, s ∈ Z+, we define the central bisnomial coefficient relatively to the parity of the number s as follows

Bsn =

(2nsn

)s, If s is odd

(n

sn/2

)s, if s is even

Pascal triangle (s = 1) is a triangular array of binomial coefficients such that, each number in the triangle is the sum ofthe two numbers directly above it. (

n

k

)=(n− 1k − 1

)+(n− 1k

).

The generalized Pascal triangle is a triangular array of bisnomial coefficients such that each number in the triangle is thesum of s+ 1 numbers directly above it. (

L

k

)

s

=s∑

m=0

(L− 1k −m

)

s

.

Definition 3. A ”suitable path” on an hypergrid takes the axes in an increasing order: it passes from the ith axe to the(i+ p)th axe, p ≥ 1, starting from the origin.

The following theorem gives an interpretation of the bisnomial coefficients using an hypergrid of dimension L, see [3].

Theorem 4. The number of suitable paths of length k tracing on L axes (directions) without exceeding the frequency of svertices on a direction, is equal to

(Lk

)s.

The mathematical formulation is

#

0i01i12i2 · · · (L− 1)iL−1

/ L−1∑

m=0

im = k; i0, i1, . . . , iL−1 ≤ s

=(L

k

)

s

.

• The path 0i01i12i2 · · · (L − 1)iL−1 takes: i0 vertices on the first direction encoded by 0, i1 vertices on the seconddirection encoded by 1, . . . , iL−1 vertices on the Lth direction encoded by L.

• The path 0i01i12i2 · · · (L− 1)iL−1 can be written under the following form

00 · · · 0︸︷︷︸i0 times

11 · · · 1︸︷︷︸i1 times

22 · · · 2︸︷︷︸i2 times

· · · (L− 1)(L− 1) · · · (L− 1)︸︷︷︸iL−1 times

.

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a3200

a3210

a2200

a2310 a1310 a0310

a3300

a3310

a2300

a3000

a1300 a0300

a3220 a0200

a2000 a1000 O

a1200

a2320 a1320 a0320

a2330 a1330 a0330

a2010 a1010 a0010

a3320

a3010

a3100 a1100 a0100a2100

a3110

a2210 a1210

a2110 a1110 a0110

a0210

a3120 a2120 a1120 a0120

a2220 a1220 a0220

a2020 a1020 a0020

a1130

a3020

a3030 a2030 a1030 a0030

a1230 a0230

a3330

a3230 a2230

a3130 a2130 a0130

@@I

?

1st 2nd

3rd

Figure 1: suitable and unsuitable paths on a cube

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• The red path is suitable because it takes the axes in an increasing order (the first, the second and the third axe startingfrom the origin O), we encode it with 000112.

• The green path is unsuitable because the order is not respected (it takes the third axe then the first axe starting from theorigin O).

• The blue path is unsuitable because the order is not respected too (it takes the second axe, the third axe then it returnsto the second axe starting from the origin O).

Remark 1. The bisnomial coefficients count the number of particular increasing words.

Example 5. The number of suitable paths of length k = 6 tracing on an hypergrid of dimension L = 4, starting from theorigin O without exceed the frequency of three units on an axe (s = 3) is equal to

(46

)3

= 44.

#

0i01i12i23i3 ; i0 + i1 + i2 + i3 = 6; i0, i1, i2, i3 ≤ 3

=(

46

)

3

.

The following codes (binary words) represent the different suitable paths starting from the origin.We have 10 paths arranged in the following set (i3 = 0)000111, 000112, 001112, 000122, 001122, 011122, 000222, 001222, 011222, 111222We have 12 paths arranged in the following set (i3 = 1)000113, 001113, 000123, 001123, 011123, 000223, 001223, 011223, 111223, 002223, 012223, 112223We have 12 paths arranged in the following set (i3 = 2)000133, 001133, 011133, 000233, 001233, 011233, 111233, 002233, 012233, 112233, 022233, 122233We have 10 paths arranged in the following set (i3 = 3)000333, 001333, 011333, 111333, 002333, 012333, 112333, 022333, 122333, 222333This example shows that there is a correspondence between the distribution of suitable paths on the hypergrid and the

distribution of the bi3nomial coefficients 10, 12, 12, 10 directly above the number 44 in the generalized Pascal triangle, seefigure 2.

We identify each suitable path 0i01i12i2 · · · (L− 1)iL−1 with its end-point ai0,i1,...,iL−1 over the grid.

Corollary 5.1. Each end-point of a suitable path can not be reached by another suitable path.

We give now our main result.We take L = 2n and k = sn for n ∈ Z+ and s odd integer.

Definition 6. We define the set M of maximum words associated to the dimension L = 2n as following

M =

0i01i12i2 · · · (2n− 1)i2n−1

/ 2n−1∑

m=0

im = sn; i0, i1, . . . , i2n−1 ≤ s,

Definition 7. The symmetrized of a point ai0,i1,...,iL−1 is the point a′i′0,i′1,...,i′L−1, such that: i′j = s− ij ,

j ∈ 0, 1, . . . , L− 1.

We can verify that if the point ai0,i1,...,iL−1 is in the setM, then its symmetrized a′i′0,i′1,...,i′L−1is also inM, so we conclude

the following result.

Theorem 8. The set M is stable by symmetry.

Theorem 9. If k 6= sn, the set

0i01i12i2 · · · (2n− 1)i2n−1

/ 2n−1∑

m=0

im = k; i0, i1, . . . , i2n−1 ≤ s,

is not stable by symmetry.

Theorem 10. Bsn is the cardinal of the set M.

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References[1] Phd Thesis H. Belbachir, USTHB, 2007.

[2] H. Belbachir, S. Bouroubi, A. Khelladi, Connection between ordinary multinomials, Fibonacci numbers, Bell polyno-mials and discrete uniform distribution,35, 2008,21–30.

[3] O. Igueroufa, H. Belbachir, JSL conference 6-7 Juin 2015, USTHB.

68


Monomorphic decomposition and profile.

Djamila OUDRAR1∗,

1 USTHB, Faculty of MathematicsBp 32, El Alia 16111, Bab Ezzouar, Algiers

[email protected]

AbstractWe present a structural approach of some results about jumps in the behavior of the profile (alias

generating functions) of hereditary classes of finite structures. Considering the notion of monomorphicdecomposition of a relational structure, we show that if a structure decomposes into finitely manyclasses then the profile of its age is polynomial, otherwise it contains a minimal substructure with thesame property. We deduce that the profile of a hereditary classe of ordered binary structures (withfinite signature) is either polynomial or bounded below by an exponential. This result was previouslyobtained for ordered graphs by Balogh, Bollobás and Morris [2]

Key words: relational structures, asymptotic enumeration, profile, ordered graphs, tournaments.

2010 Mathematics Subject Classification: 05C30, 06F99, 05A05, 03C13.

1. IntroductionThe profile of a class C of finite relational structures is the integer function ϕC which counts for each non negativeinteger n the number of members of C on n elements, isomorphic structures being identified. Numerous papersdiscuss the behavior of this function when C is hereditary (that is contains every substructure of a memberof C ) and is made of graphs (directed or not), of tournaments, of ordered sets, of ordered graphs, of orderedhypergraphs... Futhermore, it turns out that the line of study about permutations (see [1]), originating in theStanley-Wilf conjecture, solved by Marcus et Tardös (2004) [14], falls under the frame of the profile of hereditaryclasses of relational structures. The results show that the profile cannot be arbitrary, there are jumps in its possiblegrowth rate. Typically, it is polynomial or faster than every polynomial and for several classes of structures, eitherat least exponential (e.g. for tournaments [6], ordered graphs and ordered hypergraphs [3, 11], permutations [10])or at least with the growth of the partition function (e.g. for graphs [4]). For more, see the survey of Klazar [12].

Here, we present a structural approach of some jump results based on the following notion. A monomorphicdecomposition of a relational structure R is a partition of its domain V (R) into a family of sets (Vx)x∈X such thatthe restrictions of R to two finite subsets A and A′ of V (R) are isomorphic provided that the traces A ∩ Vx andA′ ∩ Vx have the same size for each x ∈ X (see [19]). We show that if a class C is made of ordered structureswhich have finite monomorphic decomposition then its profile is a polynomial. Let Sµ be the class of relationalstructures of signature µ which do not have a finite monomorphic decomposition. We show that if a hereditarysubclass D of Sµ is made of ordered relational structures then it contains a finite subset A such that every memberof D embeds some member of A. In the case of ordered binary structures, A has one thousand two hundred andforty six members. From the description of these members we show that the profile of the age A(R) of an ordered∗Speaker 69


binary relational structure R belonging to D is at least exponential. We deduce that a hereditary class of finiteordered binary structures whose profile is not bounded by a polynomial is at least exponential. Part of these resultshave been announced in [17]. They are included in chapters seven and eight of our doctoral thesis [16] defendedon september 28, 2015 at the faculty of mathematics of USTHB under the supervision of professors Moncef Abbasand Maurice Pouzet.

2. Basic notions, definitions and main resultsOur terminology follows Fraïssé [7]. A relational structure is a pair R := (V, (ρi)i∈I) where each ρi is a ni-aryrelation on V ; the set V is the domain, denoted by V (R), the sequence µ := (ni)i∈I is the signature. This is abinary relational structure, binary structure for short, if it is made only of binary relations. It is ordered if oneof the relations ρi, say for example ρ1, is a linear order. A relational structure R is embeddable in a relationalstructure R′, in notation R ≤ R′, if R is isomorphic to an induced substructure of R′. A class C of structuresis hereditary if it contains every relational structure which can be embedded in some member of C . The class offinite relational structures of signature µ is denoted by Ωµ. It is quasi-ordered by embeddability.

Let R be a relational structure. A subset V ′ of V (R) is a monomorphic part of R if for every integer k andevery pair A, A′ of k-element subsets of V (R), the induced structures on A and A′ are isomorphic wheneverA \ V ′ = A′ \ V ′. A monomorphic decomposition of R is a partition P of V (R) into monomorphic parts.A monomorphic part which is maximal for inclusion is a monomorphic component of R. The monomorphiccomponents of R form a monomorphic decomposition of R of which, every monomorphic decomposition of R isa refinement (Proposition 2.12 of [19]). Revisiting these notions, we may define this partition in a direct way asfollows.

Let x and y be two elements of V (R). Let F be a finite subset of V (R) \ x, y, we say that x and y of areF -equivalent and we set x 'F,R y if the restrictions of R to x ∪ F and y ∪ F are isomorphic. Let k be anon-negative integer, we set x 'k,R y if x 'F,R y for every k-element subset F of V (R) \ x, y. We set x '≤k,R yif x 'k′,R y for every k′ ≤ k and x 'R y if x 'k,R y for every k. This defines three equivalence relations on V (R).As it turns out:

Lemma 1. The equivalence classes of 'R are the maximal parts of R.

In the case of binary structures or of ordered structures there is a threshold phenomenon.

Lemma 2. The equivalences relations '≤6,R and 'R coincide on a binary structure. If R is a directed graph,resp. an ordered graph, we may replace 6 by 3, resp. by 2. If T is a tournament, the number of equivalences classesof '≤3,T is finite provided that the number of equivalence classes of '≤2,T is finite. There is an integer i(m) suchthat on an ordered structure of arity at most m the equivalences relations '≤i(m),R and 'R coincide.

The case of binary structures follows from a reconstruction result of Lopez [13]. The case of directed graphswas obtained independently by Boudabbous [5]. The case of ordered structures follows from a result of Ille [9].Using a result of [18] one can show that there is no threshold for ternary relations.

If a structure R has infinitely many classes and 'R coincide with '≤k,R then one can find a family of mapsf : N → V (R), gi : [N]2 → V (R) for i < k − 1 such that for every n < n′ ∈ N, f(n) and f(n′) are notgi(n, n′) : i < k − 1-equivalent. The restriction of R to the union of the images of this collection of maps hasinfinitely many equivalence classes. Ramsey’s theorem allows to find an infinite subset X ⊆ N on which the mapsof this family "invariant" with respect to R (the notion of invariance is explained in 2.2 of [6]). It turns out that ifR is an ordered binary structure, the profile of the resulting structure is at least exponential. On an other hand,it is easy to see that if an infinite relational structure has a finite monomorphic decomposition into k + 1 blocks,then its profile is bounded by some polynomial whose degree is, at most, k. In fact, and this is the main resultof [19] this a quasi-polynomial whose degree is the number of infinite monomorphic component minus 1 (that is asum ak(n)nk + · · ·+ a0(n) whose coefficients ak(n), . . . , a0(n) are periodic functions). We show that if an orderedstructure has a finite monomorphic decomposition into k + 1 blocks, then its profile is a polynomial. This yieldsthe following dichotomy result for an ordered binary structure:

Theorem 3. An ordered binary structure has either a finite monomorphic decomposition, in which case the profileof its age is polynomial, or not, in which case this profile is at least exponential.

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Let Sµ be the class of all relational structures of signature µ which do not have a a finite monomorphicdecomposition.

Conjecture 1. There is a finite subset A made of incomparable structures of Sµ such that every member of Sµ

embeds some member of A.

This conjecture holds if we replace Sµ by a hereditary class made of binary structures or of ordered binarystructures.

Examples 1. • If Sµ is made of bichains, A has twenty elements, [15].

• If Sµ is made of tournaments, A has twelve elements, [6].

• If Sµ is made of graphs, A has ten elements.

• If Sµ is made of ordered reflexive (or irreflexive) graphs, A has one thousand two hundred and forty sixelements.

In the last item A is made of members whose profile has exponential growth, in fact as fast as the Fibonaccisequence.

References[1] M.H. Albert, M.D. Atkinson, Simple permutations and pattern restricted permutations. Discrete Mathemat-

ics, 2005, 300, 1–15.

[2] J. Balogh, B. Bollobás, R. Morris, Hereditary properties of partitions, ordered graphs and ordered hypergraphs.European Journal of Combinatorics, 2006, 8, 1263–1281.

[3] J. Balogh, B. Bollobás, R. Morris, Hereditary properties of ordered graphs, in Topics in discrete mathematics,Algorithms Combin.,2006, 26, Springer, Berlin, 179–213.

[4] J. Balogh, B. Bollobás, M. Saks, V. T. Sós, The unlabelled speed of a hereditary graph property, J. Combi-natorial Theory., 2009, series B 99, 9–19.

[5] Y. Boudabbous, Personnal communication, August 2013.

[6] Y. Boudabbous, M. Pouzet, The morphology of infinite tournaments; application to the growth of their profile,European Journal of Combinatorics., 2010, 31, 461–481.

[7] R. Fraïssé, Theory of relations. Second edition, North-Holland Publishing Co., Amsterdam, 2000.

[8] C. Frasnay, Quelques problèmes combinatoires concernant les ordres totaux et les relations monomorphes,Thèse. Paris. Annales Institut Fourier Grenoble 15 (1965), pp. 415–524.

[9] P. Ille, The reconstruction of multirelations, at least one component of which is a chain. J. Combin. Theory.,1992, Ser. A, 61, no. 2, 279–291.

[10] T. Kaiser, M. Klazar, On growth rates of closed permutation classes. Permutation patterns Electron. J.Combin., 9, (2002/03), no. 2, (Otago, 2003), Research paper 10, 20 pp.

[11] M. Klazar, On growth rates of permutations, set partitions, ordered graphs and other objects. Electron. J.Combin., 2008, 15, no. 1, Research Paper 75, 22 pp.

[12] M. Klazar, Overview of general results in combinatorial enumeration, Permutation patterns, London Math.Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge, 2010, Vol. 376, 3–40.

[13] G. Lopez, L’indéformabilité des relations et multirelations binaires. Z. Math. Logik Grundlag. Math., 1978,24, no. 4, 303–317.

[14] A. Marcus, G. Tardös, Excluded permutation matrices and the Stanley-Wilf conjecture, J. Combin. Theory,2004, Ser. A (107), 153–160.

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[15] T. Monteil, M. Pouzet, From the complexity of infinite permutations to the profile of bichains. In ROGICS’08:International Conference on Relations, Orders and Graphs: Interaction with computer science, pages 1–6,(2008).

[16] D. Oudrar, Sur l’énumération de structures discrète. Une approche par la théorie des relations. Doctoralthesis. University of sciences and technology Houari Boumediene, U.S.T.H.B., Algiers (2015) 237 pages.

[17] D. Oudrar, M. Pouzet, Décomposition monomorphe des structures relationelles et profil de classes héréditaires.arXiv:1409.1432[math.CO]., 2014, 7 pp.

[18] M. Pouzet, Relations non reconstructibles par leurs restrictions, ăJ. Combin. Theory., 1979, Ser. B, 26, no. 1,22–34.

[19] M. Pouzet, N. M. Thiéry. Some relational structures with polynomial growth and their associated algebras I:Quasi-polynomiality of the profile. Electron. J. Combin., 2013, 20(2), 1– 35.

72


Coloration d’incidences et des carres des graphes circulants et desgraphes distances

Brahim BENMEDJDOUB1∗, Isma BOUCHEMAKH2 Eric SOPENA3

1 USTHB, Faculty of Mathematics, L’IFORCE LaboratoryB.P. 32 El-Alia, Bab-Ezzouar, Algiers, Algeria

[email protected] USTHB, Faculty of Mathematics, L’IFORCE Laboratory

B.P. 32 El-Alia, Bab-Ezzouar, Algiers, Algeriaisma [email protected]

3 Univ. Bordeaux, LaBRIUMR5800, F-33400 Talence, France

[email protected]

AbstractSoit G un graphe non oriente. Une incidence de G est un couple (u, e) ou u est un sommet, et e une arete

incidente au sommet u (une incidence peut ainsi etre assimilee a une demi-arete ). Deux incidences (u, e) et(v, f) sont adjacentes si (1) u = v, ou (2) e = f , ou (3) e = uv. Une k−coloration d’incidence d’un grapheG est une fonction associant une couleur prise dans l’ensemble 1, 2, ..., k a chaque incidence de G de facontelle que deux incidences adjacentes aient des couleurs distinctes. Le nombre chromatique d’incidence de G,note χi(G), est alors le plus petit entier k tel que G admet une k−coloration d’incidence.

Dans cet expose, nous allons determiner le nombre chromatique d’incidence et du carre pour certains casdes graphes circulants et distances.

Key words: Nombre chromatique; carre d’un graphe; Graphe circulant; Coloration d’incidences; Graphe distance.

2010 Mathematics Subject Classification: : 05C15,05C70.

1. IntroductionSoient n,m deux entiers positifs et soitD = a1, a2, ..., am un ensemble d’entiers strictement positifs. Un graphe circulant,note G(n,D) ou Cn(a1, a2, ..., am), est defini par l’ensemble de ses sommets V = 0, 1, ..., n − 1 et l’ensemble de sesaretes E = i, i+ aj [n], 0 ≤ i ≤ n− 1, 1 ≤ j ≤ m.Un graphe distance, note G(D) est defini par l’ensemble de ses sommets Z et l’ensemble de ses aretes E = i, i+ aj, 0 ≤i ∈ Z.Les colorations d’incidence ont ete introduites par Brualdi et Massey [3] qui ont conjecture que pour tout graphe G, χi(G) <=Delta(G)+2. Cette conjecture s’est averee fausse par la suite mais plusieurs classes de graphes verifient cette relation. C’estnotamment le cas des graphes cubiques [4] des graphes planaires exterieurs ou, plus generalement, des 2-arbres partiels [5].le carre d’un graphe G note G2, est le graphe ayant meme ensemble de sommets que G tel que deux sommets sont adjacentsdans G2 si et seulement si leur distance dans G est d’au plus 2.

∗Speaker73


2. Main results

2.1. Coloration d’incidences et des carres des graphes distances

Theorem 1. [6] Pour tout graphe G ayant au moins une arete, χ(G2) ≤ k si et seulement si G admet une K−colorationd’incidence.

Proposition 1. Soit G(D) un graphe distance:Si D = 1, a alors χi(G) = χ(G2) = 5 si et seulement si a ≡ 2[5] ou a ≡ 3[5].

Theorem 2. Soit G(D) un graphe distance. Si D = 1, a alors:

χi(G) = χ(G2) =

5 a ≡ 2[5] ou a ≡ 3[5];6 sinon.

Proposition 2. Soit G(D) un graphe distance:Si D = 1, 2, ...,m, a alors χi(G) = χ(G2) = 2m+ 3 si et seulement si a ≡ m+ 1[2m+ 3] ou a ≡ m+ 2[2m+ 3].

Theorem 3. soit G(D) un graphe distance, tel que D = 1, 2, ...m, a:

χi(G) =

2m+3 si a ≡ m+ 1[2m+ 3] ou a ≡ m+ 2[2m+ 3];2m+4 sinon.

Corollary 3.1. soit G(D) un graphe distance, tel que D = 1, 2, a:

χi(G) =

7 si a ≡ 3[7] ou a ≡ 4[7];8 sinon.

Proposition 3. Soit G(D) un graphe distance:Si D = 1, a, a+ 1 alors χi(G) = χ(G2) = 7 si et seulement si a ≡ 2[7] ou a ≡ 4[7].

Theorem 4. Soit G(D) un graphe distance tel que D = 1, 2, 3, ...a0, a1, a2, ..., am:si a1 − a0 ≤ (2a0 + 1) et ai+1 − ai ≤ (2a0 + 1) ∀i, j ∈ 1, ...,m, alors: χi(G) = χ(G2) = ∆ + 1 si et seulement si:

1. ai + aj 6= a0 + am + 1 ∀i, j ∈ 1, ...,m2. a0 + am = 2|D|

Corollary 4.1. Soit G(D) un graphe distance:Si D = 1, 3, 5, ..., 2p− 1 alors χi(G) = χ(G2) = 2p+ 1 = ∆ + 1.

Corollary 4.2. Soit G(D) un graphe distance:Si D = 1, 2, 4, ..., 2p− 2 alors χi(G) = χ(G2) = 2p+ 1 = ∆ + 1.

2.2. Coloration d’incidences et des carres des graphes circulants

Theorem 5. [1] Soit D = a. Si G(n,D) est connexe, alors

χ(G(n,D)) =

2, si n est pair;3, sinon.

Theorem 6. [1] Soit D = a, b, si G(n,D) est connexe, alors:

χ(G(n,D)) =

2, si a et b sont impairs et n est pair;4, si 3 ne divise pas n, n 6= 5, et (b ≡ ±2a[n]) ou (a ≡ ±2b[n]);4, si n = 13 et (b ≡ ±5a[13]) ou (a ≡ ±2b[13]);5, si n est pair;3, sinon.

Theorem 7. [2] Soient D = a, b, c et n ≥ 7. Si |c|n = |a+ b|n alors χ(n,D) = 4 sauf dans les cas suivants:

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1. Ni a, ni b ni c sont divisibles par 3 et n ≡ 0[3], dans ce cas χ(n,D) = 3.

2. D = 1, 2, 3 et n 6≡ 0[4], n 6= 7, 11, dans ce cas χ(n,D) = 5.

3. D = 1, 3, 4 et n ∈ 13, 17, 18, 25, ou D = 1, 7, 8 et n ∈ 19, 26, ou D = 1, 6, 7 et n = 33, ou D =1, 10, 11 et n = 37, dans tous ces cas χ(n,D) = 5.

4. D = 1, 2, 3 et n = 7 (χ(n,D) = 7) ou n = 11 (χ(n,D) = 6).

Corollary 7.1. Soit G(n,D) un graphe circulant:Si D = 1, a alors χi(G) = χ(G2) = 5 si et seulement si n ≡ 0[5] et (a ≡ 2[5] ou a ≡ 3[5]).

Theorem 8. Soit G(n,D) un graphe circulant. Si a = 3 alors:

χ(G2) =

5 si n est un multiple de 5;7 si n=7,8,9,11,13,14,16,19,21,26,31;6 sinon.


χ(G2) =

6 si n est un multiple de 6;7 sinon.


χ(G2) =

7 si n=13,14,15,16,17,18,19,20,21,25,26,27,28,29,30,31,32,37,38,39,40,43,49,50,51,61,62,73;

6 sinon.

Theorem 11. Soit G(n,D) un graphe circulant:. Si a = 6 alors:

χ(G2) =

7 si n=13,14,15,16,19,21,23,24,25,26,27,28,29,31,32,35,37,38,39,40,42,43,45,,46,48,49,50,53,54,56,57,59,61,62,65,67,73,78,79,86,97;

6 sinon.


χ(G2) =

5 si n est un multiple de 5;7 si n=19,21,26,27,31,32,37,38,43,49,56;6 sinon.

Theorem 13. Soit G(n,D) un graphe circulant. Si a ∈ 3, 7, 8 alors:

χi(G) =

5 si n est un multiple de 5;6 sinon.

Theorem 14. Soit G(n,D) un graphe circulant. Si a ∈ 4, 5, 6 alors:χi(G) = 6

3. ConclusionDans cet expose nous nous avons determine le nombre chromatique dincidence et le nombre chromatique du carre pourcertaines classe des graphes circulants et distances. Nous allons essayer de prouver la conjecture de Brualdi et Massey pourles graphes circulants G(n,D) avec D = 1, a.

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References[1] C.Heuber, On planarity and colorability of circulant graphs. Discrete math. 268(2003) 153-169.

[2] M. Meszka, R. Nedala, A. Rosa, Circulants and the chromatic index of steiner triple systems, Math. Slovaca 56 (2006)371-378, 129-146.

[3] R.A. Brualdi, J.Q. Massey. Incidence and strong edge colorings of graphs. Discrete Math. 122, 1993.

[4] M. Maydanskyi. The incidence coloring conjecture for graphs of maximum degree 3. Discrete Math. 292 (2005), 131-141.

[5] M. Hosseini Dolama, . Sopena, X. Zhu. Incidence coloring of kdenegerated graphs. Discrete Math. 283 (2004), 121-128.

[6] M. Hosseini Dolama. Contribution l’tude de quelques de problme de coloration de graphes. These de doctorat. (2005).

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The upper paired-domination subdivision numbers of graphs

Fatma MESSAOUDI1∗, Miloud MIHOUBI2

1 USTHB, RECITSPB 32 El Alia 16111 Algiers, Algeria

radia−[email protected] USTHB, RECITS

PB 32 El Alia 16111 Algiers, [email protected]

Abstract

A paired-dominating set of a graphG = (V,E) with no isolated vertex is a set S of vertices ofG such that everyvertex is adjacent to some vertex in S and the subgraph G[S] induced by S contains a perfect matching. theupper paired-domination number of G, denoted by Γpr(G) is the maximum cardinality of a paired-dominatingset ofG. The upper paired domination subdivision number sdΓpr (G) is the minimum number of edges that mustbe subdivided (each edge inG can be subdivided at most once) in order to increase the upper paired -dominationnumber. In this paper we establish upper bounds and give exact value for both the upper paired-dominationnumber and the upper paired-domination subdivision number for some graphs (trees, cycles ,bipartite graphs,gird..).

Key words: domination set, paired domination, domination subdivision number.

2010 Mathematics Subject Classification: .

1. IntroductionLet G = (V,E) be a graph without isolated vertices. we denote by n the order of G and by m the size of G. The minimumand maximum degrees of G are respectively denoted by ∆(G) and δ(G) .A leaf is a vertex of degree 1 and a support vertex is a vertex adjacent to the leaf. A support vertex is strong if it is adjacentto more than one leaf.A star K1,m is a tree obtained from the graph K1 by attaching m leaf. A corona of G is the graph constructed from a copyof G, where for each vertex v, a new vertex v

′and a pendant edge are added.For a detailed treatment of this parameter,the

reader is referred to[3].A set S of G is a dominating set of G if every vertex not in S is adjacent to a vertex in S. A matching in graph G is a set ofindependent edges in G,a perfect matching M in G is a matching in G such that every vertex of G is incident to an edge ofM .The paired domination was introduced by Haynes and Slater(1998, 1995) as a modal for assigning backups to guards forsecurity purpose,since this time many results have been obtained on this parameter [1.4.7.13].The upper paired-domination number Γpr(G) = max |S|: S a minimal paired domination set .

∗Speaker

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2. Main resultsObservations

• Every graph G without isolate vertex have a paired-domination (upper paired domination set ).

• The paired-domination number for any graph G is even number.

• The paired-domination subdivision number of the graphK2 does not change because only one edge can be subdividedthen the degree n of G must be at least equal 3.

Upper bounds:In this section,we give a value of the upper paired domination number for particular a graphs (Paths,cycles,trees, grid,

complete and bipartite complete graphs...). we show that for every cycle or path we have 1sdΓpr(G)3. we give exact value of

sdΓpr(G) for complete graph, bipartite complete graph, a star and subdivided star, also we give upper bound for this number

for trees, grid and triangular graph.

3. ConclusionIn this article,we give some definitions of particular graphs and their upper paired domination number with the subdivisionnumber for this kind of domination.

References[1] M.Atapour, A.Khodkar S.M.Sheikholeslami, Characterization of double domination subdivision numbers of

trees,webset

[2] L.S. Hopkins,Bounds on total domination subdivisio numbers,These Master, East Tennessee State University.,(2003).

[3] T.W. Haynes, S.T. Hedetniemi et P.J. Slater, Fundamentals of domination in graphs,it Marcel Decker, Inc .NewYork.,(1998)

[4] T.W. Haynes, S.T. Hedetniemi S.M.Hedetniemi,Domination and independencesubdivision numbers of graphs (2000)

[5] T. W. Haynes, S.T S .M. Hedetniemi, J.knisely L.C.Van Der Merwe,Domination subdivision numbers, discussion math-ematic graph th eory21,(2001) 239- 253

[6] T. W. Haynes P.J. Slater, dominations in graphs. Networks 32(1998)

[7] T. W. Haynes,S.T.Hedetniemi L.C. Van dermmerwe,total domination subdivision numbers,J.Comb.Math.comp.44(2003)

78


Attacks on ElGamal digital signature scheme : an overview

Omar Khadir1∗1 Laboratory of Mathematics, Cryptography and Mechanics, Fstm,

University Hassan II of Casablanca, [email protected]

AbstractThe aim of this communication is to analyse the security of an important protocol in public key cryptogra-

phy. More precisely, we review the main attacks against the ElGamal digital signature.

Key words: Public key cryptography, digital signature.

2010 Mathematics Subject Classification: 94A60

1. IntroductionPublic key cryptography started with the publication of the famous paper of Diffie and Hellman : ”New directions in cryp-tography” [1]. One of the most important field in this topic is probably the digital signature protocol. Its requirement ine-business for funds transferring, makes it a sensitive question. Let us recall the principle. For the user Alice we prepare twokind of keys. The first, y, is public and must be largely diffused to the other users. The second, x, is private and must be keptsecret. When Alice decides to sign a document M , she has to solve a difficult problem, in general a mathematical equation.This problem is depending of Alice public key y and of the document M . It is constructed in a way such that nobody, exceptAlice, can solve it. With the help of her secret key x, Alice is able to give the answer. The equation is based on a hard questionin mathematics like factorization or discrete logarithm problem. We cannot forge Alice signature, but anyone like a judge canverify that the solution she gives is valid.One of the first proposed digital signatures is Elgamal protocol.1. Alice chooses three numbers:

- p, a large prime integer.- α, a primitive element (or a generator)of the finite multiplicative group Z∗

p

- x, a random element belonging to the set 2, 3, ..., p− 2.Then she computes y = αx mod p. Alice public keys are (p, α, y), and x is her private key.2. To sign the message m, Alice needs to solve the equation :

αm ≡ yrrs [p] (1)

where r, s are the unknown variables.Alice fixes arbitrary r to be r = αk mod p, where k is chosen randomly and invertible modulo p− 1. Parameter k is called anonce. Equation (1) is then equivalent to :

m ≡ xr + ks [p− 1] (2)

As Alice knows the secret key x, and as the integer k is invertible modulo p− 1, she computes the second unknown variable

s: s ≡ m− xrk

[p− 1]3. Bob can verify the signature by checking that congruence (1) is valid for the variables r and s given by Alice.In this work we review the main attacks against this digital signature. Namely:

∗Speaker79


2. Main attacks

Attack 1:In 1996 at Eurocrypt’96, Bleichenbacher has showed, that if α divides p− 1 then he can sign any document without knowingthe secrete key x:Theorem 1. ([3]) If α is B smooth and divides p1 then it is possible to generate a valid ElGamal signature on any givenmessage.

Attack 2:In 1999, Kuwakado and Tanaka [6] proved that, when we use ElGamal method to sign two documents, if the secret noncesk1, k2 are less than the square root of the prime modulus p, then we can compute the secret key of the signer and break all thesystem. Thay gave an algorithm which works in a polynomial time.

Attack 3:In 2015 we extend this attack by proving that if we can find a positive integer i, coprime with p − 1, such that αi mod pis smooth and divides p − 1, then it is also possible to sign any given document without knowing the secrete key. Thecontributions must be in the following fields:Theorem 3.([4]) Let (p, α, y) be Alice public key in an ElGamal signature protocol. Suppose that p ≡ 1 [4]. If we cancompute a natural integer i, coprime to p1, such that ai mod p is B−smooth and divides p− 1, then it is possible to generatea digital signature for any given document without knowing Alice private key.There are many questions on ElGamal signature security, that still without answer. For example is breaking ElGamal signatureprotocol equivalent to solving the discrete logarithm problem ?

3. ConclusionIn this communication, we analyzed the security of ElGamal digital signature scheme and discussed the most importantattacks against this protocol.

References[1] W. Diffie and M. E. Hellman, New directions in cryptography, IEEE Transactions on Information Theory, 1976, 2015,

vol. IT-22, 644–654

[2] T. ElGamal, A public key cryptosystem and a signature scheme based on discrete logarithm problem, IEEE Trans. Info.Theory, IT-31, 1985, 469–472.

[3] D. Bleichenbacher, Generating ElGamal signatures without knowing the secret key, In Advances in Cryptology, Euro-crypt’96, LNCS 1070, 1996, Springer-Verlag, pp. 10− 18.

[4] O. Khadir, Insecure primitive elements in an ElGamal signature protocol, Int. J. of Math. Sci and Cryptograhy, 2015,Vol. 18, no. 3, 237–245.

[5] O. Khadir, Conditions on the generator for forging ElGamal signature scheme, Int. J. of Pure and Applied Mathematics,2011, Vol. 70, no. 7, 939–949.

[6] H. Kuwakado and H. Tanaka. On the security of the ElGamal-type signature scheme with small parameters, IEICETrans. Fundamentals Electron. Commun. Comput. Sci., 1999, E82, pp. 93–97.

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Scheduling the two-machine flow shop problem with unit-timeoperations and conflict graph

Nour El Houda TELLACHE∗, Mourad BOUDHAR

RECITS Laboratory, Faculty of Mathematics, USTHB universityBP 32 El-Alia, Bab-Ezzouar, Algiers, Algeria

[email protected], [email protected]

Abstract

This paper addresses the problem of scheduling, on a two-machine flow shop, a set of unit-time operationssubject to the constraints that some conflicting jobs cannot be scheduled simultaneously. In the context of ourstudy, these conflicts are modeled by general graphs. The problem of minimizing the maximum completiontime (makespan) is shown to be NP-hard in the strong sense. We propose heuristic and lower bound proceduresand provide a computer simulation to measure the performance of these procedures for a wide range of testproblems with known optimal solutions.

Key words: Scheduling, flow shop, conflict graph, lower bound, heuristic.

2010 Mathematics Subject Classification: 90B35.

1. IntroductionWe consider the following scheduling problem: n jobs Jj , j = 1, . . . , n has to be processed on two dedicated processorsM1, M2. Each job Jj has two operations requiring one unit of running time, and must be processed first on M1 and nexton M2. Furthermore, the jobs are subject to the constraints that some conflicting jobs cannot be executed at the same timeon different machines. These constraints can be modeled by an undirected graph G = (V,E), such that jobs represented byadjacent vertices in G must run in disjoint time intervals. The graph G is called the conflict graph and the graph G = (V,E)(called the agreement graph) denotes the complement of the conflict graph G, unless mentioned otherwise. As far as weknow, this model has not been considered in the literature yet. This problem can be solved in polynomial time if G is a clique(all the jobs are processed in disjoint time intervals), or if G is an independent set (the jobs can be processed in any orderwithout interior idle times).

In some practical applications, jobs in flow shop scheduling may require the use of additional non-renewable resources(its consumption up to any given moment is constrained; for instance: money, fuel, etc.) [5]. A job can only be processed ona machine in a given time unit after it has an exclusive lock on all required resources. Therefore, two jobs having commonresources may not run in simultaneously. This problem can be modeled as a Flow Shop problem with Conflict graph (FSC inshort).

Scheduling with conflict graphs (SWC) has been studied on identical machines: n jobs have to be scheduled on mprocessors subject to constraints represented by a conflict graph G. For two machines and arbitrary processing times, thisproblem is NP-hard even if the conflict graph is empty [8]. Baker and Coffman [1] have called Mutual Exclusion Scheduling(MES) the SWC problem in which each job requires one unit of running time. This problem corresponds exactly to a

∗Speaker81


minimum coloring of the conflict graph G such that each color appears at most m times. The MES problem is NP-hard sincethe problem of finding a minimum coloring of a graph is NP-hard [9]. For more results about the SWC problem with arbitraryprocessing times, the interested reader is referred to the papers [7, 2, 4, 3].

The remainder of this paper is organized as follows. Section 2 presents NP-hardness results. In Section 3, we discussfour lower bounds. Section 4 provides four heuristic algorithms. Section 5 is devoted to the experiments.

2. ComplexityThis section focuses on some complexity results of the FSC problem on two machines, that we obtained in previous works. Itis clear that the problem of flow shop scheduling with conflict graph and the problem of flow shop scheduling with agreementgraph are polynomially equivalent.

In [11], we showed that, for two machines and arbitrary processing times, the FSC problem is NP-hard in the ordinarysense even if the conflict graph is a collection of two disjoint cliques. Furthermore, when all the operations require one unitof running time, the FSC problem on two machines remains NP-hard (in the strong sense) for planar graphs, bipartite graphsand chordal graphs. However, there are many special graph classes on which this problem is solvable in polynomial time,including interval graphs, circular arc graphs, cocomparability graphs, cographs, trees, block graphs and bipartite permutationgraphs.

3. Lower boundsWe present in this section four different lower bounds for the makespan.

3.1. Agreement graph-based lower bounds

This lower bound is derived by considering the agreement graph G. Since non adjacent jobs must be scheduled in disjointtime intervals for any feasible solution, the sum of the processing times of the jobs (two units for each job) of a maximumindependent set (MIS) in G represents a lower bound for the makespan. The MIS problem is NP-hard, thus two greedyalgorithms, namely GMIN and GMAX, presented in [10] have been considered. The resulting lower bounds are named withLB1 and LB2 respectively.

3.2. Conflict constraints relaxation-based lower bound

This lower bound can be derived by relaxing the constraints of conflict. The problem is then reduced to the basic two-machineflow shop problem with unit-time operations. In this case: LB3 = n+ 1 is a valid lower bound for the makespan.

3.3. A maximum matching-based lower bound

This lower bound is obtained by relaxing the precedence constraints between the operations of each job. The relaxedproblem might now be viewed as a maximum matching problem in bipartite graph. Indeed, consider the bipartite graphR = (V1, V2;C) defined as follows:

• Vx = Jx1, Jx2, . . . , Jxn represents the operations of the jobs of V to be processed on machine x (x=1 or 2)

• For each two jobs Ji and Jj (i 6= j), two edges J2i, J1j and J1i, J2j exist if Ji and Jj are not in conflict.

If M is the value of a maximum matching in R, then LB4 = 2n−M is a lower bound for the makespan.

4. Heuristic algorithmsWe discuss in this section two different ways of building a priority list, and two different strategies to schedule a given prioritylist. Observe that four heuristics can be built out of this approach.

Step 1. The priority list is constructed any of the following rules.

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1. Nonincreasing order of the conflict degree.

2. Nondecreasing order of the conflict degree.

Step 2. The priority list is then scheduled with either of the following strategies.

1. Place the next job of the priority list on the first available position that generates a minimum total idle time.

2. Schedule the job with the highest priority value (from the jobs that have not been assigned yet) which is not inconflict with the job already processed. If such a job does not exist, then select the first job of the priority list.

We denote by Hij the corresponding heuristic in which priority rule i of Step 1 is applied to strategy j of Step 2.

5. Computational experimentsThe runs of the above procedures were made with Microsoft Visual Studio 2012 (using C++ language) which was executedon a Pentium(R) Dual-Core PC computer with [email protected] GHz and 1,00 Go RAM. All experiments are carried out on 270instances. The conflict graph of each instance is generated using the G(n,p) Erdos Renyi method [6]. Given n vertices, theG(n,p) method generates a graph where each element of the

(n2

)possible edges is present with probability p. For each pair

(n, p), 10 instances are built, p ∈ 0.2, 0.5, 0.8, n ∈ 20, 40, 80, 100, 150, 200, 250, 300, 400. The computational resultsare summarized in Table 1. We have observed from the implementation that the lower bounds LB1 and LB2 are weakercompared to the others. Also, H11 and H12 outperform H21 and H22 according to all the criteria and for all the instanceclasses. Therefore, in what follows we consider only LB3, LB4, H11 and H12.

Table 1: Performance of the heuristic algorithms and lower bounds.

(LB3) (LB4) (H11) (H12)p n a b c a b c a b c a b c0.2 20 100 100 0 0 0 5 100 100 0 100 100 0

40 100 100 0 0 0 2.5 100 100 0 100 100 080 100 100 0 0 0 1.25 100 100 0 100 100 0100 100 100 0 0 0 1 80 80 0.196 80 80 0.196150 100 100 0 0 0 0.667 90 100 0.065 90 100 0.065200 100 100 0 0 0 0.5 90 100 0.049 90 100 0.049250 100 100 0 0 0 0.4 100 100 0 100 100 0300 100 100 0 0 0 0.333 100 100 0 100 100 0400 100 100 0 0 0 0.25 90 90 0.024 80 80 0.049

0.5 20 100 100 0 0 0 5 70 100 1.778 70 100 1.77840 100 100 0 0 0 2.5 90 100 0.238 80 90 0.47680 100 100 0 0 0 1.25 50 90 0.728 50 90 0.609100 100 100 0 0 0 1 50 100 0.490 50 90 0.586150 100 100 0 0 0 0.667 60 90 0.263 60 90 0.263200 100 100 0 0 0 0.5 70 90 0.197 70 90 0.197250 100 100 0 0 0 0.4 50 50 0.316 50 60 0.237300 100 100 0 0 0 0.333 60 70 0.198 50 80 0.165400 100 100 0 0 0 0.25 50 90 0.149 50 100 0.124

0.8 20 50 80 4.285 0 30 5.847 0 10 11.570 30 100 3.80740 100 100 0 0 0 2.5 0 50 5.005 30 50 3.65880 100 100 0 0 0 1.25 20 90 1.805 0 70 2.046100 100 100 0 0 0 1 10 80 1.553 0 80 1.649150 100 100 0 0 0 0.667 0 70 1.370 0 80 1.367200 100 100 0 0 0 0.5 0 60 1.274 10 80 1.030250 100 100 0 0 0 0.4 0 50 1.063 10 80 0.867300 100 100 0 0 0 0.333 10 80 0.658 0 30 0.789400 100 100 0 0 0 0.25 0 60 0.643 0 70 0.569

a(%): Number of times in percentage a given heuristic coincides with the optimal solution.b(%): Number of times in percentage a given heuristic achieves the best solution.c(%): Average percentage deviation from the optimal solution.

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5.1. Discussion

We report in this section the performance of the heuristic algorithms and lower bounds developed in this paper. The evaluationshows that LB3 outperforms LB4 for all instance classes. However, LB4 shows a good performance (in terms of deviationfrom the optimal solution) especially for large instances with high densities. On the other hand, H12 seems to be moreefficient than H11 for large instances with high densities, whereas H11 performs well for small instances and problems withlow densities.

References[1] B. S. Baker and E. G. Coffman. Mutual exclusion scheduling. Theoretical Computer Science, 162:225–243, 1996.

[2] M. Bendraouche and M. Boudhar. Scheduling jobs on identical machines with agreement graph. Computers andOperations Research, 39:382–390, 2012.

[3] M. Bendraouche and M. Boudhar. Scheduling with agreements: new results. Accepted to appear in the InternationalJournal of Production Research, 2015.

[4] M. Bendraouche, M. Boudhar, and A. Oulamara. Scheduling: Agreement graph vs resource constraints. EuropeanJournal of Operational Research, 240:355–360, 2015.

[5] J. Blazewicz, W. Cellary, R. Slowinski, and J. Weglarz. Scheduling under resource constraints-deterministic models.Annals of Operations Research, 7, 1986.

[6] D. Cordeiro, G. Mounie, S. Perarnau, D. Trystram, J. M. Vincent, and F. Wagner. Random graph generation forscheduling simulations. 3rd International ICST Conference on Simulation Tools and Techniques (SIMUTools’2010)malaga: Spain, 2010.

[7] G. Even, M. M. Halldorsson, L. Kaplan, and D. Ron. Scheduling with conflicts: online and offline algorithms. Journalof Scheduling, 12:199–224, 2009.

[8] M. R. Garey and D. S. Johnson. Computers and intractability: A Guide to the Theory of NP-Completeness. New York:Freeman, 1979.

[9] R. M. Karp. Reducibilty among combinatorial problems. In R. E. Miller and J. W. Thatcher, editors, Complexity ofComputer Computations, pages 85–103. Plenum Press, NewYork, NY, 1972.

[10] S. Sakai, M. Togasaki, and K. Yamazaki. A note on greedy algorithms for the maximum weighted independent setproblem. Discrete Applied Mathematics, 126:313–322, 2003.

[11] N. E. H. Tellache and M. Boudhar. Flow-shop scheduling problem with conflict graphs. pages 33–34, Algiers, Algeria,April 20-22 2015. Operational Research Practice in Africa Conference (ORPA’2015).

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Stochastic Analysis and Bounds for the Stationary

Distribution of an M/G/1 Retrial Queue with Breakdowns

Mohamed Boualem1∗, Mouloud Cherfaoui2, Djamil Aıssani3

1 University of Bejaia, Research Unit LaMOS (Modeling and Optimization of Systems)University of Bejaia, 06000 Algeria

[email protected] University of Bejaia, Research Unit LaMOS (Modeling and Optimization of Systems)

University of Bejaia, 06000 [email protected]

2 University of Bejaia, Research Unit LaMOS (Modeling and Optimization of Systems)University of Bejaia, 06000 Algeria

lamos [email protected]

Abstract

In this paper, we use a monotonicity approach to establish insensitive bounds for some performancemeasures of an M/G/1 retrial queue with server subject to breakdowns by using the theory of stochasticorderings. We prove the monotonicity of the transition operator of the embedded Markov chain relativeto stochastic and convex orderings. We obtain comparability conditions for the distribution of thenumber of customers in the system. The main result of this paper consists in giving insensitive boundsfor the stationary distribution. Such a result is confirmed by numerical illustrations.

Key words: Retrial queue, monotonicity approach, Markov chain, stationary distribution.

2010 Mathematics Subject Classification: 60E15, 60K25.

1. Introduction

In most of the queueing literature the server is assumed to be always available, although this assumption is evidentlyunrealistic. In fact, queueing systems with server breakdowns are very common in communication systems andmanufacturing systems, the machine may break down due to the machine or job related problems. This results ina period of unavailable time until the servers are repaired. Such a system with repairable server has been studiedas a queueing model and a reliability model by many authors [4, 6].

An examination of the literature reveals the remarkable fact that the non-homogeneity caused by the flowof repeated attempts is the key to understand most analytical difficulties arising in the study of retrial queues[3]. Many efforts have been devoted to deriving performance measures such as queue length, waiting time, busyperiod distributions, and so on. However, these performance characteristics have been provided through transformmethods which have made the expressions cumbersome and the obtained results cannot be put into practice. In thelast decade there has been a tendency towards the research of approximations and bounds. Qualitative properties

∗Mohamed Boualem85


of stochastic models constitute an important theoretical basis for approximation methods. One of the importantqualitative properties and approximation methods is monotonicity which can be studied using the general theoryof stochastic ordering [7].

We consider a single server retrial queue in which external customers arrive according to a Poisson stream withrate λ > 0. We assume that there is no waiting space and therefore if an arriving customer finds the server idle,the customer obtains service immediately and leaves the system after service completion. Otherwise, if the serveris found busy or down, the customer makes a retrial at a later time and then the arriving customer becomes asource of repeated calls (a customer in retrial group). The pool of sources of repeated calls may be viewed as a sortof queue with infinite capacity. It is assumed that the retrial times for any repeated customer are exponentiallydistributed with rate α/n given that there are n customers in orbit. In this case, the retrial rate diminishes asmore customers unite to the retrial group (retrial queue with discouraged repeated demands).

The service times are independent and identically distributed with common distribution function B1(x),Laplace-Stieltjes transform LB1(s) and n-th moments β1,n. Customers leave the system forever after service com-pletion.

The server may breakdown when serving customers, and when the server fails it is sent to repair directly. Thecustomer just being served before server failure waits for the server to complete his remaining service.We supposethat the server lifetime has exponential distribution with rate ν, i.e., the server fails after an exponential time withmean 1

ν . It is assumed that the service time for a customer is cumulative and after repair the server is as goodas new. The repair times follow a general distribution B2(y) with Laplace-Stieltjes transform LB2(s) and n-thmoments β2,n.

As usual, we suppose that inter-arrival periods, retrial times, service times, server lifetimes and repair timesare mutually independent.

At an arbitrary time t, the system can be described by X(t) = (C(t), N(t), ξ1(t), ξ2(t)), where C(t) denotes theserver state (0, 1 or 2, depending if the server is free, busy or down) and N(t) is the number of repeated customersat time t. If C(t) = 1, then ξ1(t) represents the elapsed service time of the customer currently being served. IfC(t) = 2, then ξ1(t) means the elapsed service time for the customer under service and ξ2(t) symbolizes the elapsedrepair time.

In this paper, we use the general theory of stochastic ordering (see [7]) to study monotonicity properties similarto that of Boualem et al. [1, 2, 4, 5], for a single server retrial queue with server subject to active breakdowns,i.e., the service station can fail only during the service period, relative to the strong stochastic ordering, convexordering and Laplace ordering. The obtained results give insensitive bounds for the stationary distribution of theconsidered embedded Markov chain. Numerical illustrations are provided to support the results.

2. Main results

Stochastic ordering is useful for studying internal changes of performance due to parameter variations, to comparedistinct systems, to approximate a system by a simpler one, and to obtain upper and lower bounds for the mainperformance measures of systems.

In this paper, we use a monotonicity approach to establish insensitive bounds for some performance measuresof a single-server retrial queue with server subject to active breakdowns by using the theory of stochastic orderings(see [4]). The proposed technique is quite different from those in Djellab [6] and Wang et al. [8], in the sensethat our approach provides from the fact that we can come to a compromise between the role of these qualitativebounds and the complexity of resolution of some complicated systems where some parameters are not perfectlyknown (e.g. the service times and repair times distributions are unknown). We prove the monotonicity of thetransition operator of the embedded Markov chain relative to strong stochastic ordering and convex ordering. Weobtain comparability conditions for the distribution of the number of customers in the system. The main result ofthis paper consists in giving insensitive bounds for the stationary distribution of the considered embedded Markovchain. Such a result is confirmed by numerical illustrations.

In conclusion, the monotonicity approach holds promise for the solution of several systems with repeatedattempts. Hence, it is worth noting that our approach can be further extended to more complex systems (e.g.resource allocation problems in mobile networks). Moreover, the qualitative bounds given in this paper may havean interesting impact on ”robustness analysis”.

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References

[1] Boualem, M., Djellab, N., and Aıssani, D., Stochastic bounds for a single server queue with general retrialtimes, Bulletin of the Iranian Mathematical Society, 2014, 40 (1), 183–198.

[2] Boualem, M., Insensitive Bounds for the Stationary Distribution of a Single Server Retrial Queue with ServerSubject to Active Breakdowns, Advances in Operations Research, 2014, 2014, ID 985453, 1–12.

[3] Boualem, M., Cherfaoui, M., Djellab, N., and Aıssani, D., Analyse des performances du systeme M/G/1 avecrappels et Bernoulli feedback, Journal Europeen des Systemes Automatises, 2013, 47 (1-3), 181–193.

[4] Boualem, M., Djellab, N., and Aıssani, D., Stochastic approximations and monotonicity of a single serverfeedback retrial queue, Mathematical Problems in Engineering, 2012, 2012, Art. ID 536982, 1–13

[5] Boualem, M., Djellab, N., and Aıssani, D., Stochastic Inequalities for an M/G/1 Retrial Queues with Vacationsand Constant Retrial Policy, Mathematical and Computer Modelling, 2009, 50 (1-2), 207–212.

[6] Djellab, N., On the M/G/1 retrial queue subjected to breakdowns, RAIRO Operations Research, 2002, 36,299–310

[7] Muller, A., and Stoyan, D., Comparison methods for stochastic models and risk, John Wiley and Sons, LTD,2002.

[8] Wang, J., Cao, J., and Li, Q., Reliability analysis of the retrial queue with server breakdowns and repairs,Queueing Systems, 2001, 38, 363–380

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PSO and Simulated annealing for the two machines flowshopscheduling problem with coupled-operations

Meziani Nadjat1,2,∗, Boudhar Mourad2, Oulamara Ammar3

1 Abderrahmane Mira UniversityCampus d’Aboudaw, route de Tichy, Bejaia, Algeria

ro [email protected] USTHB University, RECITS laboratory, Faculty of Mathematics

BP 32, Bab-Ezzouar, El-Alia 16111, Algiers, [email protected]

1 University of LorraineIle de Saulcy, CS 10628, 57045 Metz, France

[email protected]

Abstract

In this work, we proposed heuristics, metaheuristics and an hybrid algorithm combining particle swarmoptimization with the simulated annealing (PSO-SA), to solve the flowshop scheduling problem with coupled-operations in the presence of time lags. The objective is to minimize the makespan. To verify the performanceof the algorithms, computational experiments are performed and the obtained results with the PSO-SA arecompared to the simulated annealing and PSO algorithms.

Key words: flowshop, coupled-operations, makespan, PSO, simulated annealing.

1. IntroductionThe flowshop scheduling problem which consists to process n tasks on m machines in the same order is one of the frequentlyproblem encountered an industrial processes workshop, manufacturing systems and assembly workshops. In the literature,authors studied several types of the flowshop problems with different variants. In addition to solve these problems, they devel-oped different approaches such as exact solutions, heuristics and metaheuristics. Recently, metaheuristics become a popularapproach used to solve a large number of combinatorial optimization problems in several areas. In particular, Particle SwarmOptimization (PSO) is the one of the latest metaheuristic methods in the literature which has been applied successfully tothe scheduling problems in which we find little works in the literature. In [10], Tasgetiren et al are the first to develop PSOalgorithm for solving the scheduling problem to a single machine to minimize total weighted tardiness, as they have proposedin [9] a PSO for both problems, permutation flowshop to minimize the makespan and maximum lateness, respectively. Still,for the permutation flowshop problem, Zhigang et al [12] presented an algorithm for PSO to minimize the makespan. Chang-sheng et al [2] adapted a PSO to solve the classical flowshop problem in order to minimize the completion date of processingjobs and for the same purpose, Sha and cheng [7] have suggested a hybrid PSO.

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In this work, we proposed to solve the flowshop problem on two machines with coupled-operations by developing heuris-tics and metaheuristics. The problem consists to schedule n tasks on two machines in order to minimizing the makespan,such as each task is composed on two operations on the first machine separated by an exact time lag, and only one operationon the second machine.

The motivation of the coupled-tasks problem stems from a scheduling problem of radar tasks, which consists in theemission of the pulses and the reception of answers after the time interval. This problem appears also in workshops chemicalproduction, where one machine must carry out several operations of the same task and an exact delay is imposed between theexecution of each two consecutive operations due to the chemical reactions.

The coupled-tasks scheduling problem was introduced for the first time by Shapiro [8]. Each coupled-tasks consists intwo different operations which are carried out on one machine in the order, separated by an time interval known as a time lag.In [6], Orman and Potts studied the coupled-tasks problem with one machine in order to minimize the makespan. Blazewicz etal proved in [1] that the polynomial problem 1/Coup− task, ai = bi = 1, Li = l/Cmax is NP-hard by adding a precedenceconstraint between the coupled-tasks. Yu et al [11] proved that the problem on two machines F2/Coup − Task, ai = bi =1, Li/Cmax is NP-hard. Meziani et al [5] studied the flowshop problem on two machines with the coupled-operations on thefirst machine and one operation on the second. They proved in [5] that the problems F2/Coup − Opr(1), ai, bi = Li =p, ci/Cmax and F2/Coup − Opr(1), ai = Li = p, bi, ci/Cmax are binary NP-hard. They also presented the complexity ofsome subproblems in [4].

2. Description of the problemIn this paper, we examined the flowshop scheduling problem on two machines which we described as follows. We considerthe flowshop problem with two machines for which we must schedule a set of n jobs. Each job (see Figure 1) is composedof a coupled-operations O1,j and an operation O2,j to be processed on the first and the second machine, respectively. Thecoupled-operations O1,j of job j is described by a triplet (aj , Lj , bj), where aj and bj represents the processing times of thefirst and the second operation of coupled-operations O1,j , respectively. The value Lj is the exact time lag between the twooperations of the couple O1,j . The operation O2,j is described by it processing time cj . The execution of operation O2,j startonly if the coupled-operations O1,j is completed. The objective is to minimize the makespan. The problem is denoted byF2/Coup−Opr(1), ai, bi, Li, ci/Cmax.

M1

M2

bj

cj

aj

- Lj

Figure 1: Operations scheduling

3. Proposed MethodsFor the resolution of the defined problem, we have proposed several heuristics. These heuristics are based on the uses ofpriority rules SPT and LPT, and on Johnson rule [3] to determine the sequence of jobs to be scheduled. To calculate thevalues of the makespan, we introduced a parallel algorithm by considering the interleaved jobs. These heuristics are testedby generating 100 instances such as the processing times of jobs are uniformly distributed. Then, we compared the obtainedresults, and we concluded that heuristics based on LPT rule and which uses parallel algorithm performs better than the otherheuristics. Still, to resolve the problem, we also proposed metaheuristics such as simulated annealing and Particle SwarmOptimization. Moreover, we have developed an hybrid technique by combining these two approaches. However, to evaluatethe performance of these latest algorithms, we have tested and compared their results. In conclusion, we have noted that thecombined algorithm performs better than the other approaches.

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4. ConclusionWe have interested to solve the flowshop scheduling problem on two machines with couple-operations in order to minimizethe makespan. Then, we proposed several heuristics with computational tests for which we compared their results. Also,we applied the particle swarm optimization algorithm and the simulated annealing to solve the problem. Furthermore, wedeveloped a particle swarm optimization combined with the simulated annealing for the problem. In order to evaluate theperformance of these approaches, numerical experiments are conducted and the results are compared. The computationalresults indicate that the hybridized algorithm performs better than the other proposed metaheuristics.

References[1] Blazewicz J., Ecker K., Kis T., Potts C.N., Tanas M. and Whitehead J., Scheduling of coupled tasks with unit processing

times, J. of Sched., 2010, 13, 453–461.

[2] Changsheng Z., Jigui S., Xingjun Z., Qingyun Y., An improved particle swarm optimization algorithm for flowshopscheduling problem, Infor Proce Let., 2008, 108, 204–209.

[3] Johnson S.M., Optimal two and three stage production schedules with setup time included, Nav Res Logi Quart., 1954,1, 61–67.

[4] Meziani N., Oulamara A., Boudhar M., Minimizing the makespan on two-machines flowshop scheduling problemwith coupled-tasks, Neuvime Colloque sur l’optimisation et les systmes d’Information, (COSI’2012), Tlemcen (Algrie),2012, 192–203.

[5] Meziani N., Oulamara A. and Boudhar M., NP-hardness of the two-machine flowshop problem with coupled-operations.preprint, Les annales RECITS, 2014, 1, 11–26.

[6] Orman A.J., Potts C.N., On the complexity of coupled-Task Scheduling, Dis Appl Math., 1997, 72, 141–154.

[7] Sha D.Y., Cheng Y.H., Hybrid particle swarm optimization for job shop scheudling problem, Comp and Indu Engi.,2006, 51, 791–808.

[8] Shapiro R.D., Scheduling Coupled Tasks, Nav Res Log quar., 1980, 20, 489–498.

[9] Tasgetiren M.F., Sevkli M., Liang Y.C., Gencyilmaz G., Particle swarm optimization algorithm for makespan and max-imum lateness minimization in permutation flowshop sequencing problem, In: 4th international symposium on intelli-gent manufacturing systems., Turkey, 2004, 431–441.

[10] Tasgetiren M.F., Sevkli M., Liang Y.C., Gencyilmaz G., Particle swarm optimisation algorithm for single machine totalweighted tardiness problem, In: proceeding of the IEEE confgress on evoultionary computation, Portland, 2004, 2,1412–1419.

[11] Yu W., Hoogeveen H., Lenstra J.K., Minimizing makespan in a two-machine flow shop with delays and unit-timeoperations is NP-hard, J. of Sched., 2004, 7, 333–348.

[12] Zhigang L., Xingsheng G., Bin J., A similar particle swarm optimization algorithm for permutation flowshop schedulingto minimize makespan, Appl Math and Comp., 2006, 175, 773–785.

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Scheduling with preemption and setup times

Amina HANED∗, Mourad BOUDHAR

RECITS Laboratory, USTHB University,16111, BP 32 Bab-Ezzouar, El-Alia, Algiers, Algeria

amina [email protected] , [email protected]

Abstract

We are interesting on the study of the scheduling problem on identical parallel machines, where each jobrequires a setup time. The preemption of jobs is allowed also an additional setup times are considered for thepreempted jobs. The objective is to minimize the total completion time (makespan). We prove that the problemis NP-hard. Then we present some heuristics for its resolution.

Key words: Scheduling, Parallel machines, Preemption, Setup times.

2010 Mathematics Subject Classification: 90B35.

1. IntroductionScheduling consists to assign jobs to machines in order to optimize one or more objectives under several constraints. Amongthese constraints, the constraints concerning the machines environment, the mode of treatment, the relationship between jobsand the times or delays corresponding to the transportation or preparation. The last one, preparation times, are usually calledsetup times. Setup times may have different natures: clean-up, set-up, start-up, ...

In scheduling theory, problems with setup times have a lot of interest. At the beginning, the majority of researchesassume that the setup time is negligible or included in the processing time of jobs. This assumption simplifies the analysis ofthe problem. However, it affects the quality of solution.

Since 1960s, scheduling problems with separated setup times have been discussed in several studies. In 2008, Allahverdiet al.[1] presented a survey of scheduling problems with setup times and costs and gave some industrial applications. Sadi-Nezhad and Darian [5] studied the problem in juice production lines, Pfund et al. [4] studied a scheduling problem with readytimes on identical parallel machines with sequence dependent setups in order to minimize the total weighted tardiness. Inprevious works [2, 3], we studied a scheduling problem by considering the premption of jobs and a transportation delays ofthe preempted jobs.

In this study, we consider the problem of scheduling a set of n independent jobs J1, . . . , Jn on m identical parallelmachines M1, . . . ,Mm. Each job Jj has a processing time pj which is assumed to be an integer number. Also, we assumethat all jobs are available at time t = 0. Each job requires setup time denoted sj . The preemption of jobs is allowed. Ifone job Jj is preempted on a machine Mi to continue his treatment on another machine Mk, then an additional setup time isconsidered. The objective is to minimize the total length of the scheduleCmax. This problem is noted byPm|pmtn, sj |Cmax.

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2. Main resultsWe prove that the problem P2|pmtn, sj = 1|Cmax is NP-hard by using the reduction to the 2-Partition problem with thesame part (equal size). This leads us to the conclusion that the general problem Pm|pmtn, sj |Cmax is NP-hard too.

Theorem 1. The problem P2|pmtn, sj = 1|Cmax is NP-hard.

We propose lower and upper bounds for the makespan, these bounds will be used for the resolution of the problem. Then,we give some characteristics of the feasible schedule.

We propose the following bounds:

• The lower bound LB = max⌈

1m

n∑j=1

pj + sj

⌉, max1≤j≤n

pj + sj.

• The upper bound UB =n∑

j=1

pj + sj , if all jobs are processed by the same machine without preemption.

We prove that for a feasible schedule of the problem Pm|pmtn, sj |Cmax, a job is not preempted on the same machine.

Lemma 2. For any solution of the problem Pm|pmtn, sj |Cmax, a job is not preempted on the same machine.

3. Resolution of the problem Pm|pmtn, sj|Cmax

Recall that the problem is NP-hard, then it is more interesting to use heuristics or metaheuristics that solve the problem witha large number of jobs and machines. We propose heuristics based on the Mc Naughton and list algorithms. In the first timewe describe an order of jobs according to the processing times, setup times and the sum of the processing and setup times.After that we proceed to the assignment of jobs using a heuristic HA0 which is described in the following.

Algorithm 1: HA0

• Calculate the lower bound LB = max

max

1≤j≤npj + sj ;

⌈1m

n∑j=1

pj + sj

⌉;

• Initialization : t = 0; j := 1;i := 1;while (j ≤ n) do

if (t+ pj + sj < LB) then• Assign the job Jj to the machine Mi at time t;• t := t+ pj + sj ; j := j + 1;else(1) if (t+ pj + sj = LB) then• Assign the job Jj to the machine Mi at time t and the next job to the next machine at time t = 0;• t := 0; j := j + 1; i := i+ 1;else(2) if The remaining time on the current machine is sufficient to prepare and execute one part of this jobthen• The job Jj is preempted, it is processed at the last position on the machine Mi at time t for(LB − (t+ sj)) units of time and the rest of the job at the first position on the machine Mi+1 at timet = 0;else• Find a job Jind for which (t+ sind + pind ≤ LB), go to (1) or (2);

Heuristic HA0This heuristic is based on the Mc Naughton algorithm’s. It consists to assign jobs in any order to the first machine until thelower bound. If one job can not be processed completely then it will be preempted. We verify if the remaining time on thecurrent machine is sufficient to prepare and execute one part of this job (to the setup time and the execution of a part of thejob). Otherwise we chose another job of the list. The main steps of this heuristic are done by the algorithm 1. The complexity

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of the heuristic HA0 is in O(n).

Heuristic HAiWe define several version of this heuristic according to the order of jobs as following:

• Heuristic HA1 : jobs are arranged in decreasing order of their processing time (pj ↓).• Heuristic HA2 : jobs are arranged in increasing order of their processing time (pj ↑).• Heuristic HA3 : jobs are arranged in decreasing order of their setup time (sj ↓).• Heuristic HA4 : jobs are arranged in increasing order of their setup time (sj ↑).• Heuristic HA5 : jobs are arranged in decreasing order of the sum of their processing and setup times (pj + sj ↓).• Heuristic HA6 : jobs are arranged in increasing order of the sum of their processing and setup times (pj + sj ↑).

The heuristics HAi are described as follow:Algorithm 2: HAi

Begin• Order the jobs according to one of the previous order.• Apply the algorithm HA0.End

The complexity of these heuristics is in O(nlogn).

Numerical experimentsTo evaluate the proposed heuristics, some experiments are carried out using different instances. The instances are generatedrandomly, for each instance the number of machines m is chosen in the set 2,3,5,10, the number of jobs n in the set100,500,1000. The processing times (pj) and the setup times (sj) are generated randmly using the uniform law in theintervals [1,25], [1,50], [25,50] with different combinations. For a fixedm and n we generate 100 instances, for each instancewe calculate the average and the maximum deviation of each heuristic.The results lead us to conclud that the heuristic HA5 with the decreasing order of the sum of the processing and setup timesof jobs (pj + sj ↓) is better than the others heuristics. It provides solutions with an average deviation of 0,81% from thelower bound and a maximum deviation of 2,4% for the instances with 100 jobs and 2 machines when the processing andsetup times are generated in the interval [1,50]. For the instances with 500 jobs and 2 machines, it has an average deviationof 0,11% and a maximum deviation of 0,44% for the same interval. Concerning the instances with pj in [25,50] and sj in[1,25], the heuristic HA5 has an average and maximum deviations of 0,15% and 1,96% respectively for instances with 100jobs and 2 machines. For the instances with 500 jobs and 2 machines, it has an average and maximum deviations of 0,06%and 0,4% respectively. When the processing times are generated in [1,25] and the setup times in [25,50], the heuristic HA5provides solutions with an average deviation of 1,09% and a maximum deviation of 1,9% for the instances with 100 jobs and2 machines. It has an average deviation of 0,14% and a maximum deviation of 0,39% for the instances with 500 jobs and 2machines.

4. ConclusionIn this study, we consider the scheduling problem on parallel machines with preemption and setup times. This problem hasmany applications in industry such as manufacturing printed circuit boards, juice production lines and the textile industry. Ourcontribution is the study of the complexity of the problem and the proposition of some heuristics to the resolution. Accordingto the numerical experiments we conclud that the heuristic based on the decreasing order of the sum of the processing andsetup times of jobs is better than the others heuristics. For the futur work, we propose the study of some properties of theproblem and propose a metaheuristic for the resolution.

References[1] A. Allahverdi, C.T. Ng, T.C.E. Cheng, M. Y. Kovalyov, A survey of scheduling problems with setup times or costs,

European Journal of Operational Research, 2008, 187, 9851032.

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[2] M. Boudhar, A. Haned, Preemptive scheduling in the presence of transportation times. Computers & Operations Re-search 36, no8 (2009) 2387-2393.

[3] A. Haned, A. Soukhal, M. Boudhar, N. H. Tuong, Scheduling on parallel machines with preemption and transportationdelays, Computers and Operations Research 39 (2012) 374-381.

[4] M. Pfund, J. W. Fowler, A. Gadkari, Y. Chen, Scheduling jobs on parallel machines with setup times and readytimes,Computers & Industrial Engineering 54 (2008) 764782.

[5] S. Sadi-Nezhad, S.Borhani Darian, Production scheduling for products on different machines with setup costs and times,International Journal of Engineering and Technology Vol.2 6 (2010) 410-418.

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Paths in tournaments

El Sahili Amin1, Maria Abi Aad2∗

1 Department of Mathematics, Lebanese UniversityBeirut, Lebanon

[email protected] Department of Mathematics, Lebanese University

Beirut, Lebanonmaria [email protected]

Keywords: Tournaments, Oriented paths, antidirected paths.

1. Introduction

The questions that first triggered our research and lead us to our results are the following:• Given an arbitrary orientation of a path, what can we say about its existence in a tournament?• Can we compute the parity of the number of such paths in a tournament? Are there any parameters that, oncefixed, allow us to obtain interesting and simple results in this direction?

The existence of paths in tournaments is a classical problem in graph theory, and so is the parity of the numberof a type of paths in a given tournament. Redei, in [4], proved that any tournament contains a directed path, andthe number of such paths is always odd. The problem of the existence of paths in tournaments was completelysettled by Havet and Thomasse [3] who proved that except for exactly three cases, any tournament contains anytype of oriented paths, proving an earlier conjecture of Rosenfeld [5]. The problem of parity in the general casewas treated by Forcade, who proved in [1] that except for symmetric paths, the parity of the number of paths of agiven type in a tournament depends only on the number of vertices of this tournament. The interesting case relatedto antidirected paths was first independently studied by Grunbaum, and Rosenfeld who proved that the parity ofsuch paths in an odd tournament is even [5]. Grunbaum made the following conjecture [2]:

Conjecture 1. The number of antidirected Hamiltonian paths in a tournament on at least 10 vertices is congruentto 2 (mod 4).

We disprove this conjecture by simply treating the case of antisymmetric paths, which are a generalization ofantidirected paths. Furthermore, we study the parity of paths in tournaments with a given number of backwardarcs. We prove that this parity can be expressed in a simple form, in terms of combinations, showing that it dependsonly on the number of backward arcs and the number of vertices of the tournament.

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Bibliography[1] R. Forcade, Parity of paths and circuits in tournaments, Discrete Math. 6 (1973), p. 115-118.[2] B. Grunbaum, Antidirected Hamiltonian paths in tournaments, J.Combinatorial Theory 11 (1971), p. 249-257.[3] F. Havet and S. Thomasse, Oriented Hamiltonian paths in tournaments: A proof of Rosenfeld’s conjecture, J.CombinatorialTheory, Ser. B (2000), p.243-273.[4] L. Redei, Ein kombinatorischer Satz, Acta. Sci. Math. (Szeged) 7 (1934), 39-43.[5] M. Rosenfeld, Antidirected Hamiltonian paths in tournaments, J.Combinatorial Theory, Ser. B 12 (1972), p. 93-99.

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Forwarding indices of some graph operations

Mohamed Amine Boutiche∗

University of Sciences and technology Houari BoumedieneLaboratory LaROMaD

BP 32, El Alia, 16111, Bab Ezzouar, Algiers, [email protected]

Abstract

In this paper, we compute lower bounds of vertex and edge forwarding indices for cartesian product, join,composition, disjunction and symmetric difference of graphs. Moreover, we show how those lower boundschanges with several operators on connected graphs, such as subdivision graph and total graphs.

Key words: Forwarding indices, Graph product, subdivision graphs, total graphs.

2010 Mathematics Subject Classification: 05C85, 68R10, 68W05.

1. IntroductionA routing R of a connected graph G = (V,E) of order n consists in a set of n(n− 1) elementary paths R(u, v) specified forall ordered pairs u, v of vertices of G.

To measure the efficiency of a routing deterministically, Chung, Coffman, Reiman and Simon [3] introduced the conceptof forwarding index of a routing.

The load of a vertex v (resp. an edge e) in a given routing R of G = (V,E), denoted by ξ(G,R, v) (resp. π(G,R, e)), isthe number of paths of R going through v (resp. e), where v is not an end vertex. The parameters

ξ(G,R) = maxv∈V (G)

ξ(G,R, v) and π(G,R) = maxe∈E(G)

π(G,R, e)

are defined as the vertex forwarding (resp. the edge forwarding) index of G with respect to R, and

ξ(G) = minR

ξ(G,R) and π(G) = minR

π(G,R)

are defined as the vertex forwarding (resp. the edge forwarding) index of G.The original research of the forwarding indices is motivated by the fact that maximizing network capacity can be reduced

to minimizing vertex-forwarding index or edge-forwarding index of a routing. Thus, the forwarding index problem has beenstudied widely by many researchers (see, for example, [1]).

Although, determining the forwarding index problem has been shown to be NP -complete by Saad [8]. The exactvalues of the forwarding index of many important classes of graphs have been computed [2]. For more complete resultson forwarding indices, we can refer to the survey of Xu et al. [10].

For a given connected graph G = (V,E) of order n and number of edges m, set

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A(G) =1n

∑

u∈V

∑

v∈V \u(dG(u, v)− 1)

,

B(G) =1m

∑

(u,v)∈V×V

dG(u, v).

where dG(u, v) denotes the distance from the vertex u to the vertex v inG. Let d(G, k) be the number of pairs of verticesof a graph G that are at distance k.

The following bounds of ξ(G) and π(G) were first established by Chung et al. [3] and Heydemann et al. [6], respectively.

A(G) ≤ ξ(G) ≤ (n− 1)(n− 2),

B(G) ≤ π(G) ≤ b12n2c.

2. Graph ProductsThe Cartesian product G ×H of graphs G and H is a graph with vertex set V (G) × V (H), and any two vertices (a, b) and(u, v) are adjacent in G×H if and only if either a = u and b is adjacent with v, or b = v and a is adjacent with u, see [7] fordetails.

The join G = G+H of graphs G and H with disjoint vertex sets V (G) and V (H) and edge sets E(G) and E(H) is thegraph union G ∪H together with all the edges joining V (G) and V (H).

The composition G[H] of graphs G and H with disjoint vertex sets V (G) and V (H) and edge sets E(G) and E(H) isthe graph with vertex set V (G)× V (H) and u = (u1, v1) is adjacent with v = (u2, v2) whenever (u1 is adjacent with u2) or(u1 = u2 and v1 is adjacent with v2),[7].

The disjunction G ∨ H of graphs G and H is the graph with vertex set V (G) × V (H) and (u1, v1) is adjacent with(u2, v2) whenever u1u2 ∈ E(G) or v1v2 ∈ E(H).

The symmetric difference G⊕H of two graphs G and H is the graph with vertex set V (G)× V (H) and E(G⊕H) =(u1, u2)(v1, v2)|u1v1 ∈ E(G)or u2v2 ∈ E(H) but not both. The following lemma is related to distance properties ofsome graph products.

Lemma 1. (Hossein-Zadeh et al. [4])Let G and H be graphs. Then we have:

1. |V (G×H)| = |V (G ∨H)| = |V (G[H])| = |V (G⊕H)| = |V (G)| × |V (H)|,|E(G×H)| = |E(G)| × |V (H)|+ |V (G)| × |E(H)|,|E(G+H)| = |E(G)|+ |E(H)|+ |V (G)|.|V (H)|,|E(G[H])| = |E(G)|.|V (H)|2 + |E(H)|.|V (G)|,|E(G ∨H)| = |E(G)|.|V (H)|2 + |E(H)|.|V (G)|2 − 2|E(G)|.|E(H)|,|E(G⊕H)| = |E(G)|.|V (H)|2 + |E(H)|.|V (G)|2 − 4|E(G)|.|E(H)|.

2. G×H is connected if and only if G and H are connected.

3. If (a, c) and (b, d) are vertices of G×H then dG×H((a, c), (b, d)) = dG(a, b) + dH(c, d).

4. The Cartesian product, join, composition, disjunction and symmetric difference of graphs are associative and all ofthem are commutative except from composition.

5. dG+H(u, v) =

0 u = v1 uv ∈ E(G) or uv ∈ E(H) or (u ∈ V (G), v ∈ V (H))2 otherwise

6. dG[H]((a, b), (c, d)) =

dG(a, c) a 6= c0 a = c & b = d1 a = c & bd ∈ E(H)2 a = c & bd 6∈ E(H)

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7. dG∨H((a, b), (c, d)) =

0 a = c & b = d1 ac ∈ E(G) or bd ∈ E(H)2 otherwise

8. dG⊕H((a, b), (c, d)) =

0 a = c & b = d1 ac ∈ E(G) or bd ∈ E(H) but not both2 otherwise

3. Main resultsIn this section, some exact formulae for expressions A(G) and B(G) of the Cartesian product, composition, join, disjunctionand symmetric difference of graphs are presented.

The Forwarding indices of the Cartesian product graphs were studied in [9]. In the following propositions, we computethe lower bounds of the forwarding indices A(G) and B(G) for known product graphs.

Theorem 2. Let G and H be connected graphs. ThenA(G+H) = |E(G)|+ |E(H)|,B(G+H) =

2.(|E(G)|+ |E(H)|)|E(G)|+ |E(H)|+ |V (G)|.|V (H)| + 1.

Corollary 2.1. Let nG = G1 +G2 + · · ·+Gn denotes the join of n copies of G, then

A(nG) =n∑

i=1

|E(Gi)|, and B(nG) =2.∑n

i=1 |E(Gi)||E(nG)| + 1.

Theorem 3. Let G and H be connected graphs. ThenA(G ∨H) = |V (H)||E(G)|+ |V (G)||E(H)|+ 2.|E(G)|.|E(H)|,B(G ∨H) = 1 + 2.

A(G ∨H)|E(G ∨H| .

Theorem 4. Let G and H be connected graphs. ThenA(G⊕H) = 2.|E(G)||E(H)|+ |V (H)||E(G)|+ |V (G)|.|E(H)|+ 2.|E(G)||E(H)|,B(G⊕H) = 1 + 2.

|E(G⊕H)||E(G⊕H| .

Theorem 5. Let G and H be connected graphs. Then

A(G[H]) = A(G) +|E(H)||V (H)| ,

B(G[H]) = B(G) +1

2|E(G[H])|(|V (G)||E(H)|+ 2.|V (G)||E(H)|

).

4. Graph operationsFor a connected graph G, define four related graphs as follows.

Definition 6. Let G be a graph, then

1. S(G) is the graph obtained by inserting an additional vertex in each edge of G. Equivalently, each edge of G isreplaced by a path of length 2.

2. R(G) is obtained from G by adding a new vertex corresponding to each edge of G, then joining each new vertex to theend vertices of the corresponding edge.

3. Q(G) is obtained from G by inserting a new vertex into each edge of G, then joining with edges those pairs of newvertices on adjacent edges of G.

4. T (G) has as its vertices the edges and vertices of G. Adjacency in T (G) is defined as adjacency or incidence for thecorresponding elements of G.

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The graphs S(G) and T (G) are called the subdivision and total graph of G, respectively. For more details on theseoperations we refer the reader to [11].

In this section, we compute lower bounds of forwarding indices A(G) and B(G) for some graph operations. We use thefollowing lemma.

Lemma 7. (Yan et Al 2007 [11])Let G be a graph, then

1. For any v, v′ ∈ V (G), thendS(G)(v, v′) = dT (G)(v, v′) = dR(G)(v, v′) = dQ(G)(v, v′)− 1 = dG(v, v′).

2. For any e, e′ ∈ E(G), thendS(G)(e, e′) = dT (G)(e, e′) = dR(G)(e, e′)− 1 = dQ(G)(e, e′) = dL(G)(e, e′).

3. For any e ∈ E(G), v ∈ V (G), then12.dS(G)(e, v) + 1 = dT (G)(e, v) = dR(G)(e, v) = dQ(G)(e, v).

Theorem 8. If G is the connected graph with m edges and n vertices, thenA(S(G)) = 2.A(T (G))− mn

m+ n, B(S(G)) = B(T (G))− n

2.

A(R(G)) = A(T (G)) +m(m− 1)2.m+ n

, B(R(G)) = B(T (G)) +m(m− 1)

2.|E(R(G))| .

A(Q(G)) = A(T (G)) +n(n− 1)2.m+ n

, B(Q(G)) = A(T (G)) +n(n− 1)

2.|E(Q(G))| .

Corollary 8.1. Let G be a connected graph, then

A(S(G)) = A(R(G)) +A(Q(G))−(m+ n− 1

2

),

B(S(G)) = B(R(G))− 3n+m− 16

.

References[1] Bouabdallah A., Sotteau D., On the edge-forwarding index problem for small graphs. Networks., 23(4) (1993), 249–255.

[2] Chang C.P., Sung T.Y., Hsu L.H., Edge congestion and topological properties of crossed cubes. IEEE Trans. Paralleland Distributed Systems., 11(1) (2000), 64–80.

[3] Chung F.K., Coffman E.G., Reiman M.I., Simon B., The forwarding index of communication networks. IEEE Transac-tions on Information Theory., 33(2) (1987), 224–232.

[4] Hossein-Zadeh S., Hamzeh A., Ashrafi A.R., Wiener-type invariants of some graph invariants. Filomat 23(3) (2009),103-113.

[5] Heydemann M.C., Meyer J.C., Opatrny J., Sotteau D., Forwarding indices of k-connected graphs. Discrete AppliedMathematics., 37/38 (1992), 287–296.

[6] M. C. Heydemann, J. C. Meyer and D. Sotteau, On forwarding indices of networks, Discrete Applied Mathematics, 23(1989), 103–123.

[7] Imrich W., Klavzar S., Product graphs: structure and recognition, John Wiley & Sons, New York, USA, 2000.

[8] Saad R., Complexity of the forwarding index problem. SIAM J. Discrete Math., 6(3) (1993), 418–427.

[9] Xu J., Xu M., Hou M., Forwarding indices of cartesian product graphs. Taiwanese Journal of Mathematics., 10(5)(2006), 1305–1315.

[10] Xu J., Xu M., The forwarding indices of graphs - A survey, Opuscula Mathematica 33(2) (2013), 345-372.

[11] Yan W., Yang B., Yeh Y., The behavior of Wiener indices and polynomials of graphs under five graph decorations, Appl.Math. Lett. 20 (2007) 290–295.

100


On bandwidth of some classes of graphs

Hakim Harik1,2∗, Hacene Belbachir1

1 USTHB, Faculty of Mathematics, RECITS Lab. DG-RSDTPo. Box 32, El Alia,16111, Algiers, Algeria

[email protected] CERIST

5 Rue des freres Aissou, Ben Aknoun, Algiers, [email protected]

Abstract

The bandwidth problem is a well known research area in graph theory. We investigate the bandwidth ofsome classes of graphs using some combinatorial properties.

Key words: Bandwidth, graphs, Hale’s order.

AMS Classification. 05A19; 05C15; 11B39; 05C30 .

1. IntroductionWe denote by G = (V,E) a simple undirected graph of order |V (G)| and size |E(G)|. N(v) is a set of vertices adjacent tovertex v inG and deg(v) = |N(v)| denote its degree. For two vertices u, v ∈ V (G), let dG(u, v) denote the distance betweenu and v in G and let D(G) denote the diameter of G.

A numbering of a vertex set V is any function f : V → 1, 2, . . . , |V |, which is one-to-one (and therefore onto). Anumbering uniquely determines a total order, ≤, on V as follows u ≤ v if f (u) < f (v) or f (u) = f (v) . Conversely, a totalorder defined on V uniquely determines a numbering of the graph.

The bandwidth of a numbering of a graph G = (V,E) is the maximum difference

bw (f) = maxuv∈E(G)

|f (u)− f (v)|

The bandwidth of a graph G is the minimum bandwidth over all numberings, f , of G, i.e.

bw (G) = minfbw (f)

Papadimitriou [7] proved that the problem of determining the bandwidth of a graph is NP-complete. The bandwidthproblem appears in a lot of areas of Computer Science such as VLSI layouts and parallel computing, see surveys [6] [9].

Many studies have been done towards finding the bandwidth of specific classes of graphs. Harper shows that the labelinggiven by the Hale’s order solves the bandwidth problem for Qn and gives explicit formula [6] [9].

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bw (Qn) =n−1∑

m=0

(m

m/2

)

Billera and Blanco give the bandwidth of the d− product of a path with n edges, Q(d)n [3].

bw(Q(d)

n

)=

d−1∑

i=0

R (n, i)

Where R(n, i) is number related coefficients of a polynomial.

Chinn et al give the lower bound of the bandwidth for a connected graph G.

bw (G) ≥⌈ |V (G)| − 1

D(G)

⌉

Our aim is to give an explicit expression of the bandwidth for some classes of graphs using the notion of coefficientsrelated to arithmetic triangle or quasi-arithmetic triangle obtained using recurrence rools [1], [2], [4], [8].

References[1] G. E. Andrews, J. Baxter, Lattice gas generalization of the hard hexagon model III q-trinomials coefficients, J. Stat Phys.,

47, 297–330 (1987).

[2] H. Belbachir, A. Benmezai, A q−analogue for bisnomial coefficients and generalized Fibonacci sequence, Cahier deRecherche Academie Scientifique de Paris Serie 1, 352, 167-171 (2014).

[3] L. J. Billera, S. A. Blanco, Bandwidth of the product of paths of the same length, Discrete Applied Mathematics, 161,3080–3086 (2013).

[4] R. C. Bollinger, A note on Pascal T -triangles, Multinomial coefficients and Pascal Pyramids, The Fibonacci Quaterly,24, 140–144 (1986).

[5] P.Z. Chinn, J. Chvatalova, A.K. Dewdney, N.E. Gibbs, The bandwidth problem for graphs and matrices—a survey,Journal of Graph Theory, 6, 223–254 (1982)

[6] L.H. Harper, Optimal numbering and isoperimetric problems on graphs, Journal of Combinatorial Theory, 1, 385-393(1966).

[7] C.H. Papadimitriou, The NP-completeness of the bandwidth minimization problem, Computing,16, 263–270 (1976).

[8] C. Smith, V. E. Hogatt, Generating functions of central values of generalized Pascal triangles, The Fibonacci , 17, 58–67(1979).

[9] X. Wang, X. Wu, S. Dumitrescu, On explicit formulas for bandwidth and antibandwidth of hypercubes, Discrete AppliedMathematics, 157 (8), 1947–1952 (2009).

102


The b-Chromatic Number and f -Chromatic Vertex Number ofRegular Graphs

Maidoun Mortada1∗, Amin El Sahili2, Hamamache Kheddouci3, Mekkia Kouider4

1 Faculty of Sciences-Lebanese UniversityEl Hadas-Beirut, [email protected]


[email protected] Claude Bernard University of Lyon 1

43 Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, [email protected]

4 ICJ, Department of Mathematics, University Claude-Bernard Lyon143 Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France

[email protected]

Abstract

The b-chromatic number of a graph G, denoted by b(G), is the largest positive integer k such that thereexists a proper coloring for G with k colors in which every color class contains at least one vertex adjacentto some vertex in each of the other color classes, such a vertex is called a dominant vertex. The f -chromaticvertex number of a d-regular graph G, denoted by f(G), is the maximum number of dominant vertices ofdistinct colors in a proper coloring with d + 1 colors. El Sahili and Kouider conjectured that b(G) = d + 1for any d-regular graph G of girth 5. Blidia, Maffray and Zemir [12] reformulated this conjecture by excludingthe Petersen graph and proved it for d ≤ 6. We study El Sahili and Kouider conjecture by giving some partialanswers under supplementary conditions.

Key words: Digraph, isomorphism, k-hypomorphy up to complementation, boolean sum, symmetric graph, tournament.

2010 Mathematics Subject Classification: 05C50; 05C60.

1. IntroductionA graph G is said to be k-regular if d(v) = k for all v ∈ G. A proper coloring of a graph G is a mapping c : V → S suchthat c(u) 6= c(v) whenever u and v are adjacent. The set S is the set of available colors. A proper coloring with m colorsis usually called an m-coloring. The chromatic number of G, denoted by χ(G), is the smallest integer m such that G has anm-coloring. A color class in a proper coloring of a graph G is the subset of V which contains all the vertices with same color.A vertex of color i is said to be a dominating or dominant vertex if it has a neighbor in each color class distinct from i. A

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color i is said to be a dominant color in G if there exists a dominant vertex of color i. A proper coloring of a graph is called ab-coloring, if each color class contains at least one dominant vertex. The b-chromatic number of a graph G, denoted by b(G),is the largest positive integer k such that G has a b-coloring with k colors.The concept of the b-chromatic number has been introduced by Irving and Manlove [23] when considering minimal propercolorings with respect to a partial order defined on the set of all partitions of the vertices of a graph. They proved that deter-mining b(G) is NP-hard for general graphs, but polynomial-time solvable for trees.Recently, Kratochvil et al. [11] have shown that determining b(G) is NP-hard even for bipartite graphs while Corteel,Valencia-Pabon, and Vera [25] proved that there is no constant ε > 0 for which the b-chromatic number can be approximatedwithin a factor of 120/133-ε in polynomial time (unless P=NP).Obviously, each coloring of G with χ(G) colors is a b-coloring. Also, b(G) ≤ ∆(G)+1. Therefore, for each d-regular graphG, b(G) ≤ d + 1. Since d + 1 is the maximum possible b-chromatic number of d-regular graphs, determining necessary orsufficient conditions to achieve this bound is of interest. Hoang and Kouider [7] characterized all bipartite graphs G and allP4-sparse graphs G such that each induced subgraph H of G satisfies b(H) = χ(H). If we are limited to regular graphs,Kratochvil et al. proved in [11] that for a d-regular graph G with at least d4 vertices, b(G) = d + 1. In [24], Cabello andJakovac reduced the previous bound to 2d3 − d2 + d and then El Sahili et al. [2] showed that b(G) = d + 1 for a d-regulargraph with at least 2d3 + 2d− 2d2 vertices. It was also proved in [2] that b(G) = d+ 1 for a d-regular graph G containing nocycle of order 4 and with at least d3 + d vertices. It follows from the above results that for any d, there is only a finite numberof d-regular graphs G with b(G) ≤ d. Kouider [15] proved that the b-chromatic number of a d-regular graph of girth at least6 is d + 1. El Sahili and Kouider [1] proved that the b-chromatic number of any d-regular graph of girth 5 that contains nocycle of order 6 is d + 1. In [1], El Sahili and Kouider asked whether it is true that every d-regular graph G with girth atleast 5 satisfies b(G) = d+ 1. For cubic graphs, if their girth is at least 6 or have at least 81 vertices, then by the above theirb-chromatic number is 4. Blidia, Maffray and Zemir [12] showed that the Petersen graph provides a negative answer to thisquestion since they proved that the b-chromatic number of Petersen graph is 3. They also proved that El Sahili and Kouiderconjecture is true for d ≤ 6 except for Petersen graph. Cabello and Jakovac [24] proved that a d-regular graph of girth atleast 5 has a b-chromatic number at least bd+1

2 c. Besides, they proved that every d-regular graph (d ≥ 6) that contains nocycle of order 4 and its diameter is at least d, has b-chromatic number d+ 1. S. Shaebani [26] improved this result by provingthat b(G) = d + 1 for a d-regular graph G that contains no cycle of order 4 and diam(G) ≥ 6. Also, it was shown in [26]that ifG is a d-regular graph that contains no cycle of order 4, then b(G) ≥ bd+3

2 c, and ifG has a triangle then b(G) ≥ bd+42 c.

2. Main ResultsWe study the b-chromatic number for d-regular graphs that contain no cycle of order 4 and for those of girth 5. For thispurpose, we introduce a new parameter, the f -chromatic vertex number of a d-regular graph G, which is the maximumnumber of dominant vertices of distinct colors in a (d + 1)-coloring of G. It is denoted by f(G). First we prove thatf(G) ≤ b(G) for any d-regular graph G. This result allows us to establish lower bounds of b(G) by studying f(G) whichseems more appropriate. We improve Shaebani result in case d is even by proving that f(G) ≥ dd−1

2 e + 2 for a d-regulargraph G that contains no cycle of order 4. Also we prove that b(G) = d + 1 for a d-regular graph G that contains neithera cycle of order 4 nor a cycle of order 6, and we provide a condition on the vertices of a d-regular graph G that contains nocycle of order 4 in order for b(G) to be d + 1. Finally, we show that f(G) ≥ dd−1

2 e + 4 for a d-regular graph G of girth 5and diameter 5.

References[1] A. El Sahili, M. Kouider, About b-colorings of regular graphs, Utilitas Math. Vol. 80, p.211-216, 2009

[2] A. El Sahili, M. Kouider, M. Mortada, submitted to the journal of Discrete Applied Mathematics.

[3] B. Effantin, The b-chromatic number of power graphs of complete caterpillars, J. Discrete Math. Sci. Cryptogr. 8(2005),483-502.

[4] B. Effantin, H. Kheddouci, The b-chromatic number of some power graphs, Discrete Math. Theor. Comput. Sci. 6(2003),45-54.

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[5] B. Effantin, H. Kheddouci, Exact values for the b-chromatic number of a power complete k-ary tree, J. Discrete Math.Sci. Cryptogr. 8(2005), 117-129.

[6] C. I. B. Velasquez, F. Bonomo, I. Koch, On the b-coloring of P4-tidy graphs, Discrete Appl. Math. 159(2011), 60-68.

[7] C. T. Hoang, M. Kouider, On the b-dominating coloring of graphs, Discrete Appl. Math. 152(2005), 176-186.

[8] D. Barth, J. Cohen, T. Faik, On the b-continuity property of graphs, Discrete Appl. Math. 155(2007), 1761-1768.

[9] F. Bonomo, G. Duran, F. Maffray, J. Marenco, M. Valencia-Pabon, On the b-coloring of cographs and P4-sparse graphs,Graphs Combin. 25(2009), 153-167.

[10] H. Hajiabolhassan, On the b-chromatic number of Kneser graphs, Discrete Appl. Math. 158(2010), 232-234.

[11] J. Kratochvil, Zs. Tuza, M. Voigt, On the b-chromatic number of graphs, In WG 02: Graph-Theoretic Concepts Comput.Sci., Lecture Notes in Comput. Sci. 2573(2002), 310-320.

[12] M. Blidia, F. Maffray, Z. Zemir, On b-colorings in regular graphs, Discrete Appl. Math. 157(2009), 1787-1793.

[13] M. Jakovac, S. Klavzar, The b-chromatic number of cubic graphs, Graph Combin. 26(2010), 107-118.

[14] M. Jakovac, I. Peterin, On the b-chromatic number of strong, lexicographic, and direct product, manuscript.

[15] M. Kouider, b-chromatic number of a graph, subgraphs and degrees, Res. Rep. 1392, LRI, Univ. Orsay, France, 2004.

[16] M. Kouider, M. Maheo, Some bounds for the b-chromatic number of a graph, Discrete Math. 256(2002), 267-277.

[17] M. Kouider, M. Maheo, The b-chromatic number of the cartesian product of two graphs, Studia Sci. Math. Hungar.44(2007), 49-55.

[19] R. Balakrishnan, S. Francis Raj, Bounds for the b-chromatic number of the Mycielskian of some families of graphs ,manuscript.

[18] M. Kouider, M. Zaker, Bounds for the b-chromatic number of some families of graphs, Discrete Math. 306(2006),617-623.

[20] R. Balakrishnan, S. Francis Raj, Bounds for the b-chromatic number of G- v, manuscript.

[21] R. Balakrishnan, S. Francis Raj, Bounds for the b-chromatic number of vertex- deleted subgraphs and the extremalgraphs (extended abstract), Electron. Notes Discrete Math. 34(2009), 353-358.

[22] R. Javadi, B. Omoomi, On b-coloring of the Kneser graphs, Discrete Math. 309(2009), 4399-4408.

[23] R. W. Irving, D. F. Manlove, The b-chromatic number of a graph, Discrete Appl. Math. 91(1999), 127-141.

[24] S. Cabello, M. Jakovac, On the b-chromatic number of regular graphs, submitted.

[25] S. Corteel, M. Valencia-Pabon, J-C. Vera, On approximating the b-chromatic number, Discrete Appl. Math. 146(2005),106-110.

[26] S. Shaebani, On the b-chromatic number of regular graphs without 4-cycle, submitted .

105


COLORIAGE DES NŒUDS

Hamid Abchir∗

Universit Hassan II.Casablanca, morocco

Key words: nœud, coloriage, r-colorable.

2010 Mathematics Subject Classification: 57M27, 57M25.

1. IntroductionUn entrelacs L n composantes est une sous varit de l’espace homomorphe une runion disjointe de n copies du cercle S1

(cordes noues et entrelaces). Si n = 1, L est un nœud. Etant donns deux entrelacs, est-ce qu’il y a moyen de dformer l’un“sans l’altrer” pour obtenir l’autre ? Deux entrelacs lis par une telle opration sont dits de meme type.L’tude mathmatique des types d’entrelacs est apparue au cours de la deuxime moiti du dix-neuvime sicle. Ds le dbut,l’utilisation de certaines projections planaires des entrelacs appeles diagrammes s’est avre opportune. En 1926, Reidemeistera tabli que deux diagrammes correspondent un meme entrelacs si et seulement si l’un peut etre obtenu partir de l’autre parune une suite finie de trois mouvements qu’il a introduits. Reidemeister a pu ainsi tablir une correspondance biunivoque entreles types de diagrammes et les types d’entrelacs dans l’espace ramenant ainsi l’tude des types d’entrelacs celle des types dediagrammes. Finalement, il a pu transformer des questions topologiques sur les entrelacs en problmes combinatoires.La comparaison des entrelacs repose sur la comparaison d’objets mathmatiques associs appels invariants. Les premiers invari-ants trouvs taient combinatoires. Cependant, le dveloppement de la topologie algbrique partir du dbut du vingtime a contribul’essort d’invariants plutot algbriques et entam l’intret port la combinatoire. Cette tendance s’est maintenue jusqu’au mi-lieu des annes quatre-vingt lorsque Kauffman a intrprt et calcul de manire combinatoire le puissant invariant polynomialfraıchement dcouvert par Jones. Depuis, les mathmaticiens spcialistes de la thorie des nœuds n’ont cess de porter un intretparticulier aux mthodes combinatoires. Nous proposons de mettre en valeur cet aspect de l’tude des entrelacs travers laprsentation d’invariants combinatoires lmentaires mais efficaces issus de ce qu’on appelle les r-coloriages de Fox. On selimitera au cas des nœuds.Soit r un entier naturel non nul. On considre le groupe fini Zr des entiers modulo r. Soit D un diagramme d’un nœud K. Atout arc de D, on attribue un entier dans Zr appel couleur de l’arc considr de telle facon qu’en chaque croisement, la sommedes couleurs des arcs passant en dessous moins deux fois la couleur de l’arc passant au dessus est gale 0 modulo r. Cela seramne la rsolution d’un systme d’quations dans Zr. Une solution de ce systme est appele r-coloriage de Fox de K. Il y a aumoins r solutions dont chacune consiste colorier tous les arcs de la meme manire. Ces solutions sont dites triviales.On effectue un mouvement de Reidemeister sur le diagramme D. Les r-coloriages de D avant d’effectuer le mouvementsont en correspondance biunivoque avec les r-coloriages du diagramme obtenu aprs le mouvement. Ainsi le nombre der-coloriages est un invariant du nœud.

Les r-coloriages de Fox fournissent d’autres invariants efficaces permettant de distinguer des familles de nœuds intres-santes. Entre autres, on considre ce qu’on appelle le nombre minimal de couleurs d’un nœud K modulo r not Cr(K) dfinicomme suit :

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Etant donn un entier r > 1. On considre un nœud K qui possde des r-coloriages non triviaux. Si D est un diagramme de K,on note nr,K(D) le nombre minimal de couleurs distinctes attribues aux arcs de D pour produire un coloriage non trivial deD. On note Cr(K) la plus petite valeur de ces minimums lorsque D dcrit tous les diagrammes de K [1]:

Cr(K) = minnr,K(D)|D est un diagramme de K.

On dira qu’un nœud est r-colorable s’il admet un diagramme supportant un coloriage non trivial.S; Satoh a montr en 2008 que C5(K) = 4 pour tout nœud 5-colorable [3]. Il a aussi remarqu que si p est un entier premieravec p > 3, alors tout nœud p colorable vrifie Cp(K) ≥ 4. En 2010 K. Oshiro a montr que C7(K) = 4 pour tout nœud7-colorable [2]. En 2014, T. Nakamura, Y. Nakanishi and S. Satoh on montr que C11(K) = 5 pour tout nœud 11-colorable[4]. Un rsultat similaire pour p = 13 est imminent.Nous donnerons un aperu sur ces rsultats, ceux esprs et sur une gnralisation de ces coloriages.

References[1] L. H. Kauffman and P. Lopes, On the minimum number of colors for knots, Adv. Appl. Math. 40 (2008) 36-53.

[2] K. Oshiro,Any 7 colorable knot can be colored by four colors, J. Math. Soc. Japan, Vol. 62, No 3 (2010), 963-973.

[3] S. Satoh,5-colored knot diagram with four colors, Osaka J. Math., 46 (2009), 939-948.

[4] T. Nakamura, Y. Nakanishi and S. Satoh, 11-colored knot diagram with five colors, http://arxiv.org/abs/1505.02980.

107


Sum of the Laplacian eigenvalues of a graph and Brouwer’sconjecture

Shariefuddin Pirzada1 ∗, Hilal A. Ganie2,

1 University of Kashmir, Department of MathematicsSrinagar, Kashmir, India

[email protected] University of Kashmir, Department of Mathematics

Srinagar, Kashmir, [email protected]

Abstract

Let G be a simple graph with n-vertices, m edges and Laplacian eigenvalues µ1, µ2, . . . , µn−1, µn = 0.Brouwer conjectured that Sk(G) ≤ m+

(k+12

), for all k = 1, 2, . . . , n, where Sk(G) =

∑ki=1 µi, is the sum of k

largest Laplacian eigenvalues of G. We obtain the upper bounds for Sk(G) in terms of the clique number ω, thevertex covering number τ and the diameter d of graphG and as a result we show that Brouwer’s conjecture holdsfor some classes of graphs. The Laplacian energy LE(G) of a graph G is defined as LE(G) =

∑ni=1 |µi − d|,

where d = 2mn is the average degree of G. We obtain the upper bound for Laplacian energy LE(G) of a graph

G in terms of the number of vertices n, the number of edges m, the vertex covering number τ and the cliquenumber ω of the graph.

Key words: Laplacian spectrum, clique number, vertex covering number, Laplacian energy.

2010 Mathematics Subject Classification: 05C30, 05C50.

1. IntroductionLet G(V,E) be a simple graph with n vertices and m edges having vertex set V (G) = v1, v2, . . . , vn and edge setE(G) = e1, e2, . . . , em. The adjacency matrix A = (aij) of G is a (0, 1)-square matrix of order n whose (i, j)-entry isequal to 1 if vi is adjacent to vj and equal to 0, otherwise. Let D(G) = diag(d1, d2, . . . , dn) be the diagonal matrix associ-ated to G, where di = deg(vi), for all i = 1, 2, . . . , n. The matrix L(G) = D(G)−A(G) is called the Laplacian matrix andits spectrum is called the Laplacian spectrum (L-spectrum) of the graph G. Being a real symmetric, positive semi-definitematrix, let 0 = µn ≤ µn−1 ≤ · · · ≤ µ1 be the L-spectrum of G. It is well known that µn=0 with multiplicity equal to thenumber of connected components of G and µn−1 > 0 if and only if G is connected [?].

Let Sk(G) =k∑i=1

µi, k = 1, 2, . . . , n be the sum of k largest Laplacian eigenvalues of G and let d∗i (G) = |v ∈ V (G) :

dv ≥ i|, for i = 1, 2, . . . , n. In 1994, Grone and Merris [5] conjectured the following.

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Theorem 1.1. (Grone-Merris Theorem) For any graph G and for any k, 1 ≤ k ≤ n

Sk(G) ≤k∑

i=1

d∗i (G).

This conjecture was proved by Hua Bai [1] and is nowadays called Grone-Merris theorem. As an analogue to Grone-Merristheorem, Andries Brouwer [2] conjectured the following.

Conjecture 1.2. If G is a graph with n vertices and m edges, then for any k, k = 1, 2, . . . , n

Sk(G) =k∑

i=1

µi ≤ m+(k + 1

2

).

Using computations on a computer, Brouwer [2] checked this conjecture for all graphs with at most 10 vertices. For k = 1,the conjecture follows from the well-known inequality µ1(G) ≤ n (see [3]). Also the cases k = n and k = n − 1 arestraightforward. Haemers et al. [6] showed that the conjecture is true for all graphs when k = 2 and is also true for trees.Du et al. [4] obtained various upper bounds for Sk(G) and proved that the conjecture is also true for unicyclic and bicyclicgraphs. Rocha et al. [8] obtained various upper bounds for Sk(G), which improve the upper bounds obtained in [4] for somecases and proved that the conjecture is true for all k with 1 ≤ k ≤ b g5c, where g is the girth of the graph G (the length ofsmallest cycle in G is called girth of the graph). They also showed that the conjecture is true for a for connected graph Ghaving maximum degree ∆ with p pendant vertices and c cycles for all ∆ ≥ c + p + 4. For the progress on this conjecture,we refer to [2, 6, 7, 8].

A clique of a graph G is the maximum complete subgraph of the graph G. The order of the maximum clique is calledclique number of the graph G and is denoted by ω. A subset S of the vertex set V (G) is said to be a covering set of G if everyedge of G is incident to at least one vertex in S. A covering set with minimum cardinality among all covering sets is calledminimum covering set of G and its cardinality is called vertex covering number of G, denoted by τ . The distance betweenany two vertices u and v is defined as the length of shortest path between them and the diameter d of graphG is the maximumdistance among all pair of vertices of G. If H is a subgraph of the graph G, then we denote the graph obtained by removingthe edges in H from G by G \H .

As usual, Kn and Ks,t denote, respectively, the complete graph on n vertices and the complete bipartite graph on s + tvertices. For other undefined notations and terminology from spectral graph theory, the readers are referred to [3].

The Laplacian energy LE(G) of a graph G is defined as LE(G) =∑ni=1 |µi − d|, where d = 2m

n is the average degreeof G.

2. Main resultsWe present the known upper bounds for Sk(G) in terms of clique number ω due to Du et al. [4]; the upper bound for Sσ(G)(where σ, 1 ≤ σ ≤ n− 1 is the number of Laplacian eigenvalue greater than or equal to 2m

n ), in terms of the clique numberω, the number of edges m and the maximum degree ∆ due to SP and Hilal [7]. We discuss the lower bound for Sω(G) interms of the clique number ω and the maximum degree ∆ due to SP and Hilal [7]. We obtain the upper bound for Sk(G), interms clique number ω and vertex covering number τ . We present a recent upper bound for Sk(G) in terms vertex coveringnumber τ and diameter d of the graph G.

We show that Brouwer’s conjecture holds for some classes of graphs. (1). For a connected graph G of order n ≥ 24

having clique number ω ≥ 7n8 , we have Sk(G) ≤ m +

k(k + 1)2

, for all k = 1, 2, . . . , bn2 c. (2). If G ∈ Ωn, then

Sk(G) ≤ m+k(k + 1)

2, for all k, 1 ≤ k ≤ n. (3). LetG be a connected graph of order n ≥ 2 withm edges having the vertex

covering number τ ≤ 1.3s1. IfKs1,s1 , is the maximal complete bipartite subgraph of graphG, then Sk(G) ≤ m+k(k + 1)

2,

for all k = 1, 2, . . . , n. (4). Let G be a connected bipartite graph of order n ≥ 2 with m edges having the vertex covering

number τ . If Ks1,s1 , s1 ≥ n4 is the maximal complete bipartite subgraph of graph G, then Sk(G) ≤ m +

k(k + 1)2

, for all

109


k ≤ n7 − 1 and k ≥ 6n

7 .

As an application of the above work, we obtain an upper bound for Laplacian energy LE(G) in terms clique number ωand vertex covering number τ ; and an upper bound for Laplacian energy LE(G) in terms of vertex covering number τ . andpositive integers s1 and s2.

References[1] H. Bai, The Grone-Merris conjecture, Trans. Amer. Math. Soc., 363 (2011) 4463–4474.

[2] A. E. Brouwer and W. H. Haemers, Spectra of graphs. Available from: http://homepages.cwi.nl/aeb/math/ipm.pdf.

[3] D. Cvetkovic, M. Doob and H. Sachs, Spectra of graphs-Theory and Application, Academic Press, New York, 1980.

[4] Z. Du and B. Zhou, Upper bounds for the sum of Laplacian eigenvalues of graphs, Linear Algebra Appl., 436 (2012)3672-3683.

[5] R. Grone and R. Merris, The Laplacian spectrum of a graph II, SIAM J. Discrete Math., 7 (1994) 221-229.

[6] W. H. Haemers, A. Mohammadian and B. Tayfeh-Rezaie, On the sum of Laplacian eigenvalues of graphs, LinearAlgebra Appl., 432 (2010) 2214-2221.

[7] S. Pirzada, H. A. Ganie, On the Laplacian eigenvalues of a graph and Laplacian energy, Linear Algebra Appl., 486(2015) 454–468.

[8] I. Rocha, V. Trevisan, Bounding the sum of the largest Laplacian eigenvalues of graphs, Discrete Applied Math., 170(2014) 95–103.

110


Resolution approchee d’un probleme d’ordonnancement de typeflowshop continu avec recirculation

Karim Amrouche1∗, Mourad Boudhar1, Farouk Yalaoui2

1 USTHB University, RECITS laboratory,Faculty of MathematicsBP 32 Bab-Ezzouar, El-Alia 16111, Algiers, Algeria

[email protected], [email protected] University of Technology of Troyes, LOSI laboratory, ICD UAR CNRS 6281

12 street Marie Curie, BP 2060, Troyes 10010, [email protected]

Resume

On considere un probleme d’ordonnancement de type flow shop avec recirculation sur la premiere machine,ou les operations d’une tache sont ordonnancees sur les machines d’une maniere continue. L’objectif est deminimiser le makespan. Comme ce probleme est NP-difficile au sens fort, nous proposons une metaheuristiquepour le resoudre.

Key words : flowshop, recirculation, makespan, metaheuristique.

2010 Mathematics Subject Classification : 90B35.

1. IntroductionDans ce travail, on s’interesse a un probleme d’ordonnancement d’atelier de type flowshop avec recirculation sur la

premiere machine en presence de la contrainte de sans attente (no-wait en Anglais). La contrainte sans attente signifie qu’au-cune attente entre deux operations d’une meme tache n’est permise. Donc, chaque tache Tj doit etre traitee d’une manierecontinue : d’abord sur la premiere machine, ensuite sur la deuxieme machine M2, jusqu’a la derniere machine Mm et puiselle revient pour une deuxieme execution sur la premiere machine M1 (la derniere operation). L’objectif est de minimiser ladate de fin de traitement de l’ensemble des taches. Ce probleme est note Fm |chain− reentrant, no− wait|Cmax.

On note par :aj : Le temps de traitement de la premiere operation de la tache Tj sur la premiere machine.bji : Le temps de traitement de la tache Tj sur la machine Mi, i = 2, . . . ,m.cj : Le temps de traitement de la deuxieme operation de la tache Tj sur la premiere machine.

Notons que le probleme F2|no−wait|Cmax est resolu en temps polynomial par l’algorithme de Gilmore et Gomory [3].Cependant, ce probleme est NP-difficile au sens fort lorsque le nombre de machines devient egal a trois Rock [4]. Dans Am-rouche et Boudhar [1] nous avons considere un probleme d’ordonnancement d’atelier de type flow shop avec recirculation eten presence d’un temps de latence exacte, ce probleme a ete note F2|chain− reentrant, lj = L|Cmax. Nous avons montre

∗Speaker111


M1

M2

M3

a[j]

aj-

b[j2]

bj2-

b[j3]

bj3

-

c[j]

cj -

FIG. 1 – Schema d’execution des taches dans le probleme F3|chain− reentrant, no− wait|Cmax

que ce probleme est NP-difficiles au sens fort et nous avons presente des sous problemes polynomiaux pour le resoudre.

Dans le present papier nous montrons que les sous problemes suivants :

-F2 |chain− reentrant, no− wait, bj ∈ 0, L, cj ∈ 0, L+ 1|Cmax

-F2 |chain− reentrant, no− wait, aj ∈ 0, L+ 1, bj ∈ 0, L|Cmax

-F2 |chain− reentrant, no− wait, aj = cj ∈ 0, L|Cmax

Sont NP-difficile au sens ordinaire et ceci par une reduction du probleme de 2-partition qui est NP-difficile au sens ordi-naire Garay et Johnson [2].

Pour la resolution du probleme generale Fm|chain − reentrant, no − wait|Cmax un algorithme genetique avec desexperimentations numerqiues est presente.

References[1] Amrouche K, Boudhar M. Two machines flow shop with reentrance and exact time lag. RAIRO - Operations Research,

DOI : 10.1051/ro/2015015.

[2] Garey MR, Johnson DS. Computers and intractability : a guide to the theory of NP-completeness. San Francisco : W.H.Freeman, 1979.

[3] Gilmore P.C., Gomory, R.E. Sequencing a one-state variable machine : A solvable case of the travehng salesman pro-blem. Oper. Res., 1964, 12 , 655–679.

[4] Hans Rock. The Three-Machine No-Wait Flow Shop Is NP-Complete. Journal of the Assoclatton for Computmg Ma-chinery, 1984, 31, 336–345 .

[5] Wang MY, Sethi SP, Van De Velde SL. Minimizing makespan in a class of reentrant shops. Oper. Res., 1997, 45, 702–712.

[6] V. Lev and I. Adiri, V-shop scheduling. European J. Opnl. Res., 1984, 18, 51–56.

112


Optimal control of dynamic systems with random input

Zahia Bouabbache1∗, Mohamed Aidene2

1 University M. Mammeri Tizi-Ouzou , Laboratory L2CSPAlgeria

bouabbache [email protected] University M. Mammeri Tizi-Ouzou, Laboratory L2CSP

Algeriaaidene [email protected]

Abstract

Our work is to solve a problem of optimal control with perturbed initial condition.The aim of the problemis to reach a target with a given probability. In this work, we transform a stochastic optimal control probleminto an equivalent deterministic and known problem using the inverse of the density function and they solve thislatter using the adaptive method of Gabasov and Kirillova

Key words: optimal control, support control, random variable, stochastic dynamic problem .

1. IntroductionWhen a real problem is mathematically modeled and solved by means of mathematical method it may happen that some ofthe parameters which figure in the problem are unknown, whether it appears in the objective function or in the constraints.If these parameters of unknown value can be taken as random variables the resulting problem becomes a stochastic problem. In this study, we assume that the components of the initial state are continous random variables defined on the space ofprobability (Ω,Ξ, P ), of known distribution with a probability to reach the target, so the stochastic dynamic system whichwe consider is as follows:

J(u(t)) = E(cx(tf ))→ maxu (1)x = Ax+ bu, x(t0) = xt0(ω) (2)P (Hix(tf ) ≤ gi) = αi, i = 1,m (3)d1 ≤ u(t) ≤ d2 (4)t ∈ T = [t0, tf ]

(Ip)

where, x(t) ∈ <n, x(t0) the initial state, A ∈ <n × <n, the rank of H = m < n, b ∈ <m, g ∈ <m, c ∈ <n, u(t) ∈ <,t ∈ T the control, J(u(t)) is the objective function and tf is the final time, αi is the individual probability for each of theconstraints .The idea to solve this type of problem is the transformation into equivalent problem .This transformation is achieved by thedetermination of the standardised function, witch is very complicated.For this raison we make the hypothesis that xt0(ω) follows a normal distribution of expected value xt0and covariance matrixΓxt0

.

∗Speaker113


2. problem modellingBy the Cauchy formula, the solution of system (2) can be written in the form:

x(tf ) = F (tf )(xt0 +∫ tf

t0

F−1(t)bu(t)dt) (5)

where F (t) is the solution of the system: ˙F (t) = AF (t),F (0) = In, where In represents the identity matrix of order n.Using the solution (5), the problem Ip become:

J(u(t)) = cF (tf )xt0 +∫ tf

t0cF (tf )F−1(t)bu(t)dt→ maxu

p(Hix(tf )xt0 +∫ tf

t0HiFi(tf )F−1

i (t)bu(t)dt ≤ gi) = αi

d1 ≤ u(t) ≤ d2

t ∈ T = [t0, tf ]

⇒



p(HiFi(tf )xt0 ≤ gi −∫ tf

t0HiFi(tf )F−1

i (t)bu(t)dt) = αi

d1 ≤ u(t) ≤ d2

t ∈ T = [t0, tf ]

We put vi = HiFi(tf ), and we have xt0 is the expected value vector of xt0(ω) and Γxt0its variance and covariance matrix,

then the expression of the expected value of random variable vxt0 is vxt0 and its variance is vΓxt0v wich implies that:[3]



P ( vixt0−vixt0√viΓxt0

vi≤ gi−

∫ tft0

HiFi(tf )F−1i (t)bu(t)dt−vixt0√

viΓxt0vi

) = αi

d1 ≤ u(t) ≤ d2

t ∈ T = [t0, tf ]

⇒



Φi(gi−

∫ tft0

HiFi(tf )F−1i (t)bu(t)dt−vixt0√

viΓxt0vi

) = αi

d1 ≤ u(t) ≤ d2

t ∈ T = [t0, tf ]

where Φ is the standardised normal distribution function. we know that if Φ(α) = β then α = Φ−1(β)

⇒


t0cF (tf )F−1(t)bu(t)dt→ maxu∫ tf

t0HiFi(tf )F−1

i (t)bu(t)dt = gi − vixt0 − Φ−1i (αi)

√viΓxt0

vi

d1 ≤ u(t) ≤ d2

t ∈ T = [t0, tf ]

we put: c(t) = cF (tf )F−1(t)b, ϕi(t) = HiFi(tf )F−1i (t)b, zi = gi − vixt0 −

√viΓxt0φ

−1i (αi)

Then the equivalent problem is:


t0c(t)u(t)dt→ maxu (6)∫ tf

t0ϕi(t)u(t)dt = zi (7)

d1 ≤ u(t) ≤ d2 (8)t ∈ T = [t0, tf ], i = 1,m

(Ep)

Definition 1.A control u(t) is called admissible if and only if it verified the constraints

(7)− (8)

114


Definition 2.An admissible control u0(t), at which the objective function achieves maximal value is named an optimal control and for

given value ε > 0, is called ε-optimal if it satisfies the inequality :

J(u0)− J(uε) ≤ ε

Theorem 3.The Kalman rank condition states a linear autonomous system of the form x = Ax+ bu is controllable if and only if the

matrix: C = (b, Ab,A2b, ...., An−1b) is of rank n

3. Method of resolutionIt’s a direct method based on 3 procedures:

1. Discretization

2. Adaptive method

3. Final procedure

4. Numerical example

x2(1)→ maxu

x1 = x2

x2 = uP (x1(1) ≤ 2) = 0.8x0 = x0(ω) (m,σ2)

Where: A =(

0 10 0

), b =

(01

), F (t) =

(1 t0 1

), F−1(t) =

(1 −t0 1

), Γx0 =

(3 11 3

), x0 =

(1 2

),

c =(

0 1),ε = 103,µ = 0.02, H =

(1 0

),N = 4, φ−1(0.9) = 1.56

By using the algorithm implimented under MATLAB, the result of the problem is given in the following tableJ(u) 4.3923τB 0.2679x(tf ) 8.3205 4.3923u(t) −1 1

time(s) 1.4728

5. ConclusionIn this work we have solved a terminal probleme of linear dynamic system with perturbed initial condition and with aprobability to reach a target by a method based on three procedures:Discretization, Adaptive method and Final procedure

References[1] R.Gabasov., Adaptive method of linear programming. Preprint of theUniversity of Karlsruhe, Institute of Statistics and

Mathematics. Karsruhe, Germany., 1993

[2] K.Louadj.Rsolution de problmes paramtrs de contrle optimal,universit de tizi-ouzou , 2012.

[3] B.Oksendal., Stochastic Differential Equations. An introduction with Applications,Fifth Edition, corrected printing.,2000.

[4] E. Trelat.,Thorie et application, Vuibert, collection Mathmatiques concrtes ., 2005

115


Towards a Set-String Subsequence Kernel∗

Slimane Bellaouar1†, Hadda Cherroun1 and Attia Nehar1

1 Universite Amar Telidji, Laboratoire LIMLaghouat, Algerie

s.bellaouar, a.nehar, hadda [email protected]

Abstract

Kernel methods are powerful tools in machine learning. In this paper, we present a novel approach tocompute the string subsequence kernel (SSK). Our main idea is that we can represent all string subsequencesby a weighted automaton. Thus the SSK computation of two strings can be achieved by automata intersection.The proposed approach can be used to compute efficiently the subsequence kernel between two sets of strings.

Key words: string subsequence kernel, Weighted automata .

1. IntroductionKernel methods [3] offer an alternative solution to the limitation of traditional machine learning algorithms. They allow todiscover non-linear relations and enable other data type processings (biosequences, images, graphs, . . . ).

Machine learning community devotes a great effort of research to string kernels. The philosophy of all string kernelscan be reduced to different ways to count common substrings or subsequences that occur in both strings to be compared,say s and t. In the literature, there are several approaches to improve the computation of the SSK. The first one is basedon dynamic programming; Lodhi et al. [5] apply dynamic programming paradigm to the suffix version of the SSK. Theyachieve an O(p|s||t|) complexity , where p is the length of the SSK. Later, Rousu and Shawe-Taylor [6] propose an improve-ment to the dynamic programming approach based on the observation that most entries of the dynamic programming matrix(DP) do not really contribute to the result. They use a set of match lists combined with a sum range tree. They achievean O(p|L| log min(|s|, |t|)) complexity, where L is the set of matches of characters in both strings. Beyond the dynamicprogramming paradigm, the trie-based approach [4, 6, 7] is based on depth first traversal on an implicit trie data structure.The idea is that each node in the trie corresponds to a co-occurrence between strings. But the number of gaps is restricted, sothe computation is approximate. Recently Bellaouar et al. [2] have proposed a list based approach using a layerd range tree(LRT) data structure. The Whole task takes O(|L| log |L|+ pK) time and O(|L| log |L|+K) space, where |L| is the size oflist, p is the length of the SSK and K is the total number of reported points.

Motivated by the computation of the subsequence kernel for a set of strings, in this paper we propose an adequaterepresentation of strings to be compared. Our main idea consists on mapping a machine learning problem on automata theoryfield. Precisely, we have proposed a weighted automaton construction to represent all susbequences of some string. Theintersection of such weighted automata with some filtering of redundant paths substitute the SSK computation. Moreover,using our construction to represent a set of strings can be a first step to compute their subsequence kernel.

∗This work is supported by Algeria/South Africa joint project under code: A/AS-2013-002.†Speaker

116


2. Main resultsThe philosophy of all string kernel approaches can be reduced to different ways to count common substrings or subsequencesthat occur in the two strings to compare. This philosophy is manifested in two steps:

• Project the strings over an alphabet Σ to a high dimension vector space F , where the coordinates are indexed by asubset of the input space.

• Compute the distance (inner product) between strings in F . Such distance reflects their similarity.

For the String Subsequence Kernel (SSK) [5], the main idea is to compare strings depending on common subsequences theycontain. Hence, the more similar strings are ones that have the more common subsequences. In order to measure the distanceof non contiguous elements of the subsequence, a gap penalty λ ∈]0, 1]is introduced. Formally, the mapping function φp(s)in the feature space F can be defined as follows:

φpu(s) =

∑

I:u=s(I)

λl(I), u ∈ Σp. (1)

(for the definitions of I, s(I) and l(I) see [2]). The associated kernel can be written as:

Kp(s, t) = 〈φp(s), φp(t)〉=

∑

u∈Σp

φpu(s).φp

u(t)

=∑

u∈Σp

∑

I:u=s(I)

∑

J:u=t(J)

λl(I)+l(J). (2)

Our main idea consists of mapping an SSK problem to automata theory field. To do so, we first propose a constructionalgorithm of a weighted automaton representing all subsequences of some string. Such construction is the first step to expressthe formula in (1). To describe our approache, we use two strings s = gatta and t = cata as a running example. The weightedautomaton associated to the string t is illustrated in Figure 1.

q0start q1 q2 q3 q4

q5

q6

q7

c\λ

c\λ

ε\1

a\λε\λ

a\λ

t\λε\λ

t\λ

t\λ

ε\1

a\λ

a\λt\λ

t\λ

ε\1

a\λ

Figure 1: Weighted automaton of all subsequences associated to the string t = cata

The computation of the SSK is achieved by performing the intersection of the weighted automata associated to thecompared strings A(s) ∩ A(t). However, the application of the ordinary intersection (ε-free case) would generate redundantε-paths and leads to incorrect results.

117


To overcome this problem, we have adapted a solution applied to transducers in [1]. Accordingly, we have renamed theexisting ε-labels of A(s) as ε1 and the ε-labels of A(t) as ε2 and augment A(s) with a self loop labelled with ε2 at all statesand A(t) with a self loop labelled with ε1 at all states as illustrated in Figure 2

q0start

q1

q2 q3

q4

q5

q6

q7

q8

q9

q10

c\λ

c\λε1\1

a\λε2\λ

a\λ

ε2\λa\λ

a\λ

t\λ

t\λ

ε2\λ

t\λ

t\λ

ε2\1

a\λ

a\λ

ε1\1

ε1\1

ε1\1

ε1\1

ε1\1

ε1\1

ε1\1

ε1\1

ε1\1

ε1\1

Figure 2: he augmented Weighted automaton A′(t) associated to the string t = cata

Thereafter, we have to discard redundant ε-paths by performing the result of intersection of the augmented weightedautomata with a filter automaton shown at Figure 3 ( A′(s) ∩ A′(t) ∩ F).

q0

start

q1q2ε1

Σε2

Σ

Σ ε1\1ε2\1

Figure 3: Filter automaton F discarding redundant ε-paths

So far, our computation is general, i.e. we compute the all-subsequences kernel. To deal with the SSK (formula 2),we have to restrict the computation to subsequences of a given fixed length p. To achieve this goal we have kept the sameintersection operation with a p− gram automaton (A′(s)∩A′(t)∩F ∩Pgram). Figure 4 illustrate the construction methodof such automaton for p = 2.

118


q0start q1 q2c, a, t, g\1 c, a, t, g\1

ε\1 ε\1

Figure 4: 2−gram automaton (Pgram) to restrict the computation of the SSK to the subsequences of length p = 2

3. ConclusionWe have presented a novel approach to compute the String Subsequence Kernel (SSK). We started by the construction ofthe weighted automata for all subsequences of the strings to be compared. The ε-labels of the weighted automata engenderredundant ε-paths while performing the intersection. To overcome this problem we have augmented the constructed automatawith ε1 and ε2 and we have applied the intersection with a filter automaton (F) to discard redundant ε-paths. To evaluate theSSK of length p we have reapplied the intersection with a pgram (Pgram) automaton.

The application of the intersection with the filter and the pgram automata doesn’t influence the complexity of the SSKcomputation because they contain a constant number of transitions.

A noteworthy advantage is that the variation of the pgram automaton can give rise to a variety of string kernels.We conjecture that using our approach with an intelligent intersection on the one hand and the construction of prefix

weighted automaton for a set of strings on the other hand gives efficient and competitive solution to the kernel methodcomputations.

References[1] Cyril Allauzen and Mehryar Mohri. 3-way composition of weighted finite-state transducers. In Implementation and

Applications of Automata, 13th International Conference, CIAA 2008, San Francisco, California, USA, July 21-24,2008. Proceedings, pages 262–273, 2008.

[2] Slimane Bellaouar, Hadda Cherroun, and Djelloul Ziadi. Efficient list-based computation of the string subsequencekernel. In LATA’14, pages 138–148, 2014.

[3] Nello Cristianini and John Shawe-Taylor. An introduction to support Vector Machines: and other kernel-based learningmethods. Cambridge University Press, New York, NY, USA, 2000.

[4] C. Leslie, E. Eskin, and W. Noble. Mismatch String Kernels for SVM Protein Classification. In Neural InformationProcessing Systems 15, pages 1441–1448, 2003.

[5] Huma Lodhi, Craig Saunders, John Shawe-Taylor, Nello Cristianini, and Chris Watkins. Text classification using stringkernels. J. Mach. Learn. Res., 2:419–444, March 2002.

[6] Juho Rousu and John Shawe-Taylor. Efficient computation of gapped substring kernels on large alphabets. J. Mach.Learn. Res., 6:1323–1344, December 2005.

[7] John Shawe-Taylor and Nello Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, New York,NY, USA, 2004.

119


Statistic on linear tilings and generalized q-Fibonacci polynomials

Amine Belkhir1∗, Hacene Belbachir2

1 USTHB, Faculty of MathematicsRECITS Laboratory DG-RSDT

BP 32, El Alia 16111 Bab Ezzouar, Algiers, [email protected]

2 USTHB, Faculty of MathematicsRECITS Laboratory DG-RSDT


Abstract

We introduce Φm-statistic on linear square-and-domino tilings. We prove that the distribution of this statisticover the set of tilings will give us the generalized q-Fibonacci polynomials of Cigler. Furthermore, we providebijective proofs of some identities involving the q-Fibonacci polynomials.

Key words: q-Fibonacci polynomials, statistics, tilings.

2010 Mathematics Subject Classification: 11B39, 05A15, 05A19.

1. IntroductionCigler [3] has introduced and studied a generalization of q-Fibonacci polynomials defined by

Fn+1(x, y;m) =∑

k≥0

q(k+12 )+m(k

2)(n− kk

)

q

ykxn−2k,

where(nk

)q

is the q-binomial coefficient given by

(n

k

)

q

=[n]q!

[k]q![n− k]q!.

with [n]q! :=∏ni=1[i]q and [i]q := 1 + q + q2 + · · ·+ qi−1.

The polynomials Fn(x, y;m) satisfy the two following recurrences relations

Fn+1(x, y;m) = xFn(x, qy;m) + qyFn−1(x, qm+1y;m),

andFn+1(x, y;m) = xFn(x, y;m) + qnyFn−1(x, qmy;m).

∗Speaker120


with initial values F0(x, y;m) = 0 et F1(x, y;m) = 1. The first polynomials are given by

F1(x, y;m) = 1F2(x, y;m) = x

F3(x, y;m) = x2 + qy

F4(x, y;m) = x3 + qyx+ q2yx

F5(x, y;m) = x4 + qyx2 + q2yx2 + q3yx2 + qm+3y2

Note that when m = 0 and m = 1 we obtain the classical q-Fibonacci polynomials of Cigler [3] and Carlitz [1],respectively.

Fn+1(x, y; 0) =∑

k≥0

q(k+12 )(n− kk

)

q

,

and

Fn+1(x, y; 1) =∑

k≥0

qk2(n− kk

)

q

.

Let Tn,k denote the set of linear tiling of a (1 × n)-board with cells labeled 1, 2, . . . , n arranged in a row by k dominosand n− 2k squares, where each domino covered two cells, and a square is a piece covering single cell. Shattuck and Wagner[8, 9, 10] have studied some statistics on the set of tilings Tn,k and they established the following connection

Fn+1(1, y; 1) =∑

0≤k≤bn/2c

∑

t∈Tn,k

qσ(t)yν(t).

where ν(t) is the number of dominos in the covering t and σ(t) is the sum of the numbers covered by the left halves of eachof those dominos

The aim of this work is to give a new statistic with distribution over the set of tilings Tn,k given by the polynomialsFn(x, y;m). Moreover, we provide bijective proofs of some identities involving the polynomials Fn(x, y;m).

2. Main resultsGiven a tiling t = t1t2 . . . tn where ti ∈ square, domino for 1 ≤ i ≤ n, we define the statistics inv(t) and ninv(t) ontiling t as follows.

inv(t) := #(i, j) : i < j et ti > tj,and

ninv(t) := #(i, j) : i < j et ti = tj = domino,For example, if n = 5 and t = dsdds (see Figure 1 below), then inv(t) = #(1, 2), (1, 5), (3, 5), (4, 5) = 4 and

ninv(t) = #(1, 3), (1, 4), (2, 3) = 3.

1 2 3 4 5 6 7 8

Figure 1: The tiling t = dsdds.

Definition 1. Let t = t1t2 . . . tn be a linear tiling, we define the Φm-statistic as follows

Φm(t) := (m+ 1)ninv(t) + inv(t) + ν(t).

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Theorem 2. For any n ≥ 0, Fn+1(x, y;m) is the generating polynomial of the set⋃k≥0 Tn,k according to Φm-statistic.

Fn+1(x, y;m) =∑

k≥0

∑

t∈Tn,k

qΦm(t)yν(t)xn−2ν(t).

For example, there are 5 tilings in the set⋃

0≤k≤2 T4,k,

ninv(t) = 1, inv(t) = 0, ν(t) = 2, Φm(t) = m+ 3

ninv(t) = 0, inv(t) = 2, ν(t) = 1, Φm(t) = 3

ninv(t) = 0, inv(t) = 1, ν(t) = 1, Φm(t) = 2

ninv(t) = 0, inv(t) = 0, ν(t) = 1, Φm(t) = 1

ninv(t) = 0, inv(t) = 0, ν(t) = 0, Φm(t) = 0.

Then, we have∑

0≤k≤2

∑

t∈T4−k,k

qΦm(t)yν(t)xn−2ν(t) = qm+3y2 + q3yx2 + q2yx2 + qyx2 + x4.

Corollary 2.1. The statistics Φ0, π and Φ1, σ are are equidistributed on the set Tn,k, i.e∑

t∈Tn,k

qΦ0(t) =∑

t∈Tn,k

qπ(t) and∑

t∈Tn,k

qΦ1(t) =∑

t∈Tn,k

qσ(t).

References[1] Carlitz, L. Fibonacci notes 4: q-Fibonacci polynomials, Fibonacci Quart. 13 (1975), 97–102.

[2] Cigler, J. A new class of q-Fibonacci polynomials, Electron. J. Combin. 10 (2013), Research Paper 19, 15 pp. (elec-tronic).

[3] Cigler, J. Some beautiful q-analogues of Fibonacci and Lucas polynomials, arXiv 1104.2699. (2011).

[4] Goyt, A. M., Sagan, B. E. Set partition statistics and q-Fibonacci numbers, European J. Combin., 30 (2009), 230–245.

[5] Goyt, A. M., Mathisen, D. Permutation statistics and q-Fibonacci numbers, Electron. J. Combin., 16 (2009), ResearchPaper 101, 15.

[6] Mansour, T., Combinatorics of set partitions. Discrete Mathematics and its Applications (Boca Raton). CRC Press, BocaRaton, (2013)

[7] Mansour, T., Shattuck, M. Generalizations of two statistics on linear tilings, Appl. Appl. Math., 7 (2012), 508–533.

[8] Shattuck, M., Wagner, C. G. Parity theorems for statistics on domino arrangements, Electron. J. Combin., 12 (2005), #N10.

[9] Shattuck, M., Wagner, C. G. A new statistic on linear and circular r-mino arrangements, Electron. J. Combin., (1) 13(2006).

[10] Shattuck, M., Wagner, C. G. Periodicity and parity theorems for a statistic on r-mino arrangements, J. Integer Seq., 9(2006), Article 06.3.6, 18.

122


Some results on Cauchy numbers

Mourad RAHMANI

USTHB, Faculty of Mathematics, RECITS LaboratoryP. O. Box 32, El Alia, Bab Ezzouar, 16111, Algiers, Algeria

[email protected]

AbstractIn this talk we define a new family of p-Cauchy numbers by means of the confluent hypergeometric function.We establish some basic properties. As consequence, a number of algorithms based on three-term recurrencerelation for computing Cauchy numbers of both kinds, Bernoulli numbers of the second kind are derived.

Key words: Cauchy numbers, confluent hypergeometric function, three-term recurrence relation

2010 Mathematics Subject Classification: 05A15, 11B75.

1. IntroductionThe Cauchy numbers of both kinds are introduced by Comtet [1] as integral of the falling and rising factorials

c(1)n :=

1∫

0

xn dx, and c(2)n :=

1∫

0

(x)ndx.

These numbers appears in diverse contexts, like number theory, special functions and approximation theory. Recently, manyresearch articles have been devoted to Cauchy numbers of both kinds and many generalization are introduced (poly-Cauchynumbers, hypergeometric Cauchy numbers, generalized Cauchy numbers). As an example of a recent application of theCauchy numbers of both kinds see [2], where Masjed-Jamei et al. have derived the explicit forms of the weighted Adams-Bashforth rules and Adams-Moulton rules in terms of the Cauchy numbers of the first and second kind.

In this talk, a number of algorithms based on three-term recurrence relation for calculating Cauchy numbers of both kinds arederived. In particular, we prove the following recurrence relation for computing Bernoulli numbers of the second kind [3].First, we construct an infinite matrix (bn,p)n,p≥0 as follow: the first row of the matrix is b0,p := 1 and each entry is given by

bn+1,p =1− n1 + n

bn,p −p+ 1

(p+ 2) (1 + n)bn,p+1.

Finally, the first column of the matrix are Bernoulli numbers of the second kind bn,0 := bn.

References[1] Comtet L., Advanced combinatorics,D. Reidel Publishing Co., 1974.

[2] Masjed-Jamei M., Jafari M. A., Srivastava H. M., Some applications of the Stirling numbers of the first and secondkind, J. Appl. Math. Comput., (2014), 1-22.

[3] Rahmani M., On p-Cauchy numbers, To appear in Filomat, 2015.

123


Tiling interpretation for combinatorial identity associated to orderthree Fibonacci sequence

Imene Benrabia1∗, Hacene Belbachir2

1,2 Faculty of Mathematics,USTHB, RECITS LaboratoryBP 32 El Alia, 1611 Bab Ezzouar, Algeria

[email protected]@yahoo.com

Key words: Fibonomial coefficient, Fibonacci number, Tiling.

1. Introduction

The Fibonomial coefficient(nk

)F

is define, for 1 ≤ k ≤ n, as(n

k

)

F

=FnFn−1...Fn−k+1

FkFk−1...F1,

where Fn is, for n ≥ 0, the Fibonacci sequence defined as Fn = Fn−1 + Fn−2 with F0 = 0 and F1 = 1.fn = Fn+1 counts tiling of a 1× n board with squares and dominoes[1], There are two possible ways to tile the end

of 1× n board, wether by a square and in this case it remains to tile 1× (n− 1) board in Fn−1 possible ways, or by adomino and the remaining 1× (n− 2) board can be completed with Fn−2 possible ways.

Benjamin and Plott gave a combinatorial interpretation of fibonomial coefficients using tiling by squares and dominoes[2],they established the following formula:

Theorem 1. (n

k

)

F

=∑∑

...∑

1≤x1<x2<...<xk−1≤n−1

k−1∏

i=1

Fxi−xi−1−1k−i Fn−xi−(k−i)+1.

When squares and dominoes are colored, they obtained the Lucas sequence, for n ≥ 2,Un = a.Un−1 + b.Un−2 with U0 = 0, U1 = a .Where a (resp b) is the squares(resp dominoes)’s colors number. For thesequence Un they obtained:

Theorem 2. (n

k

)

U

=∑∑

...∑

1≤x1<x2<...<xk−1≤n−1

k−1∏

i=1

bxk−1−(k−1)Uxi−xi−1−1k−i Un−xi−(k−i)+1.

∗Speaker

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2. Main results

We are proposing to demonstrate the associated formula to the interpretation of tiling with squares and linear triominos,

∑∑...∑

1≤x1<x2<...<xk−1≤n−k

k−1∏

i=1

Φk−i+1(Φk−i + Φk−i−1)xi−xi−1−1Φn−2k−xi+i+1.

Where x0 = 0 and Φn is defined by

Φn = 0 for n < 0,Φ0 = Φ1 = Φ2 = 1,Φn+1 = Φn + Φn−2.

References

[1] Arthur T. Benjamin., Jennifer J Quinn., Proofs That Really Count: The art of Combinatorial proofs, Washington DC:Mathematical Association of America, 2003.

[2] Arthur T. Benjamin., Sean S. Plott., A combinatorial approach to fibonomial coefficients, Fibonacci Quarterly., 01/2008.

124


Modeling the periodic conditional volatility as a mixture process

Faycal HAMDI∗

USTHB, RECITS LaboratoryPO box 32. El Alia, Bab Ezzouar, Algiers, Algeria

[email protected]

Abstract

There are several properties which make the mixture time series models potentially useful in economic stud-ies and financial modeling. Indeed, these models are among the most powerful tools for modeling some stylizedfeatures exhibited by many time series such as volatility clustering, multimodality, tail heaviness, change inregime and asymmetry. In this talk, we will discuss and evaluate some new dynamic mixture periodic time se-ries models, and we will show their advantage and performance in modeling the periodic conditional volatility.We will provide some probabilistic properties of these models, namely strict and second-order periodic station-arity, periodic ergodicity and existence of higher order moments. Parameters estimation via the Expectation-Maximization (EM ) algorithm and model selection method will be discussed in details. To demonstrate theusefulness of the EM algorithm, we will present some Monte Carlo experiments. Applications to the dailyStandard and Poor Composite 500 stock price index (S&P 500) and the exchange rate of the Algerian Dinar(DZD) against the U.S.-Dollar (USD) and Euro (EUR) provide some new insights into these periodic timeseries. Finally, we will discuss the limitations of the existing formulations and give suggestions for furtherresearch.

Key words: Mixture models, Periodically correlated process, Mixture PGARCH models, periodic stationarity, periodicergodicity, higher order moments, EM algorithm.

2010 Mathematics Subject Classification: 62M10; 60G10; 60J05.

References[1] A. Aknouche and M. Bentarzi (2008) On the existence of higher-order moments of periodic GARCH models. Statistics

& Probability Letters, 78, 3262 3268.

[2] A. Aknouche and A. Bibi, (2009), Quasi-maximum likelihood estimation of periodic garch and periodic armagarchprocesses. Journal of Time Series Analysis, 30, 19-46

[3] A. Aknouche and H. Guerbyenne (2009), On some probabilistic properties of double periodic AR models. Statistics &Probability Letters, 79, 407-413.

[4] M. Bentarzi and F. Hamdi (2008a), Mixture Periodic Autoregressive Conditional Heteroskedastic Models. Computa-tional Statistics & Data Analysis, 53, 1-16.

[5] M. Bentarzi and F. Hamdi (2008b), Mixture Periodic Autoregression with Periodic ARCH Errors. Advances and Ap-plications in Statistics, 8, 219-246.

∗speaker125


[6] A. Bibi and A. Aknouche (2008), On Periodic GARCH Processes: Stationarity, Existence of Moments and GeometricErgodicity. Mathematical Methods of Statistics, 17, 305-316.

[7] T. Bollerslev (1986), Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics, 51, 307-327.

[8] G. Boshnakov (2011), On First and Second Order Stationarity of Random Coefficient Models. Linear Algebra and itsApplications, 434, 415-423.

[9] P. Bougerol and N. Picard (1992), Strict stationarity of generalized autoregressive processes. Annals of Probability, 20,1714-1730.

[10] A.P. Dempster, N.M. Laird and D.B. Rubin (1977), Maximum likelihood from incomplete data via the EM algorithm(with discussion). Journal of the Royal Statistical Society, Series B, 39, 1-38.

[11] C. Francq and J.M. Zakoian (2005), The L2-structures of standard and switching-regime GARCH models. StochasticProcesses and their Applications, 115, 1557-1582.

[12] F. Hamdi and S. Souam (2013), Mixture Periodic GARCH models: Applications to Exchange Rate Modeling. 5thinternational conference on modeling, simulation and applied optimization. IEEE xplore, Print ISBN: 978-1-4673-5814-9

[13] F. Hamdi (2015), A Note on the Order Selection of Mixture Periodic Autoregressive Models, 6th international confer-ence on modeling, simulation and applied optimization. IEEE xplore, Print ISBN: 978-1-4673-6601-4.

[14] Q. Shao (2006), Mixture periodic autoregressive time series models. Statistics & Probability Letters, 76, 609-618.

[15] C.S. Wong and W.K. Li (2000), On a mixture autoregressive model. Journal of the Royal Statistical Society, Series B,62, 95-115.

[16] C.S. Wong and W.K. Li (2001), On a mixture autoregressive conditional heteroscedastic model. Journal of the AmericanStatistical Association, 96, 982–995.

[17] Z. Zhang, W.K. Li and K.C. Yuen (2006), On a mixture GARCH time-series model. Journal of Time Series Analysis,27, 577-597.

126


Periodic stationarity and existence of moments ofMarkov-switching PARMA processes

Billel ALIAT1∗, Faycal HAMDI2

1 USTHB, RECITS LaboratoryPO box 32. El Alia, Bab Ezzouar, Algiers, Algeria

[email protected] USTHB, RECITS Laboratory

PO box 32. El Alia, Bab Ezzouar, Algiers, [email protected]

Abstract

In this work, we present a generalization of the classical ARMA Markov-switching models to the periodictime-varying case. In addition of capturing regime switching often encountered during the study of financialseries, this new model also captures the periodicity feature in the autocorrelation structure. We present resultson the strict periodic stationarity, second-order periodic stationarity and the existence of higher order moments.

Key words: Markov-switching models, moment, periodic stationarity.

2010 Mathematics Subject Classification: 60J05; 60G10; 62M10.

1. IntroductionPeriodic time series models have proved useful and appropriate for modelling many time series encountered in practice, whichare characterized by a periodic autocorrelation structure. Indeed, many periodic time series models have been introduced inthe literature, among which we find the periodic autoregressive moving average (PARMA) models. These models havebeen studied extensively by many authors (see e.g., Pagano, 1978; Vicchia, 1985; Bentarzi and Hallin, 1994; Boshnakov,1996; Ula and Smadi, 1997; Lund and Basawa, 2000; Shao and Lund, 2004; Anderson and Meerschaert, 2005; Bentarziand Aknouche, 2005; Bentarzi and al., 2008; Aknouche and Hamdi, 2009; Hamdi, 2012; and many others). The succes ofPARMA models is due to their efficient representation of many stochastic phenomena exhibiting a periodic autocorrelationstructure that cannot be adequately explained by the traditional seasonal ARIMA (SARIMA) models. Another importantreason for the growing interset is that they can be exploited in the analysis of vector ARMA models (V ARMA) in order toreduce, appreciably, the number of model parameters (e.g., Pagano, 1978; Newton, 1982).

However, financial time series exhibit complex statistical dynamics which are difficult to reproduce with stochastic linearmodels. These dynamics are often referred to as the stylized facts of financial data and include, among others, the volatilityclustring, the heavy-tailed marginal distributions, the regime changes, and the multimodality. These properties make standardtime series models, such as ARMA processes, inappropriate for such series. For this reason, many non linear time seriesmodels were introduced to capture these features.

Markov-switching (MS) models have attracted a lot of atention in the econometric literature since the seminal paperof Hamilton (1989) and continue to gain popularity especially in financial time series which exhibits structural changes. In

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MS models, the parameters are allowed to depend on the state of an unobserved Markov chain. At each period of time, sothere is a certain probability to belong to a regime and a transition probability from one regime to an other. Several worksconcerning the study of Markov-switching autoregressive moving average models (MS-ARMA) were established by manyauthors, such as Hamilton (1994), Kim (1994), Krolzig (1997), Francq and Zakoıan (2001, 2002), Zhang and Stine (2001),Psaradakis and Spagnolo (2003, 2006), Douc and al. (2004), Lee (2005), Chen and Tsay (2011), Cavicchioli (2013) andmany others.

Due to the popularity of MS and PARMA models, we propose to combine these two approaches to obtain a newmodel which capture, not only, the regime changes, but also the periodicity hidden in the autocovariance structure and otherfeatures of economic series in general, and of financial time series in particular. This new model that we propose is a Markov-switching PARMA model (MS-PARMA) and can be defined as a bivariate process (yt,∆t) ; t ∈ Z, where the process∆t is a Markov chain defined on a finite state space, and yt is a PARMA process. Otherwise, a process (yt,∆t) ; t ∈ Zis said to be a Markov-switching PARMA process of orders p and q and period S, denoted by MS-PARMAS (p, q), if itsatisfies an equation of the form

yt=p∑

i=1

d∑

k=1

φ(k)t,i 1(∆t=k)yt−i+

d∑

k=1

σ(k)1(∆t=k)ηt+q∑

i=1

d∑

k=1

d∑

l=1

θ(k)t,j σ

(l)1(∆t=k)1(∆t−j=l)ηt−j , (1)

where d denotes the number of regimes and (ηt)t∈Z is an i.i.d. sequence such that E (ηt) = 0 and E(η2

t

)= 1.

The objective of this work is to study some probabilistic properties of a Markov-switching PARMA model. We providea condition of strict periodic stationarity, and we establish sufficient conditions ensuring second-order periodic stationarityand the existence of higher-order moments of the MS-PARMA model.

This work is organized as follows: in section 2, we give condition of strict periodic stationarity. In section 3 we statesufficient conditions for second-order periodic stationarity and existence of moments. In section 4, we conclude and discusspossible extensions.

2. Periodic stationarity and existence of momentsLet

φt,i (∆t) :=d∑

k=1

φ(k)t,i 1(∆t=k), θt,j (∆t) :=

d∑

k=1

θ(k)t,j 1(∆t=k), and εt :=

d∑

k=1

σ(k)1(∆t=k)ηt.

Hence, one can rewrite the model (1) as follows

yt =p∑

i=1

φt,i (∆t) yt−i + εt +q∑

i=1

θt,j (∆t) εt−j , (2)

where the parameters φt,i (∆t) , for 1 ≤ i ≤ p and θt,j (∆t) , for 1 ≤ j ≤ q, are periodic functions which depend on aMarkov chain (∆t) .

Throughout this work we make the following assumptions:H1. The processes ηt and ∆t are independent;H2. (∆t) is a homogeneous, stationnary, irreducible, and aperiodic Markov chain with finite state-space E = 1, 2, . . . , d .The stationnary probabilities of (∆t) are denoted by π (k) = P (∆t = k) , the transition probabilities are denoted by

p (k, l) = P (∆t = l | ∆t−1 = k) , and the i-step transition probabilities are denoted by p(i) (k, l) = P (∆t = l | ∆t−i = k) ,for k, l ∈ E and i ≥ 1.

Consider the matrices

P(i)ft

=

p(i) (1, 1) ft (1) · · · p(i) (d, 1) ft (1)...

. . ....

p(i) (1, d) ft (d) · · · p(i) (d, d) ft (d)

and Πft

=

π (1) ft (1)...

π (d) ft (d)

,

for any periodic function ft : E →Mn×n′ (R), whereMn×n′ (R) denotes the space of real n×n′ matrices, and any positiveintegers i, n and n′.

128


In order to facilitate the study of strict periodic stationarity of the model MS-PARMA, it is useful to write model (2)in an equivalent Markovian representation as follows:

zt = bt +Atzt−1, (3)

wherezt = (yt, ..., yt−p+1, εt, ..., εt−q+1)′ , bt =

(εt, 01×(p−1), εt, 01×(q−1)

)′,

and

At =

φt,1:p−1 (∆t) φt,p (∆t) θt,1:q−1 (∆t) θt,q (∆t)I(p−1)×(p−1) 0(p−1)×1 0(p−1)×(q−1) 0(p−1)×1

01×(p−1) 01×1 01×(q−1) 01×1

0(q−1)×(p−1) 0(q−1)×1 I(q−1)×(q−1) 0(q−1)×1

,

and where φt,1:p−1 (∆t) = (φt,1 (∆t) , ..., φt,p−1 (∆t)) and θt,1:q−1 (∆t) = (θt,1 (∆t) , ..., θt,q−1 (∆t)).The key tool in studying strict periodic stationarity would be the top Lyapunov exponent. Let us recall the definition of

the top Lyapunov exponent for i.p.d. random matrices (Aknouche and Guerbyenne, 2009). Let ‖.‖ refer to an arbitrary normin the spaceM (n) of n × n matrices. Then, the top Lyapunov exponent associated to a sequence of an i.p.d. sequence ofmatrices A := At, t ∈ Z , is defined by

γS (A) = infn∈N∗

1nE [log ‖AnSAnS−1 · · ·A1‖] , (4)

when∑S

s=1E[log+ ‖As‖

]is finite, where log+ (x) = max (log (x) , 0) .

Theorem 1. If γS (A) < 0, the model (3) admits a non anticipative s.p.s. solution (zt)t∈Z given by

zt = bt +∞∑

k=1

AtAt−1 . . . At−k+1bt−k, (5)

Moreover, the solution is unique and periodically ergodic.

Let ⊗ denotes the tensor product, ρ (A) be the spectral radius of the matrix A, and A⊗m = A ⊗ A ⊗ . . . ⊗ A. Usingthese notations, we are able to state the following result.

Theorem 2. Suppose that E (ηmt ) < ∞ and ρ

S−1∏s=0

PE(A⊗mS−s|∆S−s)

< 1, where m is a strictly positive integer. Then the

model (2) has a unique non anticipative and s.p.s. (yt) . This solution is also periodically ergodic and satisfies E (ymt ) <∞.

Corollary 2.1. If m = 2, then if ρ

S−1∏s=0

PE(A⊗2S−s|∆S−s)

< 1, then the model (2) has an s.p.s. solution such that E

(y2

t

)<

∞ for all 1 ≤ t ≤ S. Moreover, this solution is unique, periodically ergodic and is given by (5) where the series convergesin mean square.

References[1] Aknouche, A., Guerbyenne, H., (2009). Periodic stationarity of random coefficient periodic autoregressions. Statist.

Probab. Lett., 79, 990-6.

[2] Aknouche, A., Hamdi, F., (2009). Calculating the autocovariances and the likelihood calculations for periodicV ARMA models, J. Statist. Comput. Sim., 79, 227-39.

[3] Anderson, P. L., Meerschaert, M. M., (2005). Parameter estimation for periodically stationary time series. J. Time Ser.Anal., 26, 489-518.

[4] Bentarzi, M., Aknouche A. (2005). Calculation of the Fisher information matrix for periodic ARMA models. Comm.Statist. -Theor. Meth., 34, 891-903.

129


[5] Bentarzi, M., Hallin, M. (1994). On the Invertibility of Periodic Moving Average Models. J. Time Ser. Anal., 15, 263-8.

[6] Bentarzi, M., Guerbyenne, H., Hemis, R., (2008). Predictive density order selection of periodic AR models. Comm.Statist. Simulation Comput., 37, 1167-82.

[7] Boshnakov, G. N., (1996). Recursive computation of the parameters of periodic autoregressive moving-average pro-cesses. J. Time Ser. Anal., 17, 333-49.

[8] Bougerol, P., Picard, N., (1992). Strict stationarity of generalized autoregressive processes. Annals of Probability 20,1714-30.

[9] Chen, C.C., Tsay, W.J., (2011). A Markov regime-switching ARMA approach for hedging stock indices. The Journalof Futures Markets, 31, 165-91.

[10] Douc, R., Moulines, E., Ryden, T., (2004). Asymptotic properties of the maximum likelihood estimator in autoregres-sive models with Markov regime. Ann. Statist., 32, 2254-304.

[11] Francq, C., Zakoıan, J.M., (2002) Autocovariance structure of powers of switching-regime ARMA processes. ESAIM:Probab. Statist, 6, 259-70.

[12] Francq, C., Zakoıan, J.M., (2001). Stationarity of multivariate Markov-switching ARMA models. J. Econometrics, 102,339-64.

[13] Hamdi, F., (2012). Computing the exact Fisher information matrix of periodic state-space models. Comm. Statist. -Theor. Meth., 41, 4182-99

[14] Hamilton, J. D., (1994). Time series analysis. Princeton University Press, New Jersey.

[15] Kim, C.J., (1994). Dynamic linear models with Markov-switching. J. Econometrics, 60, 1-22.

[16] Krolzig, H.M., (1997) Markov-Switching Vector Autoregressions: Modelling, Statistical Inference and Application toBusiness Cycle Analysis. Berlin-Heidelberg-New York: Springer Verlag.

[17] Lee, O., (2005). Probabilistic Properties of a NonlinearARMA Process with Markov Switching. Comm. Statist. -Theor.Meth., 34, 193-204.

[18] Lund, R., Basawa, I. V., (2000). Recursive prediction and likelihood evaluation for periodic ARMA models. J. TimeSer. Anal., 21, 75-93.

[19] Pagano, M., (1978). On periodic and multiple autoregression. Ann. Statist., 6, 1310-7.

[20] Psaradakis, Z., Spagnolo, N., (2003) On the determination of the number of regimes in Markov-switching autoregressivemodels. J. Time Ser. Anal., 24, 237-52.

[21] Psaradakis, Z., Spagnolo, N., (2006) Joint determination of the state dimension and autoregressive order for modelswith in Markov regime switching. J. Time Ser. Anal., 27, 753-66.

[22] Shao, Q., Lund, R. (2004). Computation and characterization of autocorrelations and partial autocorrelations in periodicARMA models. J. Time Ser. Anal., 25, 359-72.

[23] Ula, T. A., Smadi A. A., (1997). Periodic stationary conditions for periodic autoregressive moving average processes aseigenvalue problems. Water Resources Research, 33, 1929-34.

[24] Vecchia, A. V., (1985). Maximum likelihood estimation for periodic autoregressive Moving average models. Techno-metrics, 27, 375-84.

[25] Yao, J. F., (2001). On square integrability of an AR process with Markov switching. Statist. Probab. Lett., 52, 265-70.

[26] Zhang, J., Stine, R.A., (2001). Autocovariance structure of Markov regime switching models and model selection. J.Time Ser. Anal., 22, 107-24.

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Conference on Discrete Mathematics and Computer Science

dimacos’2015

Algeria, Sidi Bel Abbes November 15 - 19, 2015.

available online at http://lrecits.usthb.dz/activites.htm

Estimation Bayesienne d’un Melange de Modeles

GARCH Periodiques par la Methode de

Griddy-Gibbs

Rokia HEMIS1, Hafida GUERBIENNE2, Faycal HAMDI3, ∗

1 Universite Ferhat Abbas de Setif, Laboratoire RECITSSetif, Algerie

[email protected] Universite USTHB,

Alger, [email protected]

3 Universite USTHB, Laboratoire RECITSAlger, Algerie

[email protected]...

Abstract

Ce travail est essentiellement consacre au probleme de l’estimation des parametres d’unmelange de modeles GARCH periodique (MPGARCH). cemelange de modeles peut etreutilise dans la modelisation des series chronologiques qui possedent certaines caracteristiquestelles que la multimodalite, le changement de regime et la periodicite dans la variance con-ditionnelle.Nous proposons une methode bayesienne pour l’estimation des parametres enutilisant l’echantillonneur de Griddy-Gibbs.

Key words: Methodes bayesiennes, modeles MPGARCH , Algorithme de Griddy-Gibbs.

2010 Mathematics Subject Classification: 62

1. Introduction

Il est bien connu que les modeles les plus populaires et les plus utilises dans la modelisation de la volatiliteinstantanee dans les series chronologiques financieres sont les modeles Autoregressifs ConditionnellementHeteroscedastiquesARCH , introduit par Engle (1982) et leur extention au modelesGARCH par Boller-slev (1986) et par la suite les modeles GARCH periodiques une nouvelle classe de modeles introduite par

∗Underline the speaker

1

Bollerslev et Ghysels (1996) qui a montre une grande capacite a capturer la periodicite dans la varianceconditionnelle. Divers modeles ont ete proposes afin de capturer differentes caracteristiques telles que lalongue memoire, le changement de regime et la periodicite dans la variance conditionnelle. Notre but estde proposer un modele qui peut presenter des series chronologiques avec une structure d’autocorrelationperiodique aussi bien que d’autres caracteristiques (telles que la multimodalite, le changement de regime)nous proposons la classe de melange des modeles GARCH periodiques notee MPGARCH.

2. Resultats

Notre travail nous a permis d’elaborer les resultats suivants:

• Definir une classe de modeles MPGARCH.

• Pour l’estimation on a propose une methode bayesienne via l’algorithme Griddy-Gibbs (produitpar Ritter et Tanner 1992).

• Nous avons determine la fonction de vraisemblance des modeles MPGARCH.

• Nous avons calcule les noyaux des lois a posteriori conditionnelles de chaque parametre necessairea l’application de notre algorithme

• L’algorithme de Griddy-Gibbs nous a permis d’Evaluer chaque fonction du noyau a partir d’unegrille de points pour approcher la fonction de repartion a posteriori en utilisant les methodesnumeriques d’integration et par la suite nous avons calcule l’inverse de la fonction de repartition aune valeur aleatoire prelevee uniformement dans [0, 1]

References

[1] Ausin M.C., Galeano P., Bayesian estimation of the Gaussian mixture GARCH model. Comput.

Statist.Data Anal.,1989, 51, 2636-2652, 2007.

[2] M. Bentarzi M., Hamdi F., Mixture Periodic Autoregressive conditional Heteroscedastic Models

Comput. Statist.Data Anal., 2008, 53, 1-16.

[3] Bollerslev T., Generalized autoregressive heterscedasticity. J. Econometrics., 1986, 51, 307-327.

[4] Ritter C., Tanner M.A., Facilitating the Gibbs sampler: the Gibbs stopper and the Griddy–Gibbs

sampler. J. Amer. Statist. Assoc., 1992, 87, 861–868.

[5] Wong C.S., Li W.K., On a mixture autoregressive conditional heteroscedastic model. J. Amer

Statist.Assoc., 2001, 96, 982- 995.

[6] Zhang Z., Li W.k., Yuen K.C., On a mixture GARCH time series model. J. Time ser. Anal., 2006,27, 577-597.

2


Periodic multivariate stochastic volatility model: structure andestimation

Nadia BOUSSAHA1∗and Faycal HAMDI2

1,2 USTHB, RECITS LaboratoryPO box 32. El Alia, Bab Ezzouar, Algiers, Algeria


Abstract

In this communication, we propose to extend univariate periodic autoregressive stochastic volatility modelto a multivariate periodic time series context. We present the structure of the proposed model and its dynamicproperties are derived and shown to be consistent with empirical properties observed in financial time serieswhose volatility displays a periodic correlation pattern. We describe also two straightforward methods for theestimation problem, where the implementation is based on periodic Chadrasekhar filter and on the ExpectationMaximization (EM) algorithm with particle filters and smoothers.

Key words: Periodic multivariate stochastic volatility model, periodic Chandrasekhar-type recursions, Expectation Maxi-mization algorithm, Particle filtering.

2010 Mathematics Subject Classification: 62M10, 60G10, 62G05.

1. IntroductionThe univariate periodic autoregressive stochastic volatility model (PAR-SV ) model introduced by Aknouche et al. (2007)provides a successful alternative to the class of periodic generalized autoregressive conditionally heteroscedastic (PGARCH)models (Bollerslev and Ghysels, 1996) in accounting for the time-varying and persistent volatility as well as for the period-icity feature in the autocorrelation structure exhibited by many nonlinear time series. Thus, a stochastic process εt; t ∈ Zhas a PAR-SV representation of period S, if it is a solution of the stochastic difference equation

εt = ηt exp

12xt,

xt = αt + βtxt−1 +Qtet,t ∈ Z, (1)

where the parameters αt, βt and Qt are periodic in t with period S (i.e., αt+τS = αt, βt+τS = βt and Qt+τS = Qt). Thesequences (ηt) and (et) are two independent sequences of i.i.d. random variables with zero mean and unit variance. Wenote that the PAR-SV model (1) has been called periodic autoregressive stochastic volatility rather that shortly ”periodicstochastic volatility”, since the log-volatility is given as a first order periodic AR model, and to differ from the PSV modelproposed by Tsiakas (2006). Nevertheless, the PAR-SV model (1) can be generalized so that ht may have a periodic au-toregressive moving average (PARMA) representation. We note also that the PAR-SV model has an additional innovativeterm in the volatility dynamics and, hence, is more flexible than PGARCH class models. Indeed, when the stochastic partof volatility, et, does not exist, model (1) reduce to PGARCH(1, 0). So the difference in modelling variance is substantialbetween PGARCH and PAR-SV approaches.

∗speaker133


It is well known that financial volatility moves together, over time, across different assets and markets. This natureplays a critical role in the construction of the appropriate model. Therefore, multivariate models for modelling time-varyingvolatility are of interest and can lead to greater statistical efficiency. However some multivariate stochastic volatility modelshave recently become a major concern in the investigation of the correlation structure of multivariate economic series ingeneral, and of multivariate financial time series in particular. Various extensions of the basic stochastic volatility (SV )model, considered by Taylor (1982, 1986), have been proposed (Harvey et al., 1994; Danıelsson, 1998; Jungbacker andKoopman, 2006; Smith and Pitts, 2006; Chan et al., 2006, Asai and McAleer, 2009; Nkemonole and Abass, 2015) in orderto capture the main stylized facts characterizing multivariate financial series such as excess kurtosis, asymmetry and heavy-tailed. Note that all of the proposed models deal with constant volatility parameters and it seems that there is no formulationwhich can adequately explain time series whose structure change over time, in particular, time series whose volatility displaysa periodic correlation pattern.

Let εt = (εt,1, ..., εt,m)′ be them×1 vector of stock returns at time t andXt be the corresponding vector of log-variances.Having introduced the univariate PAR-SV model, by Aknouche et al. (2007), to time-varying volatility, this communicationwill discuss aspects of multivariate periodic autoregressive stochastic volatility (PV AR-SV ) processes which may be definedby

εt = ηtV1/2t ,

Xt = αt + βtXt−1 + et,t ∈ Z, (2)

where Vt = diag (expXt), βt = diag (βt,1, ..., βt,m), αt = (αt,1, ..., αt,m)′. The vectors ηt = (ηt,1, ..., ηt,m)′ andet = (et,1, ..., et,m)′ represent mutually and serially uncorrelated zero-mean error processes with covariance structureE(ηtη′t+h

)= δh,0Σt,1 and E

(ete′t+h

)= δh,0Σt,2, where δ stands for the Kronecker function. The parameters αt, βt,

Σt,1 and Σt,2 are periodic in t, with period S.

References[1] Aknouche, A., Bibi, A., Hamdi, F., 2007. Periodic autoregressive stochastic volatility. Journees de Statistique, Biskra

21-22 Avril 2007.

[2] Asai, M., McAleer, M., 2005. The structure of dynamic correlations in multivariate stochastic volatility models. Journalof Econometrics, 150, 182-192.

[3] Bollerslev, T., Ghysels, E., 1996. Periodic autoregressive conditional heteroskedasticity. Journal of Business andEconomie Statistics, 14, 139-152.

[4] Chan, D., Kohn, R., Kirby, C., 2006. Multivariate Stochastic Volatility Models with Correlated Errors. EconometricReviews, 25, 245-274

[5] Danielsson, J., 1998. Multivariate stochastic volatility models: estimation and comparison with V GARCH models.Journal of Empirical Finance, 5, 155-173.

[6] Ghysels, E., Harvey, A., Renault, E., 1996. Stochastic Volatility. Amsterdam, North-Holland.

[7] Harvey, A., Ruiz, E., Shephard, N., 1994. Multivariate stochastic variance models. The Review of Economic Studies, 61,247-264.

[8] Jungbacker, B., Koopman, S. J., 2006. Monte Carlo likelihood estimation for three multivariate stochastic volatilitymodels. Econometric Reviews, 25, 385-408.

[9] Smith, M., Pitts, A., 2006. Foreign Exchange Intervention by the Bank of Japan Bayesian Analysis Using a BivariateStochastic Volatility Model. Econometric Reviews, 25, 425-451

[10] Taylor, S.J., 1982. Financial returns modelled by the product of two stochastic processes-a study of the daily sugarprices 1961-75. North-Holland, Amsterdam, 1, 203-206.

[11] Taylor, S.J., 1986. Modelling Financial Time Series. Chichester, New York, Wiley

[12] Tsiakas, I., 2006. Periodic Stochastic Volatility and Fat Tails. Journal of Financial Econometrics, 4, 90-135.

134


Inégalités de Grandes Déviations de l’Estimateur Sieves d’unOpérateur d’un Processus AR Fonctionel

Nesrine KARA TERKI1∗, Tahar MOURID2,

1 Université Abou Bekr Belkaid, Laboratoire de Statistiques et Modélisations AléatoiresTlemcen, Algerie

[email protected] Université Abou Bekr Belkaid, Laboratoire de Statistiques et Modélisations Aléatoires

Tlemcen, [email protected]

Résumé

Nous établissons des inégalités de grandes déviations pour une classe d’estimateurs sieves pour un opérateurd’un processus autorégressif fonctionel d’ordre un. Nous en déduisons la convergence presque sûre et presquecomplète des estimateurs sieves et des vitesses de convergence.

Mots clefs : autorégressif fonctionel ; estimateur sieves ; grandes déviations ; Normalité asymptotique locale ; vitesses deconvergence

Stochastic processes : .

1. IntroductionSoit (Ω,A, IP) un espace de probabilité, H un espace de Hilbert séparable réel et < ., . > et ‖ . ‖ sont le produit scalaire

et la norme associée. Un processus (Xn, n ∈ N) défini sur (Ω,A, IP) et à valeurs dans H est un Autorégressif HilbertienAR(1) si

Xn = ρ(Xn−1) + εn (1)

où ρ est un opérateur linéaire borné sur H, ‖ρ‖L < 1 et ρ ∈ Θ ⊂ H où l’espace des paramètres H est un espace deHilbert séparable de l ’espace L (H) des opérateurs linéaires bornés sur H. Soit Xn = (X0, ...Xn) les observations de (1),Pn,ρ = PX0

(X1,...,Xn,ρ)la loi conditionnelle de (X1, ..., Xn) sachant X0 et Pn,ε est la loi de probabilité du vecteur (ε1, ..., εn).

Nous notons f(Xn, ρ) := dPn,ρdPn,ε

la densité de la partie absolument continue de Pn,ρ par rapport à Pn,ε. La condition denormalité asymptotique locale (LAN) introduite par Le Cam [7] est un outil puissant pour l’estimation paramétrique. Dans[6], nous avons étudié la condition LAN quand l’operateur ρ dans (1) dépend d’un paramètre θ ∈ Rp. Des extensions de lathéorie LAN ont été faites en dimension infinie (cf. Ibragimov and Kha’sminskii [4] et les références). Dans cette Note et dansle cadre de la dimension infinie, nous établissons des inégalités de grandes déviations pour une classe d’estimateurs sievesde Grenander [2] du paramètre operateur ρ dans (1) similaire à celles qui existent dans la théorie LAN et nous donnons leursconvergence p.s. et p. co. avec des vitesses de convergence. Kallianpur et Selukar [5] ont traité l’estimation de paramètresà valeurs dans un espace de Hilbert séparable en donnant des inégalités de grandes déviations pour des estimateurs sieves

∗Speaker

135


avec des exemples. Une suite Sk, k ≥ 1 de sous ensembles de Θ est une sieves si les Sk sont des compacts croissants et⋃k Sk est dense dans Θ. Un estimateur sieves ρn,k(Xn) associé à la sieves Sk est défini par

f(Xn, ρn,k) = supρ∈Sk

f(Xn, ρ) (2)

Nous introduisons les notations suivantes.

1. [a] est la partie entière de a ∈ R .

2. λj(ρ), j ≥ 1 sont les valeurs propres de l’opérateur ρ et ‖.‖2 est la norme de Hilbert Schmidt associée.

2. Résultats.Supposons que la vraie valeur du paramètre ρ de l’equation (1) est telle que ρ ∈ H .

Sous certaines hypothèses , les résultats ci dessous donnent la convergence p.s. et p.co., des inégalités de grandes déviationset des vitesses de convergence pour les estimateurs sieves.

Théorème 2.1. L’estimateur sieves ρn,k defini par (2) satisfaiti) ρn,[ns] →n→∞ ρ, Pρ− p.s..ii) ∀h > 0, ∃N = N(ρ, h) > 0 tel que ∀n > N,

Pρ‖ρn,[ns] − ρ‖2 > h ≤ exp(−Cn1−sh2)

où 0 < s < 1/2 et la constante C > 0 ne dépend que de ρ.

Théorème 2.2. Soit 0 < β < 1/2 et 0 < s ≤ 1−2β2 . Si nβ(

∑+∞j=[ns]+1 λ

2j (ρ))1/2 →n→∞ 0, alors

i) nβ‖ρn,[ns] − ρ‖2 →n→∞ 0, Pρ− p.s..ii) ∀h > 0, ∃N = N(ρ, h) tel que ∀n > N ,

Pρnβ‖ρn,[ns] − ρ‖2 > h ≤ exp(−Cnbh2)

où b = 1− 2β − s et la constante C > 0 ne dépend que de ρ.

Références[1] Bosq D., Linear Processes in Function Spaces,Springer, New York., 2000.

[2] Grenander U., Abstract Inference. Wiley, New York, 1981.

[3] Ibragimov I.A., Has’minskii R.Z., Statistical Estimation. Asymptotic Theory, Springer, New York., 1981.

[4] Ibragimov I.A., Khas’Minskii R.Z., Asymptotically Normal Families of Distributions and Efficient Estimation, Ann.Statist. 1991,19 , 1681-1724.

[5] Kallianpur G., Selukar R.S., Estimation of Hilbert Space Valued Parameters by the Method of Sieves, Wiley EasternLimited, Publishers (1993) 325-347.

[6] Kara Terki N., Mourid T., On Local Asymptotic Normality For Functional Autoregressive Processes. Submitted 2015.

[7] LeCam L., Yang G.L., Asymptotics in Statistics, Second ed., Springer., 2000.

[8] Mourid T. Contribution à la statistique des processus autorégressifs à temps continu, Université Paris 6. 1995.

136


Sur les fonctions holomorphes discretes

Boudekhil CHAFI

Laboratory of Mathematics, University of Sidi Bel-Abbes,P.O Box 89, Sidi Bel-Abbes 22000, Algeria

Abstract

On etudie l’analogue discret des fonctions holomorphes. Ces fonctions sont appelees fonctions holomorphesdiscretes. L’etude porte sur l’analogue discret de l’operateur de Cauchy-Riemann et la notion de domained’holomorphie sur une partie de l’ensemble des entiers de Gauss Z[i] = Z + iZ considere comme un sous-ensemble de C.

Mots clefs: fonction holomorphe discrete, operateur de Cauchy-Riemann, domaine d’holomorphie.Classification Mathematique par Sujets (2010): 30G25 .

1. IntroductionSoit A un ouvert de C = R + iR. Une fonction f : A → C est holomorphe dans A si, f est de classe C1 et pour toutz = x+ iy ∈ A ,

∂f

∂x(z) =

∂f

i∂y(z) .

On note O(A) l’ensemble des fonctions holomorphes dans A.

Definition 1 (Issacs,1941, [3]). Soit A ⊆ Z[i] = Z + iZ. Une fonction f : A→ C est dite holomorphe discrete de premiereespece dans A si, pour tout z ∈ Z[i] tel que z, z + 1, z + i ∈ A, on a

f(z + 1)− f(z)1

=f(z + i)− f(z)

i.

On note O1(A) l’ensemble des fonctions holomorphes discretes de premiere espece dans A.

Definition 2 (Isaacs,1941, [3]). Soit A ⊆ Z[i] = Z + iZ. Une fonction f : A→ C est dite holomorphe discrete de secondeespece dans A si, pour tout z ∈ Z[i] tel que z, z + 1, z + i, z + 1 + i ∈ A, on a

f(z + 1 + i)− f(z)1 + i

=f(z + i)− f(z + 1)

1− i .

On note O2(A) l’ensemble des fonctions holomorphes discretes de seconde espece dans A.Ces deux notions de fonctions holomorphes discretes ont ete introduites par R. P. Isaacs [3] en 1941, qu’il appela fonctions

monodiffriques de premiere et de seconde espece; puis, dans un article [4] de 1952, il realisa une etude des fonctions holo-morphes discretes de pemiere espece. En 1944, J. Ferrand [2] etudia les fonctions holomorphes discrete de seconde espece,qu’elle appela fonctions preholomorphes. Dans un papier de 1956, R. J. Duffin [1] fit une etude des fonctions holomorphesdiscrete de seconde espece, qu’il appela fonctions analytiques discretes .

137


2. Operateur de Cauchy-RiemannDans C, l’operateur de Cauchy-Riemann s’ecrit

∂ =∂

∂x+ i

∂

∂x, avec z = x+ iy .

Soit f : A→ C, A un ouvert de C, z = x+ iy ∈ C.f est holomorphe dans A si et seulement si f est de classe C1 et ∂f(z) = 0 en tout point z ∈ A.

On poseDxf(x, y) = f(x+ 1, y)− f(x, y)Dyf(x, y) = f(x, y + 1)− f(x, y)

On introduit l’analogue discret de l’operateur de Cauchy-Riemann correspondant a la definition 1 et a la definition 2:

D1 = Dx + iDy

D2 = (1− i)Dx + (1 + i)Dy +DxDy

AlorsD1f(z) = f(z + 1) + if(z + i)− (1 + i)f(z)D2f(z) = f(z + 1 + i)− f(z) + if(z + i)− if(z + 1)

Soit f : A→ C , A ⊆ Z[i], z = x+ iy ∈ Z[i];

f est holomorphe discrete de premiere espece dans A si et seulement si D1f(z) = 0 en tout point z ∈ Z[i] tel quez, z + 1, z + i ∈ A.

f est holomorphe discrete de seconde espece dans A si et seulement si D2f(z) = 0 en tout point z ∈ Z[i] tel quez, z + 1 + i, z + i, z + 1 ∈ A.

3. Domaine d’holomorphieSoit A, B ⊆ Z[i] et A ⊆ B. On note %B

A l’application restriction

%BA : O(B) → O(A)

f 7→ f|A

Dans le cas ou A et B sont des domaines de C, l’application restriction %BA est injective . De plus, tout domaine de C est

un domaine d’existence maximale d’une fonction holomorphe. Donc, si %BA est bijective, on a B = A.

Dans le cas ou A et B sont des parties de Z[i], l’application restriction %BA peut etre injective et non surjective, ou

surjective et non injective.Kiselman [5],[6] propose la definition suivante pour caracteriser les parties de Z[i] qui peuvent se comporter comme les

domaines de C par rapport aux fonctions holomorphes.

Definition 3. Soit A ⊆ Z[i]. On dit que A est un domaine d’holomorphie dans Z[i] , si, pour tout B ⊆ Z[i], on a

A ⊆ B et %BA est bijective ⇒ B = A .

References[1] R. J. Duffin, Basic properties of discrete analytic functions, Duke Math. J., (1956), 23, pp. 335-363.

[2] J. Ferrand, Fonctions Preharmoniques et fonctions preholomorphes, 2nd ser., Bull. Sci. Math., 68, (1944), pp. 152-180.

[3] R. P. Isaacs, A finite difference function theory, Univ Nac Tucuman Revista A, 2, (1941), pp. 177-201.

138


[4] R. P. Isaacs, Monodiffric functions, In: Construction and applications of conformal maps, Proceedings of a symposium.National Bureau of Standards Appl Math Series, N. 18. Washington, DC: US Government Printing Office, (1952), pp.257-266.

[5] C. O. Kiselman, Functions on discrete sets holomorphic in the sense of Isaacs, or monodiffric functions of the first kind,Science in China, Ser. A Mathematics, (2005), Vol. 48 Supp., pp. 86-96.

[6] C. O. Kiselman, Functions on discrete sets holomorphic in the sense of Ferrand, or monodiffric functions of the secondkind, Science in China, Ser. A Mathematics, (2008), Vol. 51. N.4, pp. 604-619.

139


Shifted domino tableaux

Zakaria CHEMLI∗

University of Paris-Est Marne la vallee, LIGM5 Boulevard Descartes, 77420 Champs-sur-Marne, France

[email protected]

Abstract

We introduce new combinatorial objects called the shifted domino tableaux. We prove that these objectsare in bijection with the pairs of shifted Young tableaux. This bijection shows that these objects can be seen aselements of the super shifted plactic monoid, which is the shifted analog of the super plactic monoid.

Key words: Shifted Young tableaux, domino tableaux, shifted domino tableaux, super shifted plactic monoid.

1. IntroductionBy extending Young tableaux [6] to shifted Young tableaux (ShYT), Sagan [3] and Worley [5] developed independentlya combinatorial theory of ShYT parallel to the theory of Young tableaux. The ShYT allow to define the P-Schur and theQ-Schur functions.

We extend naturally ShYT to new combinatorial objects: The Shifted Domino Tableaux (ShDT), the shifted analog ofdomino tableaux [1]. We prove that the set of ShDT is in bijection with the set of pairs of ShYT. Our proof is inspired fromthe one of Carre and Leclerc [1] regarding the bijection between the set of domino tableaux and the set of pairs of Youngtableaux. We also extend the shifted plactic monoid [4] to dominoes and define the super shifted plactic monoid which isisomorphic to the direct product of two shifted plactic monoids.

2. Main resultsDefinition 1. Given certain partition λ paved with dominoes [1] in a certain way, a ShDT is a filling of the dominoes whichare cut by D2k [1] for any positive integer k by X , and the remaining dominoes by letters in 1′ < 1 < 2′ < 2 < · · · suchthat

- rows and columns are non-decreasing from left to right and from bottom to top;

- a letter ` ∈ 1, 2, 3, . . . appears at most once in each column;

- a letter ` ∈ 1′, 2′, 3′, . . . appears at most once in each row;

- there is no letter ` ∈ 1′, 2′, 3′, . . . on the diagonal D0.

We give below, an example of ShDT of shape (8, 5, 5, 5, 5):

∗Speaker

140


1 12’ 2 3

2’

2

4

X

5’

5X

X

X

T =

.

One of the most important result of our work is a bijection between the set of ShDT and the set of pairs of ShYT.

Theorem 1. Let λ be a valid shape of ShDT of 2-quotient (µ, ν) [2]. The set of ShDT of shape λ and the set of pairs (t1, t2)of ShYT of shape (µ, ν) are in bijection.

For example, the image of the ShDT T is the following pair of ShYT:

1 2’

X 4

, 1 2’ 2

2X

3

5’

X 5X

Theorem 1 allows us to extend the shifted plactic monoid of Serrano [4] to dominoes, which define the super shiftedplactic monoid, the shifted analog of the super plactic monoid.

Definition 2. Let A1 = a11 < a1

2 < a13 < . . . and A2 = a2

1 < a22 < a2

3 < . . . be two totally ordered infinite alphabets.The super shifted plactic monoid, denoted by SShPl(A1, A2) is the quotient of the free monoid (A1 ∪A2)∗ generated by theshifted plactic relations of ShPl(A1) and ShPl(A2) such that the letters of A1 commute with the letters of A2.

Theorem 2. Each super shifted plactic class is represented by a unique ShDT.

3. ConclusionThe shifted domino tableaux will lead as in the case of domino tableaux, to the definition of a new family of symmetricfunctions, the shifted analog of the H functions of Carre and Leclerc [1]. This analog functions can allow us to shedlights on the combinatorial properties of Q-Schur and P-Schur functions. We expect also to find a new expression of theshifted Littlewood-Richardson rule with coefficients in terms of shifted domino tableaux. Another direction, in which we caninvestigate, is to find properties and applications of the super shifted plactic monoid.

References[1] Christophe Carre and Bernard Leclerc. Splitting the square of a schur function into its symmetric and antisymmetric

parts. Journal of algebraic combinatorics, 4(3):201–231, 1995.

[2] Ian Grant Macdonald. Symmetric functions and Hall polynomials. New York, 1995.

[3] Bruce E Sagan. Shifted tableaux, Schur Q-functions, and a conjecture of R. stanley. Journal of Combinatorial Theory,Series A, 45(1):62–103, 1987.

[4] Luis Serrano. The shifted plactic monoid. Mathematische Zeitschrift, 266(2):363–392, 2010.

[5] Dale Raymond Worley. A theory of shifted Young tableaux. PhD thesis, Massachusetts Institute of Technology, 1984.

[6] Alfred Young. On quantitative substitutional analysis. Proceedings of the London Mathematical Society, 1(1):97–145,1900.

141


On the weighted sums associated to rays in Pascal’s triangle

Moussa AHMIA1∗, Laszlo Szalay2

1 University of Jijel, RECITS Laboratory, Algiers18000, Jijel, Algeria

[email protected] Institute of Mathematics, University of West Hungary

H-9400 Sopron, Ady E. ut 5, [email protected]

Abstract

In this paper, we consider two types of weighted sums of diagonal elements lying along a finite ray ingeneralized Pascal triangle. The weights are provided by a sequence Gn, in the first case it is assumed to beperiodic with shortest length d ∈ N+. Then we suppose that Gn satisfies a homogeneous linear recurrencerelation of order s ≥ 1. In both cases the weighted sums are completely described

Key words: Generalized Pascal triangle; linear recurrences, sum of elements.

1. IntroductionConsider the generalized Pascal triangle containing the element xnyk

(nk

)in the intersection of the nth row and the kth

column. In the paper of Belbachir; Komatsu and Szalay [2] the sum of elements Tn laying along the nth ray crossed thetriangle was described by the linear recurrence relation

Tn −(r

1

)xTn−1 +

(r

2

)x2Tn−2 + · · ·+ (−1)r

(r

r

)xrTn−r = yrTn−r−q, (1)

where the directed vector of the ray is given by (r, q) . (r ∈ N+, q ∈ Z, r + q > 0) . To define the intermediate rays, weneed one more parameter, p, such that 0 ≤ p ≤ r − 1. Using the notation above, the exact formula for Tn is the following.

Tn+1 =bn−p

r+q c∑

k=0

(n− qkp+ rk

)xn−p−(q+r)kyp+rk. (2)

Otherwise, if we fixe the direction (r, q) and a value of p, the general term constitutes the sum of elements laying on thecorresponding ray is noted and gived by

T(r,q,p)n+1 =

bn−pr+q c∑

k=0

T (r,q,p)(n, k) =bn−p

r+q c∑

k=0

(n− qkp+ rk

)xn−p−(q+r)kyp+rk, (3)

∗Moussa AHMIA142


where T (r,q,p)(n, k) =(n−qkp+rk

)xn−p−(q+r)kyp+rk and T0 = 0.

Now we slighly modify the problem by the sequence Gk to have the weighted sum

T(r,q,p)n+1 =

bn−pr+q c∑

k=0

(n− qkp+ rk

)xn−p−(q+r)kyp+rkGk. (4)

In general, the problem is too difficult, therefore we consider that in two specific cases.

Case 1. Gn is periodic. The length of the period is d ∈ N+.

T(r,q,p)n+1 =

bn−pr+q c∑

k=0

(n− qkp+ rk

)xn−p−(q+r)kyp+rkGk

k=ud+v=d−1∑

v=0

Gv

bn−p−v(r+q)(r+q)d c∑

u=0

(n− q(ud+ v)p+ r(ud+ v)

)xn−p−(q+r)(ud+v)yp+r(ud+v)

=d−1∑

v=0

GvT(d,r,q,p,v)n+1 .

Since T (d,r,q,p,v)n satisfies an other recurrence relation similar to (1) for any v, the sum above (as the linear combination)

do that, too.

Case 2. Gn is Multilinear recurrence given by G0 = u0, ..., Gs−1 = us−1 and Gn =∑s

j=1AjGn−j for s ≥ 2.

References[1] H. Belbachir, F. Bencherif, Linear recurrent sequences and powers of a square matrix Integers 6., 2006, A12, 17pp.

[2] H. Belbachir, T. Komatsu, L. Szalay, Linear recurrences associated to rays in Pascal’s triangle and combinatorial iden-tities, Math. Slovaca, 64 (2014), No. 2, 287-300.

[3] H. Belbachir, T. Komatsu, L. Szalay, Characterization of linear recurrences associated to rays in Pascal’s triangle,AIPConference Proc. 1264, pp. 90-99 (2010).

143


Recurrences associated to rays in negative GeneralizedArithmeticTriangle

Hacene Belbachir1, Fariza Krim2∗,

1 USTHB, Faculty of Mathematics, RECITS Laboratory, DG-RSDTBp 32, El Alia 16111, Bab Ezzouar, [email protected], [email protected]

2 USTHB, Faculty of Mathematics, LAROMADE Laboratory, DG-RSDTBp 32, El Alia 16111, Bab Ezzouar, Algiers

[email protected], [email protected]

Abstract

We consider the extension of generalized arithmetic triangle GAT, to negatively rows where the initial con-ditions are defined by real sequences (a)n∈Z and (b)n∈Z and describe the recurrence relation associated to thesum of diagonal elements laying along a finite ray.

Key words: Binomial coefficient, linear recurrence, combinatorial identities, Pascal triangle.

Definition 1. Let (an)n≥0 and (bn)n≥0 be two real sequences, A and B two reals. The GAT contains elements⟨ nk

⟩in the

nth row and kth column defined as follows

⟨nk

⟩=

A⟨n− 1k

⟩+B

⟨n− 1k − 1

⟩for 1 ≤ k ≤ n− 1

Anan for k = 0Bnbn for k = n

Our aim is to consider the negative GAT, the extension of the GAT to negatively rows. With given positive integers n and

k,⟨ −n

k

⟩denotes the kth entry of the (−n)th row.

Definition 2. Let (an)n∈Z and (bn)n∈Z be two real sequences, A and B two reals. The negative GAT contains elements⟨ −n

k

⟩in the (−n)th row and kth column defined as follows

⟨ −nk

⟩=

1A

⟨ −n+ 1k

⟩− B

A

⟨ −nk − 1

⟩for n ≥ 1, k > 0,

A−na−n for n ≥ 1, k = 0,0 for n = 0, k > 0

∗Speaker

144


Theorem 3. The negative GAT for sequences (an)n∈Z and (bn)n∈Z, real numbers A and B has as entry k in row (−n) for(n, k) 6= (0, 0),

⟨ −nk

⟩= −A−n−kBk

n−1∑

i=0

(−n+ i

k − 1

)a−i−1 + δn,kA

−na−n,

with δn,k = ⌊

n−kn

⌋for n− k ≥ 0

0 for n− k < 0 or n = 0.

1. The sum of elements lying along a finite ray in negative GAT

For a fixed direction (r, q) , r ∈ N+, q ∈ N−, and a fixed value of p, p = 0, . . . , r − 1, the sequence(G

(r,p,q)−n

)n≥1

,

G(r,p,q)−n =

bn−1−q c∑

k=0

⟨ −n− qkp+ rk

⟩.

constitues the sum of elements lying on the corresponding ray in the negative GAT.

Theorem 4. For r ≥ 1, q ∈ N−, p ∈ N, p = 0, . . . , r − 1,

G(r,p,q)−n = −

bn−1−q c∑

k=1

A−n−p−(r+q)kBp+rkn+qk−1∑

i=0

(−n− qk + i

p+ rk − 1

)a−i−1−A−n−pBp

n−1∑

i=0

(−n+ i

p− 1

)a−i−1+

⌊n− pn

⌋a−nA

−n.

First, two lemmata which will play an important role in determining the recurrence linked to G(r,p,q)−n .

Lemma 5. For n ≥ 1, r ≥ 1, α ∈ Z and (an)n∈Z sequence of real numbers, we have

r∑

j=0

(−1)j(r

j

)n+j+α−1∑

i=0

(−n− j − α+ i

r − 1

)a−i−1 = −a−n−α−r.

Lemma 6. For n ≥ 1, r ≥ 1, p ≥ 1 and (an)n∈Z sequence of real numbers, we have

r∑

j=0

(−1)j(r

j

)n+j−1∑

i=0

(−n− j + i

p− 1

)a−i−1 = −

r∑

j=0

(−1)j(r − pj

)a−n−p−j .

2. Recurrence associated to rays in negative GATTheorem 7. The terms of the sequence (G−n)n≥1 given by

G−n =bn−1−q c∑

k=0

A−n−p−(r+q)kBp+rkn+qk−1∑

i=0

(−n− qk + i

p+ rk − 1

)a−i−1.

satisfy the recurrence relation

G0−n−A

(r

1

)G0−n−1+· · ·+(−A)r

(r

r

)G0−n−r−A−n−pBp

(a−n−p −

(r − p

1

)a−n−p−1 + · · ·+ (−1)r

(r − pr

)a−n−p−r

)

= BrG−n−q−r.

145


References[1] D. Enseley, Fibonacci’s triangle and other abominations, The edge of universe: Celebrating Ten Years of Maths Horizons,

MAA (2006), pp. 287–307.

[2] G. Kallos, A generalization of Pascal’s triangle using powers of base numbers, Ann. Math. Blaise Pascal, 13 (2006)1–15.

[3] H. Belbachir, T. Komatsu, L. Szalay, Characterization of linear recurrences associated to rays in Pascal’s triangle,Diaophantine Analysis and Related Fields, T. Komatsu (Eds.), AIP Conference Proceedings, Vol. 1264, Melville, NewYork (2010), 90–99.

146


The Many Aspects of Polynomial Triangles

Nour-Eddine Fahssi

Hassan II University of Casablanca,Faculty of Sciences and Technology, Mohammedia, Morocco.

[email protected]

Abstract

By polynomial coefficients, we mean the coefficients in integral powers of a polynomial. Although thesecoefficients are considered and used in many literature, there is no systematic treatise on them. This paperaims to fill the void by bringing a fresh insight to this old subject. We describe some aspects of these extendedbinomial coefficients in different point of view, and give new results presented in a compact and self-containedmanner.

Key words: Polynomial triangles; restricted occupancy model; polynomial coefficient identities; entropy density function.

2010 Mathematics Subject Classification: : 05A10, 05A15, 05A16, 05A19.

1. IntroductionThe theme of our study is extremely simple. It consists in investigating several aspects of coefficients in integral powers ofpolynomials. These coefficients generate an array, called a polynomial triangle, whose the k-th row consists of the coefficientsof the powers of t in p(t)k, for a given polynomial p(t). The table naturally resembles Pascal’s triangle and reduces to it whenp(t) = 1 + t.

Although the subject is quite simple, it has a variety of applications in enumerative combinatorics. Historically, thisnatural extension of binomial coefficients may have been first discussed by Abraham De Moivre [4, 12] who found that thecoefficient of tn of the polynomial (

1 + t+ t2 + · · ·+ tm)k, (1)

(k ≥ 0) arises in the solution of the following problem [20, p.389]:

“There are k dice with m+1 faces marked from 1 to m+1; if these are thrown atrandom, what is the chance that the sum of the numbers exhibited shall be equalto n?”

A few decades later, Leonhard Euler [14, 15] published an analytical study of the coefficients of the polynomial (1). Theelementary properties of the array generated by these coefficients closely mimic those of binomial ones. For example, eachentry in the body of the (centered) triangle is the sum of the m entries above it, extending a well-known property of Pascal’striangle. This array, denoted in this paper by Tm, is termed the extended Pascal triangle [8], or Pascal-T triangle [30],or Pascal-De Moivre triangle [21]. In 1937, it was reintroduced by A. Tremblay [29] and in 1942 by P. Montel [23], andexplicitly discussed by John Freund in 1956 [17], as arising in the solution of a restricted occupancy problem. In fact, thecoefficient of tn in (1), denoted

(kn

)m

[3], is the number of distinct ways in which n unlabeled objects can be distributed ink labeled boxes allowing at most m objects to fall in each box (see also the Riordan monograph [25, p.104]). In statisticalphysics, the Freund restricted occupancy model is known as the Gentile intermediate statistics, called after Giovanni Gentile

147


Jr [18, 19]. This model interpolates between Fermi-Dirac statistics (binomial case : m = 1) and Bose-Einstein statistics(m =∞). It will be referred here to as Gentile-Freund statistics (GFS).

The arrays Tm have been extensively used in reliability and probability studies [4, 13, 22, 26]. Several results about theextended binomial coefficients, specially the trinomial ones (m = 2), are known [5–10, 16]. Some generalizations havebeen discussed as well. For instance, Ollerton and Shannon [24] have investigated various properties and applications ofa generalization of the triangle Tm by extending the Freund occupancy model. We underline in passing that the entries ofTm have been q-generalized by George Andrews and Rodney J. Baxter [1] for m = 2 to solve the hard hexagon model instatistical mechanics and later by Warnaar [31] for arbitrary m. This q-analog proved to have a deep connection with theRogers-Ramanujan identities and the theory of partitions [2, 3].

The Pascal-De Moivre triangles T2, T3 and T4 are recorded in Sloane’s Online Encyclopedia of Integer Sequences [28] asA027907, A008287 and A035343 respectively.

Let us now fix our terminology and notation a bit more precisely.

Definition 1. let a be a sequence ofm+1 numbers (a0, a1, . . . , am) and let pa(t) =∑m

i=0 aiti be its generating polynomial.

The polynomial coefficients associated with the vector a are defined by1

(k

n

)

a

def=

[tn] (pa(t))k, if 0 ≤ n ≤ mk

0, if n < 0 or n > mk(2)

where we have used a vector-indexed binomial symbol mimicking the notation of Andrews. When ai = 1 for all i, the binomialsymbol will be simply indexed by m. The array of polynomial coefficients will be called a-polynomial triangle or a-trianglefor short, and denoted by T(a). Rows are indexed by k and columns by n. The polynomial triangle T(a) will be calledarithmetical or combinatorial if the coefficients ai are integers.

The term “polynomial coefficients” is inspired by the designation of Louis Comtet [11, p.78] for Tm.A point that needs to be addressed is that, with the exception of the binomial triangle, T(a) is a Riordan array only if

pa(0) = 0, i.e, a0 = 0, as can be seen from the bivariate generating function :

∑

k,n

(k

n

)

a

tnuk =1

1− upa(t).

For basic informations on Riordan arrays, the reader is referred to [27].

2. Main resultsLiterature on the triangles Tm remains quite sparse, and there is no systematic interest in their properties. In this researchpaper, we plan to fill this gap by presenting a unified approach to the subject.

The following items give a sample of our results :

• Besides the weighted Gentile-Freund model, we give three combinatorial models enumerated by polynomial coeffi-cients :

1. we show that polynomial coefficients count the number of points a player can score in a game of drawingcolored balls. We discuss, by the way, a variant of the old problem of De Moivre and rediscover that polynomialcoefficients provide a natural extension of the usual binomial probability distribution based on Bernoulli trialswith more than two outcomes.

2. we give, via a bijective proof, an interpretation of (2) as number of specified colored directed lattice paths.

3. we propose an interpretation in terms of spin chain systems.

We discuss some important examples of arithmetical polynomial triangles related to restricted occupancy models.

• We give extensions of several known binomial identities. The techniques are elementary and the proofs are straight-forward, but the results seem interesting in their own right.

1We make use of the conventional notation for coefficients of entire series : [tn]∑

i aiti := an.

148


• Motivated by a generalization of the binary entropy function, we introduce and study some properties of the entropydensity function which characterizes polynomial coefficients in the thermodynamical limit (that is when k and n go toinfinity and the ratio n/k is fixed) and study its properties.

• We formulate some conjectures such as a Harper-type conjecture and an extended binomial determinant conjecture

• Finally, we prove polynomial analogs of Lucas and Babbage congruences.

References[1] G. E. Andrews and J. Baxter, Lattice gas generalization of the hard hexagon model III, q-trinomial coefficients, J. Stat.

Phys. 47 (1987) 297–330.

[2] G. E. Andrews, q-Trinomial coeffcients and the Rogers-Ramanujan identities, Analytic Number Theory (Berndt et al.,eds.), Birkhäuser, Boston, (1990), 1–11.

[3] G. E. Andrews, Euler’s ‘exemplum memorabile inductionis fallacis’ and q-trinomial coefficients, J. Amer. Math. Soc. 3(1990) 653–669.

[4] K. Balasubramanian, R. Viveros and N. Balakrishnan, Some discrete distributions related to extended Pascal triangles.Fibonacci Quart., 33 (1995) 415–425.

[5] J. D. Bankier, Generalizations of Pascal’s triangle, Amer. Math. Monthly 64 (1957) 416-419.

[6] R. C. Bollinger, A note on Pascal T-triangles, Multinomial coefficients and Pascal Pyramids, Fibonacci Quart. 24 (1986)140–144.

[7] R. C. Bollinger, C. L. Burchard, Lucas’s theorem and some related results for extended Pascal triangles. Amer. Math.Monthly 97, No.3, (1990) 198-204.

[8] R. C. Bollinger, Extended Pascal triangles. Math. Mag. 66, No.2, 87–94 (1993).

[9] R. C. Bollinger, The Mann-Shanks primality criterion in the Pascal-T triangle T3. Fibonacci Quart. 27, No.3, (1989)272–275.

[10] B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, (1990). English translationpublished by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, (1993).

[11] L. Comtet, Advanced Combinatorics, Reidel, (1974).

[12] A. De Moivre. The Doctrine of Chances: or, A Method of Calculating the Probabilities of Events in Play. 3rd ed.London: Millar, 1756; rpt. New York: Chelsea, 1967.

[13] C. Derman, G. Lieberman, and S. Ross. On the consecutive-k-of n : F System IEEE Trans. Reliability 31, (1982) 57–63.

[14] L. Euler, De evolutione potestatis polynomialis cuiuscunque (1 + x + x2 + . . . )n, Nova Acta Academiae ScientarumImperialis Petropolitinae 12 (1801), 47–57 ; Opera Omnia : Series 1, Volume 16, 28–40. A copy of the original text isavailable at the Euler Archive.

[15] L. Euler, Observationes analyticae, Novi Commentarii Academiae Scientarum Petropolitanae 11 (1767), 124–143, alsoin Opera Omnia, Series 1 Vol. 15, 50–69. A copy of the original text is available at the Euler Archive.

[16] D. C. Fielder and C. O. Alford, Pascal’s Triangle: Top Gun or Just One of the Gang ? In Applications of FibonacciNumbers 4:77. Dordrecht: Kluwer, 1990.

[17] J. E. Freund, Restricted Occupancy Theory - A Generalization of Pascal’s Triangle. Amer. Math. Monthly, 63, No. 1(1956), 20–27.

[18] G. Gentile Jr, Le statistiche intermedie e le proprietà dell’elio liquido, Nuovo Cimento 19 (1942), 109–125.

[19] G. Gentile Jr, Osservazioni sopra le statistiche intermedie, Nuovo Cimento 17 (1940) 493–497.

[20] H. S. Hall and S. R. Knight, Higher Algebra: a Sequel to Elementary Algebra for Schools. London : Macmillan and Co.1894.

149


[21] E. Larry, The Pascal-de Moivre triangles, Fibonacci Quart., 36 (1998), 20–33

[22] F. S. Makri, and A. N. Philippou. On binomial and circular binomial distributions of order k for l-overlapping successruns of length k. Statist. Papers 46.3, 411–432.

[23] P. Montel, Sur les combinaisons avec répétitions limitées, Bull. Sc. M., 66 (1942) 86–103.

[24] R.L. Ollerton and A.G. Shannon, Extensions of generalized binomial coefficients, in Howard, Frederic T., eds, Applica-tions of Fibonacci numbers. Volume 9. Proceedings of the 10th international research conference on Fibonacci numbersand their applications, Northern Arizona University, 2002. Dordrecht : Kluwer Academic Publishers. (2004) 187–199.

[25] J. Riordan, An Introduction to Combinatorial Analysis, New York. John Wiley (1964).

[26] K. Sen, M. Agarwal and S. Bhattacharya, On circular distributions of order k based on Polya-Eggenberger samplingscheme. The Journal of Mathematical Sciences 2, (2003) 34–54.

[27] L. V. Shapiro, S. Getu, W. J. Woam, and L. Woodson, The Riordan group. Discrete Appl. Math, 34 (1991) 229–239.

[28] Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences. Published electronically at https://oeis.org/

[29] A. Tremblay, Generalization of Pascal’s Arithmetical Triangle, National Mathematics Magazine, 11, No. 6, (1937)255–258.

[30] S. J. Turner, Probability via the N -th order Fibonacci-T sequence. Fibonacci Quart., 17 (1979) 23–38.

[31] S. O. Warnaar, The Andrews-Gordon identities and q-multinomial coefficients, Commun. Math. Phys. 184, (1997).arXiv:q-alg/9601012

150


Towards Integrals of Rational Tree Expressions

Younes Guellouma1∗, Ahlem Belabbaci1 , Hadda Cherroun1

1 Universite Amar Telidji, Laboratoire LIMLaghouat, Algerie

(y.guellouma, hadda cherroun, ah.belabbaci)@lagh-univ.dz

Abstract

The present paper proposes a generalization of the notion of integrals from rational string expressions torational tree expressions. We also show the link between tree integrals and tree derivatives. This generalizationis used to denote the accepted tree language of a tree automaton by a rational tree expression.

Key words: Rational tree expressions, derivatives of tree expressions, Integrals of tree expressions

2010 Mathematics Subject Classification: 68Q45.

1. IntroductionTrees are natural structures used in many fields in computer sciences like XML [12], indexing, natural language processing,code generation for compilers, term rewriting [4], cryptography [5] etc. The large use of this structure compel us to considerthe theoretical basics of such a notion. As a part of the formal language theory, trees are considered as a generalization ofstrings. Indeed in the late of 1960s [2, 8], many researches generalize strings to trees and many notions appeared like treelanguages, tree automata (TA), rational tree expressions (RTE), tree grammars, etc.

Since TA are beneficial in an acceptance point of view and the rational expressions in a descriptive one, an equivalencebetween the two representations must be resolved. Fortunately, Kleene result [11] states this equivalence between the acceptedlanguage of string automata and the language denoted by rational expressions.

Some recent work focuses on generalizing some notions of this equivalence to trees. In fact, several techniques to converta RTE into TA exist. First, Kuske and Meinecke [6] have generalized the notion of languages partial derivation [1] fromstrings to trees and propose a tree equation automaton which is constructed from a derivation of a linearized version of RTE.They have used the ZPC structure [3] to improve the complexity. After that, Mignot et al. [9] have proposed an efficientalgorithm to compute the generalized tree equation automaton. Next, Laugerotte et al. [7] have generalized the positionautomata to trees. Finally, the morphic links between these constructions have been defined in [10].

In this paper we present a generalization of integrals on RTE. Integrals on string rational expressions were first introducedby Smith et al. in [?]. This generalization allows us to generate the equivalent RTE of a tree language denoted by a given TA.

2. Derivatives of RTELet Σ =

⋃n≥0 Σn be an alphabet. A tree t over Σ is inductively defined by t = f(t1, . . . , tn) with f ∈ Σn and t1, . . . , tn

are n trees over Σ. A tree language is a subset of T (Σ).

∗Speaker

151


Given a symbol c ∈ Σ0, the c-product is the operation ·c defined for a tree t in T (Σ) and a tree language L by

t ·c L =

L if t = c,d if t = d ∈ Σ0 \ c,f(t1 ·c L, . . . , tn ·c L) otherwise (if t = f(t1, . . . , tn))

(1)

This c-product is extended for any two tree languages L and L′ by L ·c L′ =⋃

t∈L t ·c L′. In the following of this paper, weuse some equivalences over expressions using properties of the c-product.

The iterated c-product is the operation n,c defined recursively for an integer n by:

L0,c = c (2)

Ln+1,c = Ln,c ∪ L ·c Ln,c (3)

A rational expression E over Σ is inductively defined by: E = 0, E = f(E1, . . . , En), E = E1 + E2, E = E1 ·c E2,E = E∗c

1 , where f ∈ Σn, c ∈ Σ0 and Ei is a RTE with i = 1 . . . n.The language denoted by E is the tree language L(E) inductively defined by: [[0]] = ∅, [[a]] = a, [[E1 + E2]] =

[[E1]] ∪ [[E2]], [[E1.cE2]] = [[E1]].c[[E2]], [[E∗c]] = [[E]]∗c, [[f(E1, . . . , En)]] = f(s1, . . . , sn) | s1 ∈ [[E1]], . . . , sn ∈ [[En]],where f ∈ Σn, c ∈ Σ0 and Ei is a RTE with i = 1 . . . n. We say that two RTEs E and F are equivalent (denoted E ≡ F ) iif[[E]] = [[F ]].

Let N be the n-tuples set of some RTE. Then the n-tuple concatenation is defined as follows:

N .cF = (E1.cF, . . . , En.cF ) | (E1, . . . , En) ∈ NThe derivation of a RTE is a set of n-tuples defined inductively as follows:

∂0∂a

= 0 if a ∈ Σ0

∂f(E1, . . . , En)∂g

=(E1, . . . , En) if f = g0 otherwise

∂(E + F )∂g

=∂E

∂g∪ ∂F∂g

∂E∗c

∂g=

∂E

∂g.cE∗c

∂E.cF

∂g=

∂E∂g .cF si c /∈ E

∂E∂g .cF ∪ ∂F

∂g otherwise

3. Integrals of Tree ExpressionsBefore defining the integrals on RTE, we give an elementary definition of the integrals on trees.

Definition 1 Given a special symbol λ /∈ Σn, n ≥ 0 and a symbol f ∈ Σn, the integral of the tree λ(t1, . . . , tn) is defined asfollows: ∫

λ(t1, . . . , tn) df = f(t1, . . . , tn) (4)

This definition can be generalized to RTE.

Definition 2 Let N be a set of n-tuple RTE. The integral of N with respect to f ∈ Σn is∫N df =

∑

(E1,...,En)∈Nf(E1, . . . , En) (5)

152


The following lemmas can be easily checked.

Lemma 3 Let N ,M be two n-tuple expressions sets. Then we have :∫N ∪M df ≡

∫N df +

∫M df

Lemma 4[[∫N df ]] =

∫λ(t1, . . . , tn) | ∀(E1, . . . , En) ∈ N , t1 ∈ E1, . . . , tn ∈ En (6)

and

Lemma 5 Let E be a RTE. We have: ∫N .cE df =

( ∫N df

).cE

We now show that the integrals thus defined are the inverse operation of derivation. This can be proved by induction onthe structure of the RTE.

Proposition 1 Let E be a RTE, then ∫∂E

∂gdg ≡ E (7)

Proof: By induction over the structure of E, we have:

1. If E = g(E1, . . . , En) then ∂E∂g = (E1, . . . , En). (Using equations 6 and 5).

2. IfE = E1+E2 then ∂E∂g = ∂E1

∂g ∪ ∂E2∂g . We have

∫∂E∂g dg =

∫∂E1∂g ∪ ∂E2

∂g dg ≡∫

∂E1∂g dg+

∫∂E2∂g dg = E1+E2 = E

(Lemma 3).

3. If E = E∗c1 then ∂E∂g = ∂E1

∂g .cE∗c1 . We have

∫∂E∂g dg =

∫∂E1∂g .cE

∗c1 dg =

∫∂E1∂g dg.cE

∗c1 = E1.cE

∗c1 = E

(Lemma 5).

4. If E = E1.cE2, the same proof as 3.

4. Generating a RTE from a TA Using IntegralsThe basic idea of the algorithm generating a RTE from a tree automaton using integrals is to associate to each state in theunderlying automaton the RTE accepted by this state. Then, by integrating these RTEs in a defined order and using somerules for manipulating integrals (combinatory, substitution, ...) and rational expression properties, we deduce the RTE for thewhole automaton.

5. ConclusionIn this paper, we defined a generalization of the integral notion on RTE. We have shown that integral is the inverse operationof tree derivatives proposed by Kuske et al. [6]. This generalization is used to give an alternative proof of Kleene theorem byconstructing an algorithm that converts a tree automaton into a RTE.

153


References[1] Valentin M. Antimirov. Partial derivatives of regular expressions and finite automaton constructions. Theor. Comput.

Sci., 155(2):291–319, 1996.

[2] W. S. Brainerd. Tree Generating Systems and Tree Automata. PhD thesis, Purdue University, 1967.

[3] Jean-Marc Champarnaud and Djelloul Ziadi. From c-continuations to new quadratic algorithms for automaton synthesis.IJAC, 11(6):707–736, 2001.

[4] H. Comon, M. Dauchet, R. Gilleron, C. Loding, F. Jacquemard, D. Lugiez, S. Tison, and M. Tommasi. Tree automatatechniques and applications, 2007. release October, 12th 2007.

[5] Volker Diekert and Michel Habib, editors. STACS 2004, 21st Annual Symposium on Theoretical Aspects of ComputerScience, Montpellier, France, March 25-27, 2004, Proceedings, volume 2996 of Lecture Notes in Computer Science.Springer, 2004.

[6] Dietrich Kuske and Ingmar Meinecke. Construction of tree automata from regular expressions. RAIRO - Theor. Inf. andApplic., 45(3):347–370, 2011.

[7] Eric Laugerotte, Nadia Ouali Sebti, and Djelloul Ziadi. From regular tree expression to position tree automaton. InLATA, pages 395–406, 2013.

[8] M. Magidor and M. Gadi. Finite Automata Over Finite Trees. Technical reports. 1969.

[9] Ludovic Mignot, Nadia Ouali Sebti, and Djelloul Ziadi. An efficient algorithm for the equation tree automaton via thek-c-continuations. CoRR, abs/1401.5951, 2014.

[10] Ludovic Mignot, Nadia Ouali Sebti, and Djelloul Ziadi. $k$-position, follow, equation and $k$-c-continuation treeautomata constructions. In Proceedings 14th International Conference on Automata and Formal Languages, AFL 2014,Szeged, Hungary, May 27-29, 2014., pages 327–341, 2014.

[11] J.W. Thatcher and J.B. Wright. Generalized finite automata theory with an application to a decision problem of second-order logic. Mathematical systems theory, 2(1):57–81, 1968.

[12] Silvano Zilio and Denis Lugiez. Xml schema, tree logic and sheaves automata. In Robert Nieuwenhuis, editor, RewritingTechniques and Applications, volume 2706 of Lecture Notes in Computer Science, pages 246–263. Springer BerlinHeidelberg, 2003.

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Approche pour l’optimisation d’une fonction quadratique indéfinie.

Belabbaci Amel1∗, Djebbar Bachir2.

1 LIM, Université de [email protected]

2 Université des sciences et de technologie, [email protected]

RésuméDans ce travail on présente une démarche à suivre pour optimiser un programme quadratique indéfini. L’al-

gorithme commence par l’approximation de la partie concave du programme pour optimiser la partie convexe.L’optimum trouvé constitue un point de départ pour chercher l’optimum du programme indéfini.

Mots clés : programme quadratique indéfini, convexe, concave.MSC : 90C20, 90C25.

1. Introduction

La programmation quadratique indéfinie est un problème difficile de la programmation quadratique. Plusieurs méthodes ontété proposées (voir [2], [1], [3], [4], [5]). Mais, un programme indéfini peut, dans beaucoup de cas, être résolu par desméthodes simples. Il en est ainsi lorsque la matrice associée à la forme quadratique admet des valeurs propres presque toutesde même signe, c’est à dire que celles de signes contraires sont en un nombre relativement petit. C’est ce principe qu’on vautiliser pour développer notre algorithme d’optimisation.

2. Méthode proposée

On considère le problème quadratique :maxx∈Ω

f(x),

où f(x) = (α, x) + (Qx, x) et Ω = x ∈ Rn : Ax 6 b, x > 0, α et b ∈ Rn. On considère la matrice Q = (qij) de taillen× n et A = (aij) une matrice de réels de taille m× n.La fonction objectif f peut être décomposée en une partie concave ϕ(z) et une autre convexe Ψ(x). La partie concave estapproximée, on se ramène alors à l’optimisation d’une fonction convexe.Pour l’optimisation de la partie convexe, on propose un algorithme basé sur des boules concentriques. L’idée est de calculerle point critique de la fonction objectif, on cherche ensuite le sommet du convexe le plus éloigné de ce point critique.On considère le problème convexe suivant :

(PΨ)

x > 0Ax 6 bmaxΨ(x) =

∑ni=1 αixi + βix

2i

∗Speaker155


Les coefficients αi sont des nombres réels de signes quelconques et βi = 1,∀i . Soit Ω le convexe formé par les contraintesAx 6 b, x > 0, où A est une matrice réelle de taille m× n, et b un vecteur de Rm+ . Ω est un convexe fermé et borné de Rn.On pose

P (x) =∑

βi>0

αixi + βix2i

alors

P (x∗)− P (x) =n∑

i=1

−βi(x∗i − xi)2 pour tout x ∈ Ω

et

P (x∗)− P (x) = −Maxn∑

i=1

(√βix∗i −

√βixi)2

donc

P (x) = P (x∗) + Maxn∑

i=1

(√βix∗i −

√βixi)2

Posonsy∗ = (y∗i ) = (αi/2

√βi)i

ety = (y0)i = (

√βixi)i

On voit que y0 est le point le plus éloigné du convexe Ω = y ∈ Rn : yi =√βixi, x = (xi) ∈ Ω.

Pour trouver l’optimim d’une fonction convexe, on doit donc suivre les étapes suivantes :1. Calculer le point critique x∗ ;2. Voir s’il existe un β

′i 6= 1 alors le convexe doit être transformé.

3. Choisir un sommet du convexe, et calculer la distance entre ce sommet et x∗ ;4. Comparer la distance calculée avec celles de ses voisins par rapport à x∗ ;5. Voir si cette distance est maximale alors stop c’est le sommet optimal ;6. Sinon passer au sommet de distance maximale, éliminer tous les autres sommets. Soit s le sommet choisi.7. Calculer la distance entre s et x∗, retourner à l’étape 4.

La solution optimale de la partie convexe sera considérée comme une approximation initiale. Une méthode séquentielle(voir [6]) est utilisée ensuite pour trouver la solution optimale du problème indéfini.

La démarche proposée est convergente. En effet, pour l’algorithme qui optimise la fonction convexe le sommet le pluséloigné du point critique est toujours atteint (puisque le convexe est fermé). Pour la méthode séquentielle choisie, on doits’assuser de sa convergence pour l’utiliser.

3. Exemple numérique

Soit à chercher : maxx∈Ω

f(x)CT + 12x

TQx où Ω = x : Ax 6 b, x > 0xi > 1 pour i = 1, · · · 5.Les données de ce programme :

Q = (1, 1, 1, 1, 1,−1)I

C = (10.5, 7.5, 3.5, 2.5, 1.5, 10)

b = (6.5, 20)T

A =(

3 3 3 2 1 010 0 10 0 0 1

)

L’optimum de la fonction convexe est atteint au sommet (1, 1, 0, 0, 0.5, 0). Avec une méthode séquentielle, on arrivera à unesolution approchée au point (1, 1, 0, 0, 0.5, 5).

156


Références[1] P.-A.Absil and André L.Tits, Newton-KKT interior-point methods for indefinite quadratic programming,Comput Optim

Applic (2007) 36 :5–41,DOI 10.1007/s10589-006-8717-1.

[2] M.J.Best and B.Ding, A Decomposition Method for Global and Local Quadratic Minimization,Journal of Global Opti-mization 16 : 133–151, 2000.

[3] H.An Le Thi, T.P.Dinh and N.D.Yen, Properties of two DC algorithms in quadratic programming,J Glob Optim (2011)49 :481–495, DOI 10.1007/s10898-010-9573-1.

[4] R. Cambini and C. Sodini, A computational comparison of some branch and bound methods for indefinite quadraticprograms, CEJOR (2008) 16 :139–152, DOI 10.1007/s10100-007-0049-4.

[5] S.A.Vavasis, Approximation algorithms for indefinite quadratic programming, Mathematical Programming 57 (1992)279-311.

[6] N.Bakhvalov, Méthode numérique, Mir Moscou, traduction française, 1976.

157


Frequent Pattern Mining : a Reduction to Acyclic WeightedAutomata Determinisation

Slimane Oulad-Naoui∗, Hadda Cherrroun

Laboratoire d’Informatique et de Mathématiques, Université Amar TelidjiLaghouat, Algeria

s.ouladnaoui, [email protected]

Résumé

In [3] we have introduced a new model for the Frequent Itemset (FI) mining problem based on a formalseries over the counting semiring (N,+,×, 0, 1), whose range constitutes the itemsets and the coefficients aretheir supports. In this paper, we show that the discovery of FI for a dataset D turns out to be a determinisationof the associated ε-moves augmented prefixial weighted automaton, with the respect of the support-threshold s.

Key words : Frequent Itemsets, Formal Series, Weighted Automata, Unification.

2010 Mathematics Subject Classification : 68, 40, 93.

1. IntroductionIn [3] we developed a general unifying model able to express the work done so far in the field in the main state of the artapproaches [1, 2, 4]. The introduced model enjoy many capitals characteristics such as : the completeness while remainingsimple and intuitive, extensibility, and efficiency. In this work, we show that the mining problem of a transaction datasetwith the respect to a given support threshold s is equivalent to compute the deterministic weighted automaton of the definedprefixial weighted automaton associated with D taking into account the frequentness criterion.

2. Problem StatementLet A = a1, a2, . . . , am be an alphabet of m symbols called items. An itemset is a subset of A, if k is its cardinal it

is called a k-itemset. A transaction ti is a nonempty set of items identified by its unique identifier i. A dataset D is a set ofn transactions, which we denote as a multi-set : D = t1, t2, . . . , tn. In a dataset D, the support of an itemset x, denotedsprt(x,D), is the number of transactions containing x, i.e : sprt(x,D) = |tk ∈ D | x ⊆ tk|. An itemset x is frequent ifits support exceeds a specified minimum support-threshold s. The problem of mining FI consists to discover the set F of allitemsets whose support is greater than the given minimum support-threshold s.

3. The Polynomial ModelThe main idea in this modeling is to encode an itemset by a word, and all its subsets by a polynomial. After defining thepolynomial of a dataset, the question is then to extract from this polynomial all the terms where the support-criterion holds.

∗Speaker158


First, let us assume, without loss of generality, that the alphabet A is sorted according to an arbitrary total order, wherewe can write : A = a1, a2 . . . am, with : ε < a1 < a2 < . . . < am. We represent a k-itemset x = ai1 , ai2 , . . . , aik

by the word w(x) of length k, built by the concatenation of its items according to the predefined order. We will write :w(x) = ai1ai2 . . . aik

, such that ai1 < . . . < aik. In what follows, we confuse an itemset x and its word representation

w(x).

Definition 1 (Itemset Subsequence Polynomial). Let x = ai1ai2 . . . aikbe a k-itemset, The subsequence polynomial Sx

associated with x is defined as follows : Sx = (ai1 + 1)(ai2 + 1) . . . (aik+ 1), with : Sε = 1

Hereafter, we denote, for each a ∈ A, by a the polynomial (a + 1). So, the polynomial Sx associated with a k-itemsetx will be denoted : Sx = ai1ai2 . . . aik

. So, Sx is the polynomial that represents all the subsets of x. For example, weassociate with the itemset x = abc the polynomial Sx = abc = (a + 1)(b + 1)(c + 1), that gives us the polynomial :1 + a+ b+ c+ ab+ ac+ bc+ abc.From the itemset subsequence polynomial, we can derive the subsequence polynomial associated with a dataset D.

Definition 2 (Dataset Subsequence Polynomial). Let D = t1, . . . , tn be a dataset. The subsequence polynomial SD asso-ciated with D is the sum of the n subsequence polynomials of its transactions : SD =

∑ni=1 Sti

It is obvious to see that the terms of the polynomial SD have the form 〈SD, w〉w, where w is an itemset and 〈SD, w〉a coefficient in N representing its support in the database. Indeed, an itemset have 1 as coefficient in the polynomial of thetransaction ti where it appears and, consequently, its coefficient in the database is then the number of the transactions where itoccurs. To illustrate this concept let us consider a running example by taking the following dataset abde, bce, abde, abce,bcd,abcde.

Definition 3 (Itemset Prefixial Polynomial). Let x = ai1ai2 . . . aikbe a k-itemset, the prefixial polynomial associated with x

is Px defined as follows : Px =∑

u∈Pref(x) u, where Pref(x) is the set of all the prefixes of x.

Definition 4 (Dataset Prefixial Polynomial). LetD = t1, . . . , tn be a dataset. The prefixial polynomial PD associated withD is the sum of the n prefixial polynomials of its transactions :

PD =∑n

i=1 Pti

Definition 5 (Prefixial Weighted Automaton (PWA)). Let PD be the prefixial polynomial of a datasetD. The related prefixialweighted automaton PD = (Q,A, µ, λ, γ) is defined as follows :

– Q = range(PD),– µ(u) = 1, for u = ε, and 0 otherwise, for u ∈ Q,– λ(u, a, ua) = 1, for u and ua ∈ Q, and a ∈ A,– γ(u) = 〈PD, u〉, for u ∈ Q.

Definition 6. Let D be a dataset, and PD the associated prefixial polynomial. The prefixial-bar polynomial PD is :PD = 〈PD, ε〉+

∑u∈A∗a∈A〈PD, ua〉ua

4. Main ResultsLemma 7. For a dataset D, the automaton PD realizes the polynomial PD.

Proposition 1. Let A and B be two PWAs associated respectively with datasets X and Y . There exists a PWA C for thedataset X ∪ Y derived from A and B.

Proposition 2. Let D = t1, . . . , tn be a dataset. Let PD and SD respectively the associated prefixial-bar and the subse-quence polynomials. Then : PD = SD

159


0

6

1

4

2

4

3

2

4

1

5 1

6 1

7

2

8 2

9

2

10

2

11 1

12 1

a/1

b/1

c/1 d/1

e/1

e/1

d/1

e/1

b/1

c/1

d/1

e/1

FIG. 1 – A PWA associated with our running example dataset.

Algorithm 1 DISCOVER-FI(S, s, w,F)

Require: a PWA of D, the support-threshold s, a setof states Qw, and an itemset w

Ensure: The set of all FIfor all a > w|w| do

(Qwa, 〈SD, wa〉)← EXTEND(Qw, w, a)if 〈SD, wa〉 ≥ s thenF ← F ∪ (wa, 〈S, wa〉)DISCOVER-FI(S, s, wa,F)

end ifend for

function EXTEND(Qw,w,a)P ← Qw, R← ∅while P 6= ∅ do

q ← pick a state from Pif i(q) = a then

R← R ∪ qelse if i(q) < a then

P ← P ∪ δ+(q)end if

end whilereturn (R, γ(R))

end function

5. The Mining AlgorithmOnce the PWA associated with D has been built, it serves as a structure for the problem space exploration. In the miningphase, we explore the automaton using a DFS traversal as exhibited in Algorithm 1. We begin with the invocation DISCOVER-FI(S, s, q0, ε, ∅) (q0 is the initial state). At each step, and starting from the set of states Qw, an itemset w is extended byconcatenation with its successors by the function EXTEND(Qw, w, a). This call returns the set Qwa of all paths labeled wawith their coefficients. The support of the concerned itemset wa is then the sum of the coefficients of the elements in the

160


FIG. 2 – two automata (a) and (b) and the merging one (c) by determinization.

returned set Qwa, (γ(R) =∑

r∈R

γ(r)). If this extension succeeds with a frequent itemset, the process will continue taking into

account the last reached set of states Qwa ; Otherwise the returned couple is (∅, 0). Note that i(q) stands for the item-label ofthe transition leading to the state q, with i(q0) = ε, and δ+(q) for the successor states of the state q.

Proposition 3. Algorithm 1 can be done in O(∑

w∈Fa>w|w|

Cwa) time and O(|Q|) space, where Q is the set of states of the PWA,

and F is the set of FI in the dataset, and Cwa is the time required to compute the set Qwa by extending the set Qw.

Theorem 8. Let D = t1, . . . , tn be a dataset, and ε− PD the associated PWA augmented with ε moves. The discovery ofall FI of D with the respect of a support threshold s is equivalent to the computation of the deterministic weighted automatonof ε−PD conditioned by the support s.

6. Comparison and UnificationOur model uses formal series, which are mappings between a monoid M and a semiring K. The appropriate choice of Mand K, and the automaton characteristics which realizes it is driven by the targeted application and needed performances. Weare convinced that this framework can be generalized for mining other elaborated items such as sequences, trees or graphs,provided that much more work must be carried out to define monoids of these elements with the appropriate operations andthe corresponding implementations by means of specific automata. In what follows, we compare our model against the mainstate of the art techniques, and explain how these ones can be derived from it.Level-wise Approaches : Our model can be modified to fit a similar principle of [1] if we adopt a breadth-first determinisationof the defined automaton.Vertical Approaches : The main benefit of the vertical approach [4] is speedy in the support calculation via set intersections.However, the drawback is when the intermediate results become too big. Our method, in contrast, is based on a simple outputweight read or their summation without need of any additional memory. We can see this approach as a formal series on thepowerset semiring of the set T of transactions (2T ,∪,∩, ∅, T ). Consequently it is equivalent to a DFS determinisation of ourautomatonProjection Approaches : While FPGrowth [2] uses a heavily intermediate memory, and time overhead, for conditionaldatabases construction, our model do not require any additional memory other that necessary for the automaton. Further, theopen ordering adopted in our model leads to significant time improvement both in the construction phase (only one scanis required), and the mining one, since there is no need to repetitive resorting, neither database projections. To the end ofunification, we can view these approaches as a sequence of right derivations by the set of items A of our dataset subsequencepolynomial SD, or like the exploration of the mirror of the automaton.

7. ConclusionWe proved that the FI mining problem can be stated as a determinisation of an acyclic weighted automaton which

realizes a polynomial over the counting semiring. This formalism gives a generic view of the problem and leads to efficient

161


implementation in both time and space. We argue that this formalism open many other extensions to mine more complexstructures which constitute our next investigation.

Références[1] Agrawal R., Srikant R., Fast Algorithms for Mining Association Rules in Large Databases, VLDB’94, Proceedings of

20th Int. Conf. on VLDB, Sept. 12-15, 1994, Santiago de Chile, Chile

[2] Han J., Pei J., Yin Y., Mining Frequent Patterns without Candidate Generation, Proceedings of the 2000 ACM SIGMODInt. Conf. on Management of Data, May 16-18, 2000, Dallas, Texas, USA.

[3] Oulad-Naoui S., Cherroun H., Ziadi D., A Unifying Polynomial Model for Efficient Discovery of Frequent Itemsets,DATA 2015 - Proceedings of 4th Int. Conf. on Data Management Technologies and Applications, Colmar, Alsace,France, 2015

[4] Zaki, M.J., Scalable algorithms for association mining, IEEE Trans. Knowl. Data Eng., Vol. 12, N.3,372-390, 2000.

162


Note on some restricted Stirling numbers of the second kind

Mohammed Said Maamra1∗ and Miloud Mihoubi 2

1 USTHB , RECITS LaboratoryBP 32 El-Alia, 16111, Algiers, Algeria


BP 32 El-Alia, 16111, Algiers, [email protected]

AbstractThe aim of this work is to establish some properties of the coefficients of the chromatic polynomials of

special graphs. An application on (restricted) Stirling numbers of the second kind is considered.

1. Introduction and main results

The object of our investigations in this article is to establish the connection of chromatic polynomials of somegraphs and special (restricted) Stirling numbers of the second kind. We introduce variations of the Stirling numbersof the second kind counting the number of partitions with special conditions and we give the relations of these num-bers to the chromatic polynomials of special graphs and some of their properties. For a given graph G = (V,E) of

order n ≥ 1, the useful representation of the chromatic polynomial ofG used here isP (G,λ) =n∑

i=χ(G)

αi (G) (λ)i,

see [2, Thm. 1.4.1], where αi (G) is the number of ways of partitioning V into i nonempty sets and χ (G) is thechromatic number of G .For any graph H, we give some recurrence relations for the coefficients αk (G ∪H) for some particular cases ofgraphs G and some results on log-concavity and Polya frequency for sequences related to these coefficients. Wealso present an application on special (restricted) Stirling numbers of the second kind.Indeed, let H be any graph of h vertices, h ≥ 1 and (Gn;n ≥ 0) be a sequence of graphs with the order of Gn isn and G0 is the graph with no vertices with P (G0, λ) := 1.

The main results are based on the following theorem.

Theorem 1. Let n, k be nonnegative integers. Assume that

P (Gn, λ) = (λ− sn−1)P (Gn−1, λ) , n ≥ n0,

for some nonnegative integer n0 and real numbers s0, . . . , sn−1 such that s0 = 0.

∗speaker

163


Then, for χ (Gn) ≤ k ≤ n we obtain

αk (Gn) = (k − sn−1)αk (Gn−1) + αk−1 (Gn−1) , n ≥ n0.

To give some special cases of Theorem 1, let On be the empty graph of order n, Kn the complete graph oforder n and Tn the tree of order n.

Proposition 1. Let n, k be integers. Then

a- For χ (H) ≤ k ≤ n+ h we obtain

αk (On ∪H) = kαk (On−1 ∪H) + αk−1 (On−1 ∪H), n ≥ 1.

b- For max (n, χ (H)) ≤ k ≤ n+ h we obtain

αk (Kn ∪H) = (k − n+ 1)αk (Kn−1 ∪H) + αk−1 (Kn−1 ∪H) , n ≥ 1.

c- For max (χ (Tn) , χ (H)) ≤ k ≤ n+ h we obtain

αk (Tn ∪H) = (k − 1)αk (Tn−1 ∪H) + αk−1 (Tn−1 ∪H), n ≥ 2.

Some properties of the sequences defined in Proposition 1 are given by the following proposition.

Proposition 2. For n, k ≥ 0, let (U (n, k))), (V (n, k))) and (W (n, k))) be sequences of nonnegative numberswith U (n, k) = V (n, k) = W (n, k) = 0 when k > n and for 0 ≤ k ≤ n, these sequences are defined by

U (n, k) = αk+h (On ∪H) , V (n, k) = αk+1+h (Tn+1 ∪H) , W (n, k) = αk+h (Kn ∪H)

and let

Un (q) =∑n

k=0 U (n, k) qk and Vn (q) =∑n

k=0 V (n, k) qk.

Then, the sequences (U (n, k) , 0 ≤ k ≤ n) and (V (n, k) , 0 ≤ k ≤ n) are log-concave and Polya frequencysequences, the sequences of polynomials (Un(q)) and (Vn(q)) are q-log-convex sequences and the sequence(W (n, k) , 0 ≤ k ≤ n) is a Polya frequency sequence.

As it is known, the Stirling number of the second kindnk

and the r-Stirling number of the second kind

nk

r

count, respectively, the number of partitions of an n-set into k nonempty sets and the number of partitions of ann-set into k nonempty sets such that the r first elements are in different sets. Below, a restriction of these numbersis considered.

Indeed, for any integer p ≥ 1, we set rp := (r1, . . . , rp) , |rp| :=∑p

i=1 ri and consider the graph Kn,rp =Kr1,...,rp ∪On with chromatic number χ

(Kn,rp

)= max (p, 1) = p. We start by giving the following definition.

Let R1, . . . , Rp be subsets of the set [n] with |Ri| = ri and Ri ∩ Rj = Ø for all i, j = 1, . . . , p, i 6= j. TheK (r1, . . . , rp)-Stirling number of the second kind, denoted by

nk

K(rp)

:=nk

K(r1,...,rp)

, counts the number ofpartitions of the set [n] into k nonempty subsets such that if x ∈ Ri and y ∈ Rj with i 6= j, then x and y belong todifferent subsets of the partition.

164


From this definition, we may state the following:

nk

K(r1,...,rp)

= 0, if n < |rp| or k < p,nk

K(r1,...,rp)

=nk

|rp| if r1, . . . , rp ∈ 0, 1 ,

nk

K(r1,...,rp)

=nk

K(rσ(1),...,rσ(p)) for all permutations σ on the set [p].

Furthermore, for ri ≥ 1, i = 1, . . . , p, the coefficient αk(Kn,rp

)represents the number of ways of partition-

ing the set of n + |rp| vertices of Kn,rp into k nonempty sets, and by the definition of the graph Kn,rp , any twoelements x of the i-th block of the subgraph Kr1,...,rp and y of the j-th block of Kr1,...,rp , with i 6= j, can’t be inthe same subset. So, αk

(Kn,rp

)=n+|rp|

k

K(rp)

.

Other interesting applications can be taken on the graphs Kr1∪ · · · ∪ Krp ∪ On and Tr1 ∪ · · · ∪ Trp ∪ On.

The application on the graph Kr1∪ · · · ∪ Krp ∪ On gives the (r1, . . . , rp)-Stirling numbers of the second kind

introduced by Mihoubi et al. in [4, 3].

References

[1] A. Z. Broder, The r-Stirling numbers. Discrete Math. 49 (1984) 241–259.

[2] F. M. Dong, K. M. Koh and K. L. Teo, Chromatic Polynomials and Chromaticity of Graphs. World Scientific,British library, (2005).

[3] M. S. Maamra and M. Mihoubi, The (r1, . . . , rp)-Bell polynomials. Integers 14 (2014) Article A34.

[4] M. Mihoubi and M. S. Maamra, The (r1, . . . , rp)-Stirling numbers of the second kind. Integers 12 (2012)Article A35.

165


The Generalized r-Lah Numbers Revisited

Imad Eddine Bousbaa1∗, Hacene Belbachir2, Amine Blekhir3 ,

1 USTHB, Faculty of Mathematics, RECITS Laboratory, DG-RSDTPo. Box 32, El Alia, 16111, Algiers, Algeria

[email protected] USTHB, Faculty of Mathematics, RECITS Laboratory, DG-RSDT

Po. Box 32, El Alia, 16111, Algiers, [email protected]

3 USTHB, Faculty of Mathematics, RECITS Laboratory, DG-RSDTPo. Box 32, El Alia, 16111, Algiers, Algeria

[email protected]

AbstractWe provide a new combinatorial interpretation of the two-parameter polynomial generalization of r-Lah

numbers using a weighted ordered partitions approach. Moreover, using combinatorial interpretation basedon inclusion-exclusion principal, we give an explicit formula and some combinatorial properties. Finally, weprovide a exponential generating function.

Key words: Generalized Stirling numbers, Inclusion-Exclusion principal, r-Lah numbers, combinatorial interpretation.

2010 Mathematics Subject Classification: 05A19, 11B37, 11B83, 11B75.

1. IntroductionThe r-Lah numbers, denoted

⌊nk

⌋r, are coefficients in the expression of the shifted raising and falling factorial of x, see [1],

(x+ r)n =n∑

k=0

⌊n+ r

k + r

⌋

r

(x− r)k, (1)

with xn = x(x+ 1)(x+ 2) · · · (x+ n− 1), xn = x(x− 1)(x− 2) · · · (x− n+ 1).The r-Lah number

⌊nk

⌋r

is interpreted combinatorially as the number of ways to partition a set of n elements into knonempty lists such that r elements must be in distinct lists.They satisfy the following recurrence relation

⌊n

k

⌋

r

=⌊n− 1k − 1

⌋

r

+ (n+ k − 1)⌊n− 1k

⌋

r

, (2)

with initial values⌊nk

⌋r

= δk,r for n = r and⌊nk

⌋r

= 0 for n < k or k < r.Many properties of the r-Lah numbers can be found in [1, 3, 7]. Recently, Shattuck [6] considered a two-parameter

polynomial generalization of the r-Lah numbers, denoted by Gα,β(n, k; r) and defined, for 1 ≤ k ≤ n+ 1, by the recurrencerelation

Gα,β(n+ 1, k; r) = Gα,β(n, k − 1; r) + (αn+ βk + (α+ β)r)Gα,β(n, k; r), (3)

∗speaker166


with initial values Gα,β(0, k; r) = δk,0 and Gα,β(n, 0; r) =∏n−1j=0 (α(j + r) + βr).

Note that, when α = β = 1 we obtain the r-Lah numbers and for r = 0, the numbers Gα,β(n, k; 0) are reduced tothe numbers

⌊nk

⌋α,βstudied in [2]. Also, for (α, β, r) = (α, 0, r), (0, α, r) and (α, α, r) we obtain the translated r-Whitney

numbers of three kinds respectively [2].Our aim is to give a simple combinatorial interpretation of Gα,β(n, k; r) which allows us to provide proofs of some

combinatorial relations given in [6]. Furthermore, we propose an explicit formula and some other combinatorial identities.Finally, the exponential generating function is established.

2. Main resultsLet [n] = 1, 2, . . . , n, [k, n] = k, k+ 1, . . . , n and Ωr(n, k) be the set of all possible ways to distribute the set of [n+ r]into k + r ordered lists, one element at a time, such that

• r distinguished elements have to be in distinct ordered lists,

• the first element putted in the list have a weight 1,

• we assign a weight of β to the element inserted as head list,

• we assign a weight of α to the element inserted after an other one.

We will use the notation ” ·/ · / · · · /· ” to represent a distribution of elements into ordered lists. Given a distributionε ∈ Ωr(n, k), we define the weight of ε, denoted by w (ε), to be the product of the weights of its elements. Since the totalweight of Ωr(n, k) is given by the sum of weights of all distributions.

Theorem 1. For any non-negative integers n, k, we have

Gα,β(n, k; r) =∑

ε∈Ωr(n,k)

ω(ε). (4)

The explicit formula of the generalized r-Lah numbers is expressed as follows

Theorem 2. For any n ≥ k ≥ r, we have

Gα,β(n, k; r) =1

βk(k + r)!

k+r∑

j=0

(−1)j(k + r

j

)(k + r − j)r (β(k + r − j) + αr|α) n. (5)

The generalized r-Lah numbers are linked to the symmetric functions, they satisfy

Theorem 3. We have

Gα,β(n+ k, n; r) =∑

r≤i1≤···≤ik≤n+r

k∏

j=1

((α+ β)ij + α(j − 1)). (6)

The generalized r-Lah numbers have the following exponential generating function.

Theorem 4. For αβ 6= 0, we have

∑

n≥kGα,β(n, k; r)

xn

n!=

1k!

(1− αx)−(α+β)r/α

((1− αx)−β/α − 1

β

)k. (7)

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References[1] H. BELBACHIR, A. BELKHIR: Cross recurrence relations for r-Lah numbers. Ars Combin., 110 (2013), 199–203.

[2] H. BELBACHIR, A. BELKHIR, I. E. BOUSBAA: Combinatorial approach of certain generalized Stirling numbers.arXiv:1411.6271.

[3] H. BELBACHIR, I. E. BOUSBAA: Combinatorial identities for the r-Lah numbers. Ars Combin., 110 (2014)

[4] H. BELBACHIR, I. E. BOUSBAA: Translated Whitney and r-Whitney numbers: A combinatorial approach. J. of integerSeq., 16:13.8.6 (2013)

[5] L. C. HSU, P. J. -S. SHIUE: A unified approach to generalized Stirling numbers. Adv. in Appl. Math. 20 (1998),366–384.

[6] M. SHATTUCK: Generalized r-Lah numbers. arXiv:1412.8721.

[7] G. NYUL, G. RACZ: The r-Lah numbers. Discrete Math. (2014), http://dx.doi.org/10.1016/j.disc.2014.03.029 (inpress).

168


An identity on pairs of Appell-type polynomials

Yamina Saidi1∗ and Miloud Mihoubi2

1 USTHB, RECITSPB 32 El Alia 16111 Algiers, Algeria

y [email protected] USTHB, RECITS

PB 32 El Alia 16111 Algiers, [email protected]

Abstract

In this paper, we define a sequence of polynomials P (α)n (x | A,H) depending only on the choice of two

analytic functions A and H in a neighborhood of zero. For a pair of compositional inverses A and B, we willshow the identity P (α)

n (x | B,H B) = P(n+1−α)n (1− x | A,A′H) , which generalize the Carlitz’s identity

on Bernoulli polynomials.

Key words: Polynomial pairs, The higher order Bernoulli polynomials, Recurence relations.


1. IntroductionLet A and H be two analytic functions around zero with A (0) = 0, A′ (0) 6= 0 and let P (α)

n (A,H) be defined by(

t

A (t)

)αH (t) =

∑

n≥0

P (α)n (A,H)

tn

n!. (1)

This definition is motivated by the work of Tempesta [4] on the generalized higher-order Bernoulli polynomials. The higher-order Bernoulli polynomials of the first kind B(α)

n (x) [2] correspond to the choice A (t) = exp (t) − 1, H (t) = exp (xt)and the higher order Bernoulli polynomials of the second kind b(α)

n (x) [3] correspond to the choice A (t) = ln (1 + t) ,H (t) = (1 + t)x .In this paper, we show that the Carlitz’s identity B(n+1−α)

n (x) = n!b(α)n (x− 1) [1, Eqs. (2.11), (2.12)] can be generalized

to a large class of polynomials P (α)n (x | A,H) introduced below.

∗speaker

169


2. Main resultsTheorem 1. Let H, A, B be analytic functions around zero with (A B) (t) = (B A) (t) = t. Then

P (α)n (B,H B) = P (n+1−α)

n (A,A′H) ,

where (A′H) (z) := H (z)DzA (z) and Dz = ddz .

To present some applications of Theorem 1, we give the following definition.

Definition 2. Let A and H be two analytic functions around zero with A (0) = 0, A′ (0) 6= 0 and let α, x be real numbers.

A sequence of polynomials(P

(α)n (x | A,H)

)is said to be of Appell type if

(t

A (t)

)α(A′ (t))xH (t) =

∑

n≥0

P (α)n (x | A,H)

tn

n!.

When we replace H (t) by (A′ (t))xH (t) in Theorem 1 we get

Corollary 2.1. Let α, x be real numbers and let A, B be analytic functions around zero with (A B) (t) = (B A) (t) = t.Then, it holds

P (α)n (x | B,H B) = P (n+1−α)

n (1− x | A,H) , n ≥ 1.

In particular, for H (t) = 1 in Corollary 2.1, the polynomials P (α)n (x | A) defined by

(t

A (t)

)α(A′ (t))x =

∑

n≥0

P (α)n (x | A)

tn

n!(2)

satisfyP (α)n (x | B) = P (n+1−α)

n (1− x | A) , (3)

and for H (t) =(

tB(t)

)β(B′ (t))y in Corollary 2.1 with β, y real numbers, the polynomials P (α,β)

n (x, y | A,B) definedby (

t

A (t)

)α(A′ (t))x

(t

B (t)

)β(B′ (t))y =

∑

n≥0

P (α,β)n (x, y | A,B)

tn

n!

satisfyP (α,β)n (x, y | A,B) = P (n+1−β,−α)

n (1− y,−x | A,B) = P (−β,n+1−α)n (−y, 1− x | A,B) .

3. Connection to the partial Bell polynomialsLet Bn,k (x1, x2, . . .) := Bn,k (xj) be the partial Bell polynomials defined by

∑

n≥kBn,k (x1, x2, . . .)

tn

n!=

1k!

∑

j≥1

xjtj

j!

k

.

Proposition 1. The sequence(P

(α)k (A,H)

)satisfies the following recurrence relations

P (n+1+α)n (A,A′H) =

n∑

k=0

Bn,k (b1, b2, . . .)P(α)k (A,H) ,

P (α)n (A,A′H) =

n∑

k=0

(n

k

)an−k+1P

(α)k (A,H) ,

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where the sequences (an;n ≥ 1) and (bn;n ≥ 1) are defined by

∑

n≥1

antn

n!= A (t) and

∑

n≥1

bntn

n!= B (t) , a1b1 = 1.

For H (t) = 1 in Proposition 1 we get

Corollary 2.2. Let A be an analytic function around zero with A (0) = 0, A′ (0) 6= 0 and(

t

A (t)

)α(A′ (t))x =

∑

n≥0

P (α)n (x | A)

tn

n!.

Then, if we denote byB for the compositional inverse ofA, the sequence of polynomials(P

(α)n (x | A)

)satisfies the following

recurrence relations

P (n+1+α)n (x+ 1 | A) =

n∑

k=0

Bn,k (b1, b2, . . .)P(α)k (x | A) ,

P (α)n (x+ 1 | A) =

n∑

k=0

(n

k

)an−k−1P

(α)k (x | A) .

References[1] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers. Util. Math., 1979, 15, 51–88.

[2] D.C. Kurtz, A note on concavity properties of triangular arrays of numbers. J. Combin. Theory Ser., 1972, A 13, 135–139.

[3] S. Roman, The umbral calculus. Dover Publ. Inc. New York, 2005.

[4] P. Tempesta, On Appell sequences of polynomials of Bernoulli and Euler type. J. Math. Anal. Appl., 2008, 341, 1295–1310.

171


Generalized Geometric Polynomials and Applications to Series withZeta Values

Ayhan Dil1∗, Khristo N. Boyadzhiev2

1Middle East Technical University,Department of Mathematics, 06531 Ankara, Turkey

[email protected] Northern University,

Department of Mathematics and Statistics, Ada, Ohio 45810,[email protected]

Abstract

We provide several properties of the general geometric polynomials. Further, the general geometric poly-nomials are used to obtain closed form evaluations of binomial series, Riemann’s zeta function, Hurwitz zetafunction and Lerch Trancendent.

Key words: Geometric polynomials, binomial series, Hurwitz zeta function, Riemann zeta function, Lerch Transcendent.

2010 Mathematics Subject Classification: 11B83, 11M35, 33B99, 40A25.

1. Introduction

Let S(n, k) be the Stirling numbers of the second kind. The geometric polynomials

ωn(x) =n∑

k=0

S(n, k)k!xk

were discussed and used in [3, 4, 5, 6, 7]. The polynomials ωn can be extended to a more general form dependingon a parameter

ωn,r(x) =1

Γ(r)

n∑

k=0

S(n, k)Γ(k + r)xk (1)

for every r > 0, where ωn,1 = ωn . The polynomials ωn,r(x) have the property(xd

dx

)m 1(1− x)r+1

=∞∑

k=0

(k + r

k

)kmxk

=1

(1− x)r+1ωm,r+1

(x

1− x

)(2)

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for any m, r = 0, 1, 2, ... and also participate in the series transformation formula

∞∑

k=0

(k + r

k

)f(k)xk =

1(1− x)r+1

∞∑

m=0

f (m)(0)m!

ωm,r+1

(x

1− x

)

for appropriate entire functions f(z), see [3].On the other hand, considering the Pochhammer symbol

(x)n = x (x+ 1) . . . (x+ n− 1) =Γ (x+ n)

Γ (x)

we have

wn,r (x) =n∑

k=0

S (n, k) (r)k xk. (3)

The purpose of this talk is to present further properties and applications of the polynomials ωn,r. To provesome of these properties we shall use the close relationship of ωn,r to the exponential polynomials

ϕn(x) =n∑

k=0

S(n, k)xk

which were studied in [2, 3, 5]. The new properties are collected in Section 2, while the applications are in Section3. In Section 3 we evaluate in closed form several series with zeta values. For example, the following series isevaluated in closed form (for any |x| < 2 and for arbitrary integers r ≥ 0, p > 0 )

∞∑

n=1

(n+ r

n

)np ζ(n+ 1)− 1xn.

References

[1] Bruce Berndt, Ramanujan’s Notebooks, vol. V, Springer, Berlin, 1998

[2] Khristo N. Boyadzhiev, Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals,Abstract and Applied Analysis, Volume 2009 (2009), Article ID 168672.

[3] Khristo N. Boyadzhiev, A series transformation formula and related polynomials, Int. J. Math. Math. Sci.2005:23 (2005), 3849-3866.

[4] Khristo N. Boyadzhiev, Power Series with Binomial Sums and Asymptotic Expansions, Int. Journal of Math.Analysis,Vol. 8 (2014), no. 28, 1389-1414.

[5] Ayhan Dil, Veli Kurt, Investigating geometric and exponential polynomials with Euler-Seidel matrices, J.Integer Seq. 14 (2011), no. 4, Article 11.4.6.

[6] Ayhan Dil, Veli Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series I,Integers 12 (2012), Paper No. A38, 18 pp. 11B73.

[7] Ayhan Dil, Veli Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number seriesII, Appl. Anal. Discrete Math.5 (2011) 212-229.

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[8] Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics, Addison-Wesley Publ. Co.,New York, 1994.

[9] Fritz Oberhettinger, Tables of Mellin Transforms, Springer, Berlin. 1974.

[10] Aleksandar Petojevic, A Note about the Pochhammer Symbol, Math. Morav. Vol. 12 -1 (2008), 37–42.

174


Existence and Compactness Results for System of DifferenceEquations

Abdelghani Ouahab

Laboratoire de mathematiques (LDM), University of Sidi Bel-AbbesP.O. Box 89, 22000 Sidi Bel-Abbes, Algeria

agh [email protected]

Abstract

In this work we present some existences and uniqueness results and compactness of some class ofsystems of difference equations. The approach is based on a fixed point theory in generalized metricspace in Prevo sense.

Key words and phrases: Difference equation, fixed point, generalized metric space, solution set, compact-ness.AMS (MOS) Subject Classifications: 34A37, 39A10.

1. Introduction

Let R and N be the sets of real numbers and integers, respectively. For N(a, b) with a < b, define N(a, b) =a, a+ 1, . . . b+ 1. We consider the following system of difference equation:

∆x(k) = f(k, x(k), y(k)), k ∈ N(a, b− 1) = a, a+ 1, . . . , b− 1,∆y(k) = g(k, x(k), y(k)), k ∈ N(a, b− 1),

x(a) = x0,y(a) = y0,

(1.1)

where f, g : N(a, b− 1)× Rm → Rm are given functions.Due to the wide application in many fields such as astrophysics, gas dynamics, fluid mechanics, rel-

ativistic mechanics, nuclear physics and chemically reacting systems in terms of various special forms off(k, x, y), etc., the theory of difference equations has been widely studied since 90’s of the last century (see,for example, [1]). Also, in recent years, there are many other literature dealing with various boundary valueconditions of difference equations. We refer to [3, 5] and references therein. However, we note that theseresults were usually obtained by analytic techniques and various fixed point theorems. For example, the up-per and lower solution method, the conical shell fixed point theorems, the Brouwer and Schauder fixed pointtheorems and topological degree theory (see, for example [1, 2] and the references therein. Our purpose inthis paper is to show that the existence of solutions of the impulsive differential inclusions considered is aproblem equivalent to some abstract equations in generalized Banach space. The goal of this paper is tosolve some a class of system of difference equations by using fixed point theory.

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2. Generalized metric and Banach spaces

In this section we define vector metric spaces and generalized Banach spaces and prove some properties. If,x, y ∈ Rn, x = (x1, . . . , xn), y = (y1, . . . , yn), by x ≤ y we mean xi ≤ yi for all i = 1, . . . , n. Also|x| = (|x1|, . . . , |xn|) and max(x, y) = max(max(x1, y1), . . . ,max(xn, yn)). If c ∈ R, then x ≤ c meansxi ≤ c for each i = 1, . . . , n. For x ∈ Rn, (x)i = xi, i = 1, . . . , n.

Definition 1. Let X be a nonempty set. By a vector-valued metric on X we mean a map d : X ×X → Rn

with he following properties:

(i) d(u, v) ≥ 0 for all u, v ∈ X; if d(u, v) = 0 then u = v

(ii) d(u, v) = d(v, u) for all u, v ∈ X

(iii) d(u, v) ≤ d(u,w) + d(w, v) for all u, v, w ∈ X.

Note that for any i ∈ 1, . . . , n (d(u, v))i = di(u, v) is a metric space in X.

We call the pair (X, d) generalized metric space .For r = (r1, r2, . . . , rn) ∈ Rn+,we will denote by

B(x0, r) = x ∈ X : d(x0, x) < r

the open ball centrad in x0 with radius r and

B(x0, r) = x ∈ X : d(x0, x) ≤ r

the closed ball centered in x0 with radius r.

Definition 2. Let E be a vector space on K = R or C. By a vector-valued norm on E we mean a map‖ · ‖ : E → Rn

+ with the following properties:

(i) ‖x‖ ≥ 0 for all x ∈ E ; if ‖x‖ = 0 then x = 0

(ii) ‖λx‖ = |λ|‖x‖ for all x ∈ E and λ ∈ K

(iii) ‖x+ y‖ ≤ ‖x‖+ ‖y‖ for all x, y ∈ E.

The pair (E, ‖ · ‖) is called a generalized normed space. If the generalized metric generated by ‖ · ‖ (i.ed(x, y) = ‖x− y‖) is complete then the space (E, ‖ · ‖) is called a generalized Banach space, where

‖x− y‖ =

‖x− y‖1. . .

‖x− y‖n

.

Definition 3. A square matrix of real matrix of real numbers is said to be convergent to zero if and only ifits spectral radius ρ(M) is strictly less than 1 ,In other words ,this means that all the eigenvalues of M arein the open unit disc.

176


3. Fixed point results

The classical Banach contraction principle was extended for contractive maps on spaces endowed withvector-valued metric space by Perov in 1964 [7], Perov and Kibenko [6] and Precup [6]. For a version ofSchauder fixed point, see Cristescu [4]. The purpose of this section is to present the version of Schaefer’sfixed point theorem and nonlinear alternative of Leary-Schauder type in generalized Banach spaces.

Theorem 4. [7]Let (X, d) be a complete generalized metric space with d : X × X −→ Rn and letN : X −→ X be such that

d(N(x), N(y)) ≤Md(x, y)

for all x, y ∈ X and some square matrix M of nonnegative numbers. If the matrix M is convergent to zero,that is Mk −→ 0 as k −→∞, then N has a unique fixed point x∗ ∈ X

d(Nk(x0), x∗) ≤Mk(I −M)−1d(N(x0), x0)

for every x0 ∈ X and k ≥ 1.

Theorem 5. [4] Let E be a generalized Banach space, C ⊂ E be a nonempty closed convex subset of Eand N : C → C be a continuous operator with relatively compact range. Then N has at least fixed point inC.

As a consequence of Schauder fixed point theorem we present the version of Schaefer’s fixed pointtheorem.

Theorem 6. Let (E, ‖ · ‖) be a generalized Banach space and N : E → E is a continuous compactmapping. Moreover assume that the set

A = x ∈ E : x = λN(x) for some λ ∈ (0, 1)

is bounded. Then N has a fixed point.

4. Existence and Uniqueness Result

Let us introduce the following hypothesis:

(H1) There exist nonnegative numbers ai and bi for each i ∈ 1, 2|f(k, x, y)− f(k, x, y)| ≤ a1|x− x|+ b1|y − y||g(k, x, y)− g(k, x, y)| ≤ a2|x− x|+ b2|y − y|

for all x, y, x, y ∈ Rm.

For our main consideration of Problem (1.1), a Preov fixed point is used to investigate the existence anduniqueness of solutions for system of impulsive stochastic differential equations.

Theorem 7. Assume that (H1) is satisfied and the matrix

M = (b− 1)(a1 b1a2 b2

)∈M2×2(R+).

If M converges to zero. Then the problem (1.1) has unique solution.

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In ordinary differential equations the Arzela-Ascoli theorem plays an important role. In this section wegive the discrete version of the Arzela-Ascoli theorem. The topology on N(0, b + 1) will be the discretetopology. Let (E, | · |) be a Banach space, we denote the space of continuous functions on N(0, b+ 1) by

C(N(a, b− 1), E) = y : N(a, b− 1)→ E, is continuous

with norm‖y‖∞ = sup

k∈N(0,b+1)|y(k)|

is Banach space. Now we set and prove the discrete Arzela-Ascoli Theorem.

Theorem 8. Let A be a closed subset of C(N(0, b+ 1), E). If Ω is uniformly bounded and the set

y(k) : y ∈ Ω

is relatively compact for each k ∈ N(0, b+ 1), then Ω is compact.

We shall also need the following existence principles.

Theorem 9. Let f, g : N(a, b− 1)× Rn × Rn −→ Rn are continuous functions. Assume that condition

(H5) There exist p1, p2 ∈ C(N(a, b− 1),R+) such that

|f(k, x, y)| ≤ p1(k)(|x|+ |y|), k ∈ N(a, b− 1), (x, y) ∈ Rn × Rn,

and|g(k, x, y)| ≤ p2(k)(|x|+ |y|), k ∈ N(a, b− 1), (x, y) ∈ Rn × Rn.

holds. Then the problem (1.1) has at least one solution. Moreover, the solution set S(x0, y0) is compact andthe multivalued map S : (x0, y0)( S(x0, y0) is u.s.c.

Now we shall consider the discrete system (1.1) where f, g : N(a) × Rn × Rn → Rn are continuousfunctions. For the second result we shall also use the following discrete version of Avramescu criteria ofcompactness in BC(N,Rn).

Theorem 10. A set Ω ⊂ BC(N, E) is relatively compact if the following conditions hold:

(i) for every k ∈ N the set y(k) : x ∈ Ω is relatively compact in E,

(ii) the functions from Ω are equiconvergent at infinity, i.e. for every ε > 0 there exists k(ε) ∈ N such thatif |y(k)− y(∞)| ≤ ε for all k > kε and y ∈ Ω.

Now we consider the following Cauchy problem with parameter

∆x(k) = f(k, x(k), y(k), α), k ∈ N(a),∆y(k) = g(k, x(k), y(k), α), k ∈ N(a),

x(a) = x0,y(a) = y0,

(4.1)

where α ∈ Rm is a parameter such that |α−α0| ≤ δ and α0 is a fixed vector in Rm and f, g : N(a)×Rn×Rn × Rm → Rn are given functions.

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Theorem 11. Let Let f, g : N(a, b− 1)× Rn × Rn −→ Rn are continuous functions satisfies

(H6) There exist p1, p2 ∈ C(N(a, b− 1),R+) such that

|f(k, x, y)| ≤ p1(k)(|x|+ |y|), k ∈ N(a, b− 1), (x, y) ∈ Rn × Rn,

and|g(k, x, y)| ≤ p2(k)(|x|+ |y|), k ∈ N(a, b− 1), (x, y) ∈ Rn × Rn.

with ∞∑

l=a

(p1(k) + p2(k)) <∞.

Then problem (4.1) has at least one solution. Moreover, the solution set S(x0, y0) is compact and themultivalued map S : (x0, y0)( S(x0, y0) is u.s.c.

References

[1] R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods, and Applications, Secondedition. Monographs and Textbooks in Pure and Applied Mathematics, 228. Marcel Dekker, Inc., NewYork, 2000. Ann. Math. 43 (1942) 730–738.

[2] R.P. Agarwal, D. O’Regan and P.J.Y. Wong, Positive solutions of differential, difference and integralequations Kluwer Academic Publishers, Dordrecht, 1999

[3] D. Bai, Y. Xu, Nontrivial solutions of boundary value problems of second-order difference equations,J. Math. Anal. Appl. 326 (1) (2007) 297302.

[4] R. Cristescu, Order Structures in Normed Vector Spaces, Editura Stiin tifica si Enciclopedica,Bucuresti, 1983 (in Romanian).

[5] X. He and X. Wu, Existence and multiplicity of solutions for nonlinear second order difference bound-ary value problems, Comput. Math. Appl. 57 (2009), 1-8.

[6] R. Precup, Methods in Nonlinear Integral Equations, Kluwer, Dordrecht, 2000.

[7] A.I. Perov, On the Cauchy problem for a system of ordinary differential equations, Pviblizhen. Met.Reshen. Differ. Uvavn., 2, (1964), 115-134. (in Russian).

179


Associated Stirling numbers for some families of graphs

Assia Medjerredine1 ∗, Hacene Belbachir2

1,2 Faculty of Mathematics,USTHB, RECITS LaboratoryBP 32 El Alia, 1611 Bab Ezzouar, Algeria

[email protected]@gmail.com

Abstract

The Stirling number of the second kind

Gk

for a graph G is the number of independent partitions of V (G)

into k subsets, it can be graph-theoretically interpreted as the number of non isomorphic colorings of G withexactly k colors. The Bell number BG for a graph G is the number of unordered independent partitions ofV (G). We have already established a convolution sum and interpreted some identities and recurrence relationsrelated to these numbers for some families of graphs. As an extension of our work we introduce the graphical

s-associated Stirling numbers of the second kind

Gk

(s)that count the number of independent partitions of

V (G) into k subsets such that each subset contains at least s elements and we investigate these numbers forother classes of graphs.

Key words: Stirling numbers, Bell numbers, Associated Stirling numbers, Independent partitions.

1. IntroductionLet G be a simple (finite) graph. A partition of V (G) is called an independent partition if each block is an independent vertexset (i.e. adjacent vertices belong to distinct blocks). Then for a positive integer k ≤ |V (G)|, let the Stirling number of thesecond kind

Gk

for graph G be the number of independent partitions of V (G) into k subsets, moreover let

G0

= 0, and

define the Bell number BG for graph G as the number of independent partitions of V (G). Then

BG =|V (G)|∑

k=0

G

k

.

It’s known that for any simple graph G,

Gk

= 0 if 0 ≤ k ≤ χ(G) − 1, where χ(G) is the chromatic number of G.

Moreover

G|V (G)|

= 1 and

G

|V (G)|−1

= |V (G)|(|V (G)| − 1)/2− |E(G)|.

Stirling and Bell numbers can be computed using a recurrence relation on graphs (see Theorem 1 by Duncan and Peele[2]).

Theorem 1. If G is a simple graph, e ∈ E(G) and 0 ≤ k ≤ |V (G)| − 1, then

Gk

=

G−ek

−

G/ek

and BG = BG−e −

BG/e, where G− e and G/e are the simplified graphs obtained by deleting and contracting edge e from G, respectively.

Stirling and Bell numbers of graphs were defined by Duncan and Peele [3], Kereskenyi-Balogh and Nyul [5] summarizedknown properties related to these numbers. Duncan [2] described these numbers for graphs having two components.

∗Assia Medjerredine180


2. Main resultsFor vertex-disjoint graphs G and H , their join G ∨ H is the supergraph of G ∪ H in which each vertex of G is adjacent toevery vertex of H and both G and H are induced subgraphs. Duncan [5] showed that BG∨H = BG.BH . We can see forinstance some join graphs in the figure 1.

Figure 1: Join of some well-known graphs

Theorem 2. Let G and H be simple graphs of order n ≥ 1, p ≥ 1 respectively, with χ(G) + χ(H) ≤ k ≤ n+ p. Then

G ∨Hk

=

k∑

j=0

G

j

H

k − j

.

Corollary 2.1. Let G1, ..., Gr be simple graphs and denote by G1 + ... + Gr their join, with |Gi| = ni and χ(Gi) = χi,

k ∈[

r∑i=1

χi,r∑

i=1

ni

].

Then, G1 + ...+Gr

k

=

∑

k1+...+kr=k

G1

k1

G2

k2

...

Gr

kr

.

Theorem 3. Let us consider γp,k =Gu1,u2,...,up

k

/ Gu1,u2,...,up

denotes the graph G with p pendant vertices with (p ≤ n),|G| = n, χ(G) + p ≤ k ≤ n+ p and add the restriction that each vertex of G is adjacent to at most one pendant vertex. Werefer to this class of graphs as p-pendant graphs and we use the notation G(p). Then,

γp,k =k−1∑

i=0

γ0,k−i

k−1∑

j1+j2+...+ji+1=p−i

(k − 1)j1 ...(k − i+ 1)ji+1 .

The following figure shows some p-pendant graphs.Knowing the Stirling and the Bell numbers of some fewer graphs; En, Kn, Sn, Pn, Pn, Cn, Cn, Km,n that denote

the empty graph, the complete graph, the star graph, the path graph, the complementary of the path graph, the cycle graph,

181


Figure 2: Some p-pendant graphs

the complementary of the cycle graph and the complete bipartite graph, respectively, we compute the Stirling and the Bellnumbers of some join of two special graphs listed above. The application of the last theorem allows us to compute the Stirlingnumbers of the second kind for other families of graphs. Let us now add a restriction to graphical Stirling numbers of thesecond kind on the number of vertices by blocks, so that each block contains at least s elements and we call them for s ≥ 1and |V (G)| ≥ sk, the associated Stirling numbers of the second kind and we use the notation

Gk

(s). So,

Gk

(s)counts

the number of independent partition of V (G) into k subsets such that each subset contains at least s elements. In term ofcoloring, it can be interpreted as the number of ways to color G with exactly k colors such that each color is assigned at leasts times.

It’s clear that for G = En (empty graph with n vertices),

En

k

(s)is the ordinary s− associated Stirling number of the

second kind

nk

(s).

For s = 1, these numbers are reduced to graphical Stirling numbers of the second kind.The number of ways to color a path graph of n vertices Pn with exactly k colors such that each color is assigned at least

twice can be computed using the following recurrence relation, for k ≥ 2 and n ≥ 2k :

Pn

k

(2)

= (n− 2)Pn−2

k − 1

(2)

+ (k − 1)Pn−1

k

(2)

.

Theorem 4. Let Pn be a path graph of order n ≥ 2, for s ≥ 1, k ≥ 2 and n ≥ sk we have,

Pn

k

(s)

=(n− 2s

)Pn−2

k − 1

(s)

+ (k − 1)Pn−1

k

(s)

.

References[1] Belbachir H., Bousbaa I. E., Associated Lah numbers and r-Stirling numbers, (2014), arXiv:1404.5573.

[2] Duncan B., Bell and Stirling numbers for disjoint unions of graphs, Congr. Numer., 206 2010, 215-217.

[3] Duncan B., Peele R., Bell and Stirling numbers for graphs, J. Integer Seq., 12 (2009), 09.7.1.

182


[4] Howard F. T., Associated Stirling numbers, Fibonacci Quart., 18(4):303-315, (1980).

[5] Kereskenyi-Balogh Z., Nyul G., Stirling numbers of the second kind and Bell numbers for graphs, J. Combinatorics.,58 (2) (2014), 264-274.

183


The r-Jacobi-Stirling numbers of the second kind

Miloud Mihoubi1, Asmaa Rahim2∗

1 USTHB, RECITS LaboratoryUSTHB, Faculty of Mathematics, PB 32 El Alia 16111 Algiers, Algeria.


USTHB, Faculty of Mathematics, PB 32 El Alia 16111 Algiers, [email protected]

Abstract

In this paper, we study the r-Jacobi-Stirling numbers of the second kind introduced by Gelineau in his Phdthesis. We give, upon using combinatorial and analytic arguments, the ordinary generating function of thesenumbers, two recurrence relations, their exact expressions and the log-concavity.

Key words: The r-Jacobi-Stirling of the second kind; Generating function; recurrence relations; Log-concavity.

2010 Mathematics Subject Classification:11B75;03E05.

1. IntroductionLet α and β be real numbers such that α > −1 and β > −1. The n-th Jacobi polynomial P (α,β)

n (x) is defined as the uniquepolynomial solution with degree n satisfies the following Jacobi ordinary differential equation of second order:

(1− x2

)y′′ (x) + (β − α− (α+ β + 2)x) y′ (x) + n (n+ α+ β + 1) y (x) = 0 (1)

with inial conditions

P (α,β)n (1) =

(α+ 1n

)and P (α,β)

n (−1) = (−1)n(β + 1n

). (2)

We introduce Jacobi differential operator l(α,β) [y] (x) to be

l(α,β) [y] (x) =1

w(α,β) (x)(−w(α+1,β+1) (x) y′ (x)

)′, w(α,β) (x) = (1− x)α (1 + x)β . (3)

Then, the polynomial solution y = P(α,β)n (x) of (1) satisfies

l(α,β) [y] (x) = n (n+ α+ β + 1) y (x) . (4)

For the n-th composite powers of the Jacobi operator l(α,β) given inductively by

l(1)(α,β) [y] = l(α,β) [y] , l(2)(α,β) [y] = l(α,β)

(l(α,β) [y]

), . . . , l

(n)(α,β) [y] = l(α,β)

(l(n−1)(α,β) [y]

),

∗Speaker184


the authors Everitt-Kwon et al. [3, Theorem 4.2] proved that

l(n)(α,β) [y] (x) =

1w(α,β) (x)

n∑

k=0

(−1)k JS (n, k;α, β)(w(α+k,β+k) (x) y(k) (x)

)(k)

, n ≥ 0, (5)

where the coefficients JS (n, k;α, β) are the Jacobi-Stirling numbers of the second kind.The same authors proved in [3, 4.4] that the Jacobi-Stirling number JS (n, k;α, β) has the following exact expression

JS (n, k;α, β) =1k!

k∑

j=0

(−1)k−j(k

j

)α+ β + 2j + 1

(α+ β + k + j + 1)k+1

(j (j + α+ β + 1))n , (6)

where (x)n := x (x− 1) · · · (x− n+ 1) if n ≥ 1 and (x)0 := 1.We remark that the last expression of JS (n, k;α, β) depends only on α+ β, so if set z := α+ β + 1 and JS (n, k;α, β) :=JS (n, k; z) , the identity given in (6) becomes

JS (n, k; z) =1k!

k∑

j=0

(−1)k−j(k

j

)z + 2j

(z + k + j)k+1

(j (j + z))n , z > −2. (7)

Using different methods, Everitt et al. [3, Theorem 4.1] and Y. Gelineau et al. [4, Section 4.2] showed that

Xn =n∑

k=0

JS (n, k; z) (X)k,z , (8)

where (X)k,z :=∏k−1i=0 (X − i (i+ z)) if k ≥ 1 and (X)0,z := 1.

The equation (8) shows the the Jacobi-Stirling numbers satisfy the following recurrence relation:

JS (0, 0; z) = 1,JS (n, 0; z) = JS (0, k; z) = 0 if n, k ≥ 1,JS (n, k; z) = JS (n− 1, k − 1; z) + k (k + z) JS (n− 1, k; z) if n, k ≥ 1.

To give a combinatorial interpretation of the Jacobi–Stirling number JS (n, k; 2γ − 1) , let [±n] be the set ±1,±2, ...,±n andwe will use definition of the jacobi-Stirling set partition [2, definition 4.1] given as follows.

Definition 1. For all positive integers n, j, γ, a Jacobi–Stirling set partition of [±n] into γ zero blocks A1, . . . , Aγ and jnonzero blocks B1, . . . , Bk is an ordinary set partition of [±n] into j + γ blocks for which the following conditions hold:

(1) The blocks A1, . . . , Aγ , called the zero blocks, are distinguishable, but the blocks B1, . . . , Bk are indistinguishable.(2) The zero blocks may be empty, but all other blocks are nonempty.

(3) ∀i ∈ [n] = 1, 2, . . . , n , −i, i *γ∪j=1

Aj ,

(4) ∀j ∈ [k] , ∀i ∈ [n] , we have −i, i ⊂ Bj ⇔ i = minBj .

Andrews et al. showed that the Jacobi–Stirling number JS (n, k; 2γ − 1) is the number of Jacobi–Stirling set partitions of[±n] into γ zero blocks and j nonzero blocks.[2, Theorem 4.1].In his Phd Thesis, Gelineau [5, Section 1.4] has introduced the r-Jacobi-Stirling numbers of the second kind JSr (n, k; z) ,n ≥ r ≥ 0, as follows:

Definition 2. The r-Jacobi–Stirling number JSr (n, k; 2γ − 1) is the number of Jacobi–Stirling set partitions of [±n] into γzero blocks and k nonzero blocks such that minBj = j for j = 1, . . . , r.

The author gives some interesting properties of these numbers. In this paper, we study the r-Jacobi-Stirling numbers of thesecond kind by giving, with combinatorial and analytic proofs, their ordinary generating function, two recurrence relations,their exact expressions and the log-concavity.

185


2. Main resultsTo start, we give in the following theorem the ordinary generating function.

Theorem 3. For all positive integers n, k, the r-Jacobi–Stirling number JSr (n, k; z) has ordinary generating function tobe

∑

n≥0

JSr (n+ k, k; z) tn =

(k∏

i=r

(1− i (i+ z) t)

)−1

.

Proposition 1. For all positive integers n, k, r, the r-Jacobi–Stirling numbers JSr (n, k; z) satisfy

JSr (n, k; z) = JSr−1 (n, k; z)− (r − 1) (r − 1 + z) JSr−1 (n− 1, k; z) .

Proposition 2. For all positive integers n, k, the r-Jacobi–Stirling numbers JSr (n, k; z) satisfy

JSr (n, k; z) = JSr (n− 1, k − 1; z) + k (k + z) JSr (n− 1, k; z) , 0 ≤ r ≤ k ≤ n, r < n.

Theorem 4. For non-negative integers n, k, r, the r-Jacobi–Stirling numbers have the following expression:

JSr (n+ r, k + r; z) =1k!

k∑

j=0

(−1)k−j(k

j

)2j + 2r + z

(j + k + 2r + z)k+1

((j + r) (j + r + z))n .

In particular, for r = 0 or r = 1, we get

JS (n, k; z) =1k!

k∑

j=0

(−1)k−j(k

j

)2j + z

(j + k + z)k+1

(j (j + z))n .

Corollary 4.1. For non-negative integers n, k, r, the r-Jacobi–Stirling numbers can be expressed in terms of the Jacobi–Stirling numbers as follows:

JSr (n+ r, k + r; z) =n−k∑

i=0

(n

i

)(r (r + z))i JS (n− i, k; z + 2r) .

Corollary 4.2. For non-negative integers n, r, the r-Jacobi–Stirling numbers satisfy

(X + r (r + z))n =n∑

k=0

JSr (n+ r, k + r; z) (X)k,z+2r .

Theorem 5. For all positive integer n and real number z ≥ −1, the polynomial

Pn (x; r) =n∑

k=0

JSr (n+ r, k + r; z)xk

has only real simple non-positive zeros, and then, the r-Jacobi-Stirling numbers of the second kind are strictly log-concave(thus unimodal).

References[1] G.E. Andrews, W. Gawronski, L.L. Littlejohn, The Legendre–Stirling numbers, Discrete Mathematics, 311 (2011)

1255–1272.

[2] G. E. Andrews, E. S. Egge, W. Gawronskic and L. L. Littlejohn, The Jacobi–Stirling numbers, Journal of CombinatorialTheory, Series A, 120 (2013) 288–303.

186


[3] W. N. Everitt, K. H. Kwon, L. L. Littlejohn, R. Wellman and G. J. Yoon, Jacobi-Stirling numbers, Jacobi polynomi-als, and the left-definite analysis of the classical Jacobi differential expression, Journal of Computational and AppliedMathematics, 208 (2007) 29–56.

[4] Y. Gelineau and J. Zeng, Combinatorial interpretations of the Jacobi-Stirling numbers, The electronic journal of combi-natorics 17 (2010), #R70.

[5] Y. Gelineau, Etude combinatoires des nombres de Jacobi-Stirling et d’Entringer, These de Doctorat, Universite ClaudeBernard-Lyon 1 (2010).

[6] G. H. Hardy, J. E. Littlewood, G. Ploya, Inequalities, (Cambridge: The University Press, 1952).

[7] P. Mongelli, Total positivity properties of Jacobi-Stirling numbers, Advances in Applied Mathematics, 48 (2012) 354–364.

187


The Stirling Numbers with constraint

Abdelghani Mehdaoui∗, Hacene Belbachir, Amine Belkhir

U.S.T.H.B, Laboratory RECITS, DG-RSDTBP 32, El Alia 16111 Bab Ezzouar, Algiers, Algeria

[email protected]

[email protected]

[email protected]

Abstract

Using a combinatorial approach, we introduce the Stirling numbers with constraint. We give the recurrencerelation and some identities in bijection with the chromatic polynomial.

Key words: Stirling numbers, Stirling numbers with constraints, Chromatic polynomial.2010 Mathematics Subject Classification: 05A18, 05C15.

1. Introduction

The Stirling numbers of second kind, denoted by S(n, k), are defined as follows

xn =n∑

k=0

S(n, k)x(x− 1) · · · (x− k + 1).

It well known that S(n, k) is the number of partitions of the set 1, 2, · · · , n into k non empty subsets. They satisfy therecurrence relation, for any 1 ≤ k ≤ n

S(n, k) = S(n− 1, k − 1) + kS(n− 1, k),

white initial values S(n, 0) = δn,0. In [5], the authors have introduces a generalization of Stirling numbers, denoted S2(n, k),and defined as the number of partitions of 1, 2, · · · , n into k nonempty subsets such that for any i, j in a given subset wehave |i−j| ≥ d. They give the recurrence relation and some identities and properties. They focused on the case d = 2, wherethey found some properties in bijection with the chromatic polynomial.

Theorem 1. [5] For any n ≥ k ≥ 2

S2(n, k) = S2(n− 1, k − 1) + (k − 1)S2(n− 1, k).

∗Speaker188


Theorem 2. [5] Let Pn denote the path on n-vertices

χ(Pn, k) = k!S2(n, k).

where χ(G, k) denote the chromatic polynomial using exactly k colors.

Our aim is to introduce Stirling numbers with new constraint. We denote these numbers by Sd(n, k) where n ≥ 1, Basedon the interpretation, we derive recurrence relation and we establish combinatorial identities.

2. Main results

We start by introducing the Stirling numbers with constraint.

Definition 1. For a positive integer d, let Sd(n, k) denote the number of partitions of 1,2,· · · ,n into exactly k nonemptysubsets such that for each subset ε we require min|i− j|, n− |i− j| ≥ d, where i,j ∈ ε.

Notice the case d = 1 yields the classical Stirling numbers of the second kind, S(n, k).

If we want to represent all elements in the form of a graph such that each element is represented by a vertex and twovertices are adjacent if it do not match the condition, we obtain:

1

2

3

I

n− 2

n− 1

n

I + 1

We first focus on the numbers S2(n, k), their representation in the form of a graph is:

1

2

3

n

n− 1

n− 2

It satisfy the following recurrence relation:

Theorem 3. For any n ≥ k

S2(n, k) = S2(n− 1, k − 1) + (k − 2) ·S2(n− 1, k) + S2(n− 2, k − 1)

= +(k − 1) ·S2(n− 2, k).

189


n\ k 0 1 2 3 4 5 6 7 80 11 0 12 0 0 13 0 0 0 14 0 0 1 2 15 0 0 0 5 5 16 0 0 1 10 20 9 17 0 0 0 21 70 56 14 18 0 0 1 42 231 294 126 20 1

Table 1: The triangle of the numbers S2(n, k) (0 ≤ k ≤ n ≤ 8).

Theorem 4. Let Cn denote a cycle with length n, we have

χ(Cn, k) = k!S2(n, k).

The explicit formula of S2(n, k) is given by

Theorem 5. For n ≥ 2 and k ≤ n, we have

S2(n, k) =1k!

k−2∑

j=0

(−1)j

(k

j

)[(k − j − 1)n + (−1)n(k − j − 1)] .

References

[1] Z. K. Balogh, N. Gbor, Stirling numbers of the second kind and Bell numbers for graphs. Australas. J. Combin. 58(2014), 264–274.

[2] L. Comtet, Advanced Combinatorics, D. Reidel Publishing Company, 1974.

[3] Donald E. Knuth and O. P. Lossers, Partitions of a Circular Set, The American Mathematical Monthly, 2007, 114(3),265–266.

[4] B. Duncan and R. Peele, Bell and Stirling numbers for graphs, J. Integer Seq. 12 (2009), Article 09.7.1.

[5] A. Mohr, T. D Porter, Applications of Chromatic Polynomials Involving Stirling Numbers, J. Combin. Math. Combin.Comput. 70 (2009), 57–64.

[6] A. O. Munagi, Set partitions with successions and separations, Int. J. Math. Math. Sci. (2005), 451463.

[7] A. O. Munagi, Extended set partitions with successions. European J. Combin. 29 (2008), no. 5, 1298–1308

190


Existence Results for Couple Stochastic Difference Equations withDelay

Tayeb Blouhi∗and Abdelghani Ouahab

Laboratory of Mathematics, Univ Sidi Bel AbbesPoBox 89, 22000 Sidi-Bel-Abbes, [email protected] , agh [email protected]

Abstract

We present some existence and uniqueness and compactness of solution sets for system of stochastic differ-ence . Our approach based on Perov and Schauder type fixed point theorem fixed point in generalized Banachspaces.

Key words: Stochastic difference equations, Matrix convergent, Generalized Banach space, Fixed point.

2010 Mathematics Subject Classification:39A10,60H99,47H10.

1. IntroductionDifference equations usually appear in the investigation of systems with discrete time or in the numerical solution of systemswith continuous time [1].In recent years, the stability investigation of stochastic difference equations has been interesting tomany investigators, and various advanced results on this problem have been reported [2] and [3]. Consider the followinglinear stochastic difference equation

xi+1 = G1(k, xi−h, . . . , xi, yi−h, . . . , yi)ξi, i ∈ Z,yi+1 = G2(k, xi−h, . . . , xi, yi−h, . . . , yi)ξi, i ∈ Z,x(i) = ϕi,

y(i) = ϕ′i,

(1)

This paper is concerned with the existence of solutions for the perturbed of stochastic difference equation of the followingtype:

xi+1 = F 1(k, xi−h, . . . , xi, yi−h, . . . , yi) +G1(k, xi−h, . . . , xi−h, yi−h, . . . , yi)ξi, i ∈ Z,yi+1 = F 2(k, xi−h, . . . , xi, yi−h, . . . , yi) +G2(k, xi−h, . . . , xi−h, yi−h, . . . , yi)ξi, i ∈ Z,x(i) = ϕi, i ∈ Z0,

y(i) = ϕ′i, i ∈ Z0

(2)

has been investigated widely. Here i is a discrete time,i ∈ Z0 ∪ Z, Z0 = −h, . . . , 0, Z = 0, 1, 2, . . ., h is a givennonnegative integral number, the sequence of real numbers xi, yi is a solution of (2), F,G : Z × Rh+1 × Rh+1 → R.Let Ω, P,Fn be a basic probability space, fi ∈ Fi, be a sequence of Fn-algebras, E be the mathematical expectation,ξ0, ξ1, . . . be a sequence of mutually independent random variables, ξi ∈ R, ξi be fi+1-adapted and independent on Fi+1 ,E(ξi) = 0, E(ξi) = 1, i ∈ Z.

∗Speaker191


2. PreliminariesIn this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.

Let (Ω,F ,P) be a complete probability space with a filtration (F = Fn)n∈N0 satisfying the usual conditions (i.e. rightcontinuous and F0 containing all P-null sets). An R-valued random variable is an F-measurable function xk : Ω → R andthe collection of random variables

S = xk(ω) = x(k, ω) : Ω→ R| k ∈ N0is called a stochastic process. Generally, we just write xk instead of xk(ω).Let C = C(N0,Rn) denote the class of maps x continuous on N0 (discrete topology)with the norm

‖x‖ = maxk∈Z

E|x(k)|2.

when the initial condition ϕ = (ϕ−h, . . . , ϕ0)T satisfies ‖ϕ‖2 = maxk∈Z0

E|ϕi|2. Note that C is a Banach space.By a solution

of (1),we mean a pair (x, y) ∈ C × C such that it satisfies problem (1).

3. Generalized metric and Banach spacesIn this section we define vector metric spaces and generalized Banach spaces and prove some properties. If, x, y ∈ Rn,x = (x1, . . . , xn), y = (y1, . . . , yn), by x ≤ y we mean xi ≤ yi for all i = 1, . . . , n. Also |x| = (|x1|, . . . , |xn|)and max(x, y) = max(max(x1, y1), . . . ,max(xn, yn)). If c ∈ R, then x ≤ c means xi ≤ c for each i = 1, . . . , n. Forx ∈ Rn, (x)i = xi, i = 1, . . . , n.

Definition 1. Let X be a nonempty set. By a vector-valued metric on X we mean a map d : X ×X → Rn with he followingproperties:

(i) d(u, v) ≥ 0 for all u, v ∈ X; if d(u, v) = 0 then u = v

(ii) d(u, v) = d(v, u) for all u, v ∈ X(iii) d(u, v) ≤ d(u,w) + d(w, v) for all u, v, w ∈ X.

Note that for any i ∈ 1, . . . , n (d(u, v))i = di(u, v) is a metric space in X.

We call the pair (X, d) generalized metric space .For r = (r1, r2, . . . , rn) ∈ Rn+,we will denote by

B(x0, r) = x ∈ X : d(x0, x) < rthe open ball centrad in x0 with radius r and

B(x0, r) = x ∈ X : d(x0, x) ≤ rthe closed ball centered in x0 with radius r.

Definition 2. Let E be a vector space on K = R or C. By a vector-valued norm on E we mean a map ‖ · ‖ : E → Rn+ with

the following properties:

(i) ‖x‖ ≥ 0 for all x ∈ E ; if ‖x‖ = 0 then x = 0

(ii) ‖λx‖ = |λ|‖x‖ for all x ∈ E and λ ∈ K

(iii) ‖x+ y‖ ≤ ‖x‖+ ‖y‖ for all x, y ∈ E.

The pair (E, ‖·‖) is called a generalized normed space. If the generalized metric generated by ‖·‖ (i.e d(x, y) = ‖x−y‖)is complete then the space (E, ‖ · ‖) is called a generalized Banach space, where

‖x− y‖ =

‖x− y‖1. . .

‖x− y‖n

.

Notice that ‖ · ‖ is a generalized Banach space on E if and only if ‖ · ‖i, i = 1, . . . , n are norms on E.

192


Definition 3. A square matrix of real matrix of real numbers is said to be convergent to zero if and only if its spectral radiusρ(M) is strictly less than 1 ,In other words ,this means that all the eigenvalues of M are in the open unit disc.

Lemma 4. [?] Let M be a square matrix of nonnegative numbers. The following assertions are equivalent:

(i) M is convergent towards zero;

(ii) the matrix I −M is non-singular and

(I −M)−1 = I +M +M2 + . . .+Mk + . . . ;

(iii) ‖λ‖ < 1 for every λ ∈ C with det(M − λI) = 0

(iv) (I −M) is non-singular and (I −M)−1 has nonnegative elements;

Some examples of matrices convergent to zero are the following:

1) A =(a 00 b

), where a, b ∈ R+ and max(a, b) < 1

2) A =(a −c0 b

), where a, b, c ∈ R+ and a+ b < 1, c < 1

3) A =(a −ab −b

), where a, b, c ∈ R+ and |a− b| < 1, a > 1, b > 0.

The classical Banach contraction principle was extended for contractive maps on spaces endowed with vector-valued metricspace by Perov in 1964 [4], Perov and Kibenko [?] and Precup [?]. For a version of Schauder fixed point, see Cristescu[5]. The purpose of this section is to present the version of Schaefer’s fixed point theorem and nonlinear alternative ofLeary-Schauder type in generalized Banach spaces.

Theorem 5. [4]Let (X, d) be a complete generalized metric space with d : X ×X −→ Rn and let N : X −→ X be suchthat

d(N(x), N(y)) ≤Md(x, y)

for all x, y ∈ X and some square matrix M of nonnegative numbers. If the matrix M is convergent to zero, that is Mk −→ 0as k −→∞, then N has a unique fixed point x∗ ∈ X

d(Nk(x0), x∗) ≤Mk(I −M)−1d(N(x0), x0)

for every x0 ∈ X and k ≥ 1.

Theorem 6. [5] Let E be a generalized Banach space, C ⊂ E be a nonempty closed convex subset of E and N : C → C bea continuous operator with relatively compact range. Then N has at least fixed point in C.

4. Existence and uniquenessLet us introduce the following hypothesis:

(H1) There exist nonnegative numbers a(i), a(i) and b(i), b(i) for each i ∈ Z

∣∣∣F 1(k, xi−h, . . . , xi, yi−h, . . . , yi)− F 1(k, xi−h, . . . , xi, yi−h, .., yi)∣∣∣ ≤

h∑

j=0

aj(i)|xi−j − xi−j |

+h∑

j=0

bj(i)|yi−j − yi−j |

∣∣∣F 2(k, xi−h, . . . , xi, yi−h, . . . , yi)− F 2(k, xi−h, . . . , xi, yi−h, . . . , yi)∣∣∣ ≤

h∑

j=0

aj(i)|xi−j − xi−j |

+h∑

j=0

bj(i)|yi−j − yi−j |

193


where a(i) =h∑

j=0

aj(i), a(i) =h∑

j=0

aj(i) and b(i) =h∑

j=0

bj(i), b(i) =h∑

j=0

bj(i) for all x, y, x, y ∈ R.

(H2) There exist positive constants α(i), α(i) and β(i), β(i) for each i ∈ Z

∣∣∣G1(k, xi−h, . . . , xi, yi−h, . . . , yi)−G1(k, xi−h, ..xi, yi−h, . . . , yi)∣∣∣ ≤

h∑

j=0

αj(i)|xi−j − xi−j |

+h∑

j=0

βj(i)|yi−j − yi−j |

∣∣∣G2(k, xi−h, . . . , xi, yi−h, . . . , yi)−G2(k, xi−h, . . . , xi, yi−h, . . . , yi)∣∣∣ ≤

h∑

j=0

αj(i)|xi−j − xi−j |

+h∑

j=0

βj(i)|yi−j − yi−j |

where α(i) =h∑

j=0

αj(i), α(i) =h∑

j=0

αj(i) and β(i) =h∑

j=0

βj(i), β(i) =h∑

j=0

βj(i) for all x, y, x, y ∈ R.

For our main consideration of Problem (1), a Preov fixed point is used to investigate the existence and uniqueness of solutionsfor system stochastic difference equations.

Theorem 7. Assume that (H1)− (H2) are satisfied and the matrix

M =

h∑

j=0

aj(i)a(i) +h∑

j=0

αj(i)α(i)h∑

j=0

bj(i)b(i) +h∑

j=0

βj(i)β(i)

h∑

j=0

aj(i)a(i) +h∑

j=0

αj(i)α(i)h∑

j=0

bj(i)b(i) +h∑

j=0

βj(i)β(i)

,

if M converges to zero. Then the problem (1) has a unique solution.

5. Expectance and compactnessIn this section we present the existence result under a nonlinearity F i and Gi, i = 1, 2 satisfying a Nagumo type growthconditions:

(H)4 There exist a function pl, pl ∈ l1(Z,R+) and nondecreasing function ψi : [0,∞)→ [0,∞) for each l = 1, 2 such taht

E(|F l(k, xi−h, . . . , xi, yi−h, . . . , yi)|)2 ≤h∑

j=0

pl(j)ψ(E|xi−j |2 + E|yi−j |2)

and

E(|Gl(k, xi−h, . . . , xi, yi−h, . . . , yi)|)2 ≤h∑

j=0

pl(j)ψ(E|xi−j |2 + E|yi−j |2)

Theorem 8. Let N(0, b+ 1) = 0, 1, . . . , b+ 1, A be a closed subset of C(N(0, b+ 1), E). If Ω is uniformly bounded andthe set

y(k) : y ∈ Ωis relatively compact for each k ∈ N(0, b+ 1),, then Ω is compact.

Now, we present our result of the section.

Theorem 9. Assume that the condition (H)4 holds. Then the problem (1) has at least one solution. Moreover, the solutionset S(x0, y0) is compact.

194


References[1] V.B. Kolmanovskii, L.E. Shaikhet, Control of systems with aftereffect, in: Translations of Mathematical Monographs,

vol. 157, American Mathematical Society, Providence, RI, 1996.

[2] N. Kuchkina, L. Shaikhet, Optimal control of Volterra type stochastic difference equations, Comput. and Math. Appl.36 (1998), 251-259. 251259.

[3] Goodarz Ahmadi, On the mean square stability of linear difference equations, Appl. Math. Comput. 5 (1979) 233-241.

[4] A.I. Perov, On the Cauchy problem for a system of ordinary differential equations, Pviblizhen. Met. Reshen. Differ.Uvavn., 2, (1964), 115-134. (in Russian).

[5] R. Cristescu, Order Structures in Normed Vector Spaces, Editura Stiin tifica si Enciclopedica, Bucuresti, 1983 (inRomanian).

195


Some asymptotic normality result of k-Nearest Neighbor estimator

Wahiba Bouabsa1∗, Mohammed Kadi Attouch2,

1 Djilali Liabes University, Statistique Stochastique Process LaboratoryBp 89, Sidi Bel Abbs, Algeria

wahiba [email protected] Djilali Liabes University, Statistique Stochastique Process Laboratory

Bp 89, Sidi Bel Abbs, Algeriaattou [email protected]

...

Abstract

In this paper, we study the nonparametric estimator of the conditional mode using the k-Nearest Neighbors(k-NN) estimation method for a scalar response variable given a random variable taking values in a semi-metricspace. We establish the asymptotic normality for independent functional data, and propose confidence bandsfor the conditional mode function. Some simulations for real data application have been driven to show howour methodology can be implemented.

Key words: Functional data, k-NN estimator, the conditional mode estimation, small balls probability, asymptotic normality.


1. IntroductionThe first result of the asymptotic normality on the kernel estimator comes from Masry (2005), he considers the case of α-mixing data but he did not give the explicit expression of the dominant asymptotic terms of bias and variance. After Ferratyet al.(2007) gives the explicit expression of the asymptotic law (that is the dominant terms of bias and variance) In the caseof set of independent data . The results exposed in the papers of Delsol (2007a, 2008b) make the link between these articles,they generalize the results of Ferraty et al.(2007) in the case of data α-mixing, Attouch and Benchikh (2012) established theasymptotic normality of robust nonparametric regression function.For the k-NN conditional mode estimator Attouch and Bouabca (2013) obtained the almost complete convergence with ratesin independent and identically distributed (i.i.d.) functional data case.Let (Xi, Yi), i = 1, .., n be n copies of random vector identically distributed as (X,Y ) where X is valued in infinite dimen-sional semimetric vector space (F ,d) and Y’s are valued in IR, In most practical applications, S is a normed space which canbe of infinite dimension (e.g. Hilbert or Banach space) with norm ‖.‖ so that d(x, x′) = ‖x− x′‖.We will denote the conditional distribution function of Y by:

∀y ∈ IR F x(y) = IP(Y ≤ y|X = x).

Then we will denote by fx (respfx(j)) the conditional density (resp its jth order derivative) of Y .

∗speaker196


The conditional distribution in Ferraty et al.(2006b) is defined as follows:

F x(y) =

n∑

i=1

K(h−1K d(x,Xi))G(h−1

G (y − Yi))

n∑

i=1

K(h−1K d(x,Xi))

∀y ∈ IR. (1)

K is a kernel function, G is a distribution function (df) and h = hK := hK,n and hG = hG,n are sequence of positive realnumbers which goes to zero as n goes to infinity.Under a differentiability assumption of G(.), we can obtain the conditional density estimation function as:

fx(y) =

h−1G

n∑

i=1

K(h−1K d(x,Xi))G(1)(h−1

G (y − Yi))

n∑

i=1

K(h−1K d(x,Xi))

(2)

and its jth partial derivative with respect to y as:

fx(j)(y) =

h−j−1G

n∑

i=1

K(h−1K d(x,Xi))G(j)(h−1

G (y − Yi))

n∑

i=1

K(h−1K d(x,Xi))

(3)

G(j) is the jth order derivative function of G. We assume that fx(.) has a unique mode, denoted by θ(x) which is defined by

fx(θ(x)) = supy∈S

fx(y). (4)

A kernel estimator of the conditional mode θ(x) is defined as the random variable θ(x) which maximizes the kernel estimatorfx(.) of fx(.)

fx(θ(x)) = supy∈S

fx(y). (5)

Let consider quantities

fx(y) =fx

N (y, h)

F xD(h)

. (6)

Where

fxN (y, h) =

1nhG φ(h)

n∑

i=1

K(h−1K d(x,Xi))G(1)(h−1

G (y − Yi)), (7)

F xD(h) =

1n φ(h)

∑

i=1

K(h−1K d(x,Xi)), (8)

where φ(.) is a function supposed to be strictly positive and it will be described later.An analogous estimator to (7) was already given in Ferraty et al.(2006a) in the general setting, remark that (7) and (8) areunbiased estimators of a1f

x(y)F xD(h) = a1f

xN (y, h) and a1F

xD respectively where a1 and F x

D will be described later.The k-Nearest Neighbour estimator is a weighted average of response variables in the neighborhood of x, this estimationmethod take into account the local structure of the data. The k-NN kernel estimate has a significant advantage over theclassical kernel estimate. The main advantage of the k-NN method is in the nature of the smoothing parameter. Indeed, in theclassical kernel method, the smoothing parameter is the bandwidth hn, which is a real positive number and in k-NN method,the smoothing parameter kn takes its values in discrete set.

197


Now we focuses on the estimation of the jth order derivative of the conditional density f on x by the k nearest neighbors(k-NN), where fx(j)

kNN (y) is defined by:

fx(j)kNN (y) =

h−j−1G

n∑

i=1

K(H−1n,kd(x,Xi))G(j+1)(h−1

G (y − Yi))

n∑

i=1

K(H−1n,kd(x,Xi))

. (9)

Hn,k(.) is defined by:

Hn,k(x) = min

h ∈ IR+/

n∑

i=1

IIB(x,h)(Xi) = k

.

Hn,k is a positive random variable (r.v) which depends (X1, ..., Xn).From now on, when we refer to the bandwidth of the k-NN conditional mode estimation, we mean the number of neighboursk we are considering.A natural θkNN (x) is given by:

fx(θkNN ) = supy∈IR

fxkNN (y). (10)

so that

fxkNN (y) =

fxN (y,Hn,k)

F xD(Hn,k)

. (11)

Note that this estimate θkNN (x) is not necessarily unique, so the remainder of the paper concerns any value θkNN (x)satisfying (10).We consider two types of non-parametric models. The first one is called continuity-type, which means that the densityfunction f is continuous and the second one, the Holder-type, which means that the density function f is Holder continuouswith constant b1 , b2 > 0.Our main goal is to establish the asymptotic normality of the estimator in (10) when suitably normalized. As a consequencewe get the asymptotic normality of a predictor and propose confidence bands for the conditional mode function with thek-NN method.

Theorem 1. Assume that (H1) and (H6) hold, then for any x ∈ Υ,we have

(h3

G k (a1 fx(2)(θ(x)))2

σ2(x, θ(x))

)1/2 (θkNN (x)− θ(x)

) D→ N (0, 1) as n→∞. (12)

D→ means the convergence in distribution, Υ = x : x ∈ S, , fx(θ(x)) 6= 0

σ2(x, θ(x)) = a2fx(θ(x))

∫

IR

(G(2)(t)

)2

dt (with aj = Kj(1)−∫ 1

0

(Kj(s))′ζ0(s)ds, for, j = 1, 2). (13)

References[1] Attouch, M., A.Laksaci, E.Ould-Saıd: Asymptotic distribution of robust estimator for functional nonparametric models.

Comm. Statist. Theory and Methods 38 (8), 1317-1335, 2009.

[2] Attouch, M., t.Benchikh: Asymptotic distribution of robust k-nearest neighbour estimator for functional nonparametricmodels. Mathematic Vesnic (64), No. 4, pp. 275-285, 2012.

[3] Attouch M., W.Bouabca: The k-nearest neighbors estimation of the conditional mode for functional data. Rev. RoumaineMath. Pures Appl., (58), No. 4, pp. 393-415, 2013.

198

Conference on Discrete Mathematics and Computer Science

dimacos’2015

Algeria, Sidi Bel Abbes November 15 - 19, 2015.

available online at http://lrecits.usthb.dz/activites.htm

Existence of mild solutions to stochastic delay

evolution equations with a fractional Brownian

motion and impulses

A.Boudaoui1, T.Caraballo2, A.Ouahab1, ...∗

1Laboratory of Mathematics, Univ Sidi Bel AbbesPoBox 89, 22000 Sidi-Bel-Abbes, Algeria.


2 Depto. Ecuaciones Diferenciales y Analisis Numerico,Universidad de Sevilla, Campus de Reina Mercedes

41012-Sevilla, [email protected]

...

Abstract

In this paper, we study the existence of mild solutions to stochastic impulsive evolutionequations with time delays, driven by fractional Brownian motion with the Hurst indexH > 1

2 via a new fixed point analysis approach.

Key words: Fractional Brownian motion, fixed point, mild solutions, stochastic functional differentialequation.

35R60, 60H15, 60G18, 35A01: .

1. Introduction

Stochastic partial functional differential equations with finite delays driven by a fractional Brownianmotion are very important as stochastic models of biological, chemical, physical and economical systems.

In this paper, our main objective is to establish sufficient conditions for the existence of mild solutionsof the following first order stochastic impulsive functional equation with time delays:

dy(t) = [Ay(t) + f(t, yt)]dt+ g(t)dBHQ (t), t ∈ J := [0, T ]; (1)

∗A.Boudaoui

1

y(t+k )− y(tk) = Ik(y(tk)), k = 1, . . . ,m (2)

y(t) = φ(t), for a.e. t ∈ [−r, 0], (3)

in a real separable Hilbert space H with inner product (·, ·) and norm ‖·‖, where A : D(A) ⊂ H −→ H, isthe infinitesimal generator of a strongly continuous semigroup of bounded linear operators U(t), 0 ≤ t ≤ T.BHQ is a fractional Brownian motion on a real and separable Hilbert space K, with Hurst parameterH ∈ (1/2, 1), and with respect to a complete probability space (Ω,F ,Ft, P ) furnished with a family ofright continuous and increasing σ-algebras Ft, t ∈ J satisfying Ft ⊂ F . Also r > 0 is the maximumdelay, and the impulse times tk satisfy 0 = t0 < t1 < t2 < . . . , tm < T . As for yt we mean the segmentsolution which is define in the usual way, that is, if y(·, ·) : [−r, T ] × Ω → H, then for any t ≥ 0,yt(·, ·) : [−r, 0]× Ω→ H is given by

yt(θ, ω) = y(t+ θ, ω), for θ ∈ [−r, 0], ω ∈ Ω.

Before describe the properties fulfilled by the operators f, g and Ik, we need to introduce some notationand describe some spaces.

Let DH the following Banach space defined by

DH = φ : [−r, 0]→ H, φ is continuous everywhere except for a finitenumber of points t at which φ(t−) and φ(t+) exist and satisfy φ(t−) = φ(t),

endowed with the L2−norm:

‖φ‖2 =∫ 0

−r‖φ(t)‖2dt.

Also we define D0H as the space of all piecewise continuous processes φ : [−r, 0]×Ω→ H such that φ(θ, ·)

is F0–measurable for each θ ∈ [−r, 0] and∫ 0

−rE‖φ(t)‖2dt <∞. In the space D0

H, we consider the norm:

‖φ‖2D0H

=∫ 0

−rE‖φ(t)‖2dt.

Now, for a given T > 0, we define

DTH =y : [−r, T ]× Ω→ H, yk ∈ C(Jk,H) for k = 1, . . .m, y0 ∈ D0

H,and there exist y(t−k ) and y(t+k )

with y(tk) = y(t−k ), k = 1, · · · ,m, supt∈[0,T ]

E(|y(t)|2) <∞ and∫ 0

−rE‖φ(t)‖2dt <∞

,

endowed with the norm‖y‖DT

H= supt∈[0,T ]

(E(|y(t)|2))12 + ‖y‖D0

H,

where yk denotes the restriction of y to Jk = (tk−1, tk], k = 1, 2, · · · ,m, and J0 = [−r, 0].Then we will consider our initial data φ ∈ D0

H.Let K be another real separable Hilbert and suppose that BHQ is a K-valued fractional Brownian

motion with increment covariance given by a non-negative trace class operator Q (see next section formore details), and let us denote by L(K,H) the space of all bounded, continuous and linear operatorsfrom K into H.

Then we assume that f : J × D0H → H , g : J → L0

Q(K,H) Here, L0Q(K,H) denotes the space of all

Q-Hilbert-Schmidt operators from K into H, which will be also defined in the next section.As for the impulse functions we will assume that Ik ∈ C(H,H) (k = 1, . . . ,m), and ∆y|t=tk =

y(t+k )− y(t−k ), y(t+k ) = limh→0+

y(tk + h) and y(t−k ) = limh→0+

y(tk − h).

The plan of this paper is as follows. In Section 2 we introduce notations, definitions, and preliminaryfacts which are useful throughout the paper. In Section 3 we prove existence of mild solutions for problem(1)–(3). Our approach to prove the existence of mild solutions is based on a fixed point theorem of Burtonand Kirk [5] for the sum of a contraction map and a completely continuous one. Finally, in Section 4, anexample is given to demonstrate the applicability of our results.

2

2. Main results

Now we can define the concept of mild solution to our problem.

Definition 1. Given φ ∈ D0H, an H−valued stochastic process y(t), t ∈ [−r, T ] is called a mild solution

of the problem (1)-(3) if y(t) is measurable and Ft-adapted, for each t > 0, y(t) = φ(t) on [−r, 0] and ysatisfies the integral equation

y(t) = U(t)φ(0) +∫ t

0

U(t− s)f(s, ys)ds+∫ t

0

U(t− s)g(s)dBHQ (s)

+∑

0<tk<t

U(t− tk)Ik(y(tk)), t ∈ [0, T ].(4)

Notice that this concept of solution can be considered as more general than then classical concept ofsolution to equation (1)–(3). A continuous solution of (4) is called a mild solution of (1)–(3).

Our main result in this section is based on the following fixed point theorem due to Burton and Kirk[5].

Theorem 2. Let X be a Banach space, and Φ1,Φ2 : X → X two operators satisfying:

1. Φ1 is a contraction, and

2. Φ2 is completely continuous

Then, either the operator equation y = Φ1(y) + Φ2(y) possesses a solution, or the set Ξ =u ∈ X :

λΦ1(uλ ) + Φ2(u) = u, for some λ ∈ (0, 1)

is unbounded.

We are now in a position to state and prove our existence result for the problem (1)-(3). First wewill list the following hypotheses which will be imposed in our main theorem.

• (H1) operator A : D(A) ⊂ H → H is the infinitesimal generator of a strongly continuous semigroupof bounded linear operators U(t), t ∈ J which is compact for t > 0 in H, and there exists aconstant M such that ‖U(t)‖2 ≤M for all t ∈ [0, T ].

• (H2) There exist constants ck > 0, k = 1, . . . ,m with 8Mm

m∑

k=1

ck < 1 such that

|Ik(y)− Ik(x)|2 ≤ ck|y − x|2, for all y, x ∈ H.

• (H3) For each t ∈ J the functions f(t, ·) : D0H −→ H is continuous, and for each y ∈ H the function

f(·, y) : J → H is strongly Ft-measurable.

• (H4)The function g : J −→ LQ(K,H) satisfies

∫ T

0

‖g(s)‖2L0Qds <∞.

• (H5) For the initial value φ ∈ D0H, there exists a continuous nondecreasing function ψ : [0,∞) →

[0,∞) and p ∈ L1(J, IR+) such that E|f(t, y)|2 ≤ p(t)ψ(‖y‖2D0H

), for a.e. t ∈ J and y ∈ D0H

with ∫ ∞

TC0

du

ψ(u)> TC1

∫ T

0

p(t)dt,

where

C0 =

4M

(E|φ(0)|2 + 2m

m∑

k=1

E|Ik(0)|2 + 2HT 2H−1

∫ T

0

‖g(s)‖2L0Qds

)

(1− 8Mm

m∑

k=1

ck

) ,

3

C1 =4MT(

1− 8Mm

m∑

k=1

ck

) .

Theorem 3. Assume that hypotheses (H1), (H2), (H3), (H4) and (H5) hold. Then, problem (1)-(3)possesses at least one mild solution on [−r, T ].

References

[1] E. Alos, O.Mazet and D.Nualart, Stochastic calculus with respect to Gaussian processes. Ann Probab,29 1999, 766-801

[2] D.D. Bainov and P.S. Simeonov, Systems with Impulsive effect, Horwood, Chichister, 1989.

[3] M. Benchohra, J. Henderson and S. K. Ntouyas, Impulsive Differential Equations and Inclusions,Hindawi Publishing Corporation, Vol 2, New York, 2006.

[4] B. Boufoussi, S. Hajji, Neutral stochastic functional differential equations driven by a fractionalBrownian motion in a Hilbert space. Statist Probab Lett, 2012, 82: 1549-1558

[5] T.A. Burton and C. Kirk, A fixed point theorem of Krasnoselskiii-Schaefer type, Math. Nachr. 189(1998), 23-31.

[6] G. Cao, H. He and X. Zhang, Successive approximations of infinite dimensional SDES with jump.Stoch Syst, 2005, 5: 609-619

[7] T. Caraballo, M. J. Garrido-Atienza, T. Taniguchi, The existence and exponential behavior of so-lutions to stochastic delay evolution equations with a fractional Brownian motion. Nonlinear Anal,2011, 74: 3671-3684

[8] T. Caraballo, D. Mamadou Abdoul, Neutral stochastic delay partial functional integro-differentialequations driven by a fractional Brownian motion .Front. Math. China 2013, 8(4): 745-760

[9] J. R. Graef, J. Henderson and A. Ouahab, Impulsive differential inclusions. A fixed point approach.De Gruyter Series in Nonlinear Analysis and Applications 20. Berlin: de Gruyter, 2013.

[10] T. E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differentialequations. Stochastics, 2005, 77: 139-154

[11] R. Grimmer, Resolvent operators for integral equations in a Banach space.Trans Amer Math Soc,1982, 273: 333-349

[12] F. Jiang and Y. Shen A note on the existence and uniqueness of mild solutions to neutral stochasticpartial functional differential equations with non-Lipschitz coefficients. Comput Math Appl, 2011,61: 1590-1594

[13] V. Lakshmikantham, D. Bainov and P. Simeonov, Theory of Impulsive Differential Equations.WorldScientific Press, Singapore. 1989.

[14] Y. Mishura Stochastic Calculus for Fractional Brownian Motion and Related Topics. Lecture Notesin Mathematics, Vol 1929. Berlin: Springer-Verlag, 2008

[15] D. Nualart The Malliavin Calculus and Related Topics. 2nd ed. Berlin: Springer- Verlag, 2006 760

[16] A. M. Samoilenko and N. A. Perestyuk: Impulsive Differential Equations, World Scientific, Singapore1995

[17] Tindel S, Tudor C, Viens F. Stochastic evolution equations with fractional Brownian motion. ProbabTheory Related Fields, 2003, 127(2): 186-204

[18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations.Springer-Verlag, New York, 1983.

4


Some applications of the translated Whitney numbers

Meriem Tiachachat1∗ and Miloud Mihoubi2

1 USTHB, Faculty of Mathematics, RECITS’s laboratory.USTHB, PB 32 El Alia 16111, Algiers, Algeria.

[email protected] UUSTHB, Faculty of Mathematics, RECITS’s laboratory.

USTHB, PB 32 El Alia 16111, Algiers, [email protected]

Abstract

The aim of this work is to give an application of the translated Whitney numbers on the values of Bernoullipolynomials. The obtained formulas generalize the known expressions of the Bernoulli numbers .

Key words: Whitney numbers, translated Whitney numbers, Bernoulli polynomials.


1. IntroductionThe r-Whitney numbers of Dowling lattice of the first kind and second kind wm,r (n, k) , Wm,r (n, k) can be defined by theirexponential generating functions to be

∑

n≥kwm,r (n, k)

tn

n!=

1k!

(ln (1 +mt)

m

)k(1 +mt)−

rm , (1)

∑

n≥kWm,r (n, k)

tn

n!=

1k!

(exp (mt)− 1

m

)kexp (rt) . (2)

For more information about these numbers, one can see [2, 3, 4, 5, 6, 7].

Belbachir and Bousbaa [1] introduced a new class of Whitney numbers called a translated Whitney numbers w(α) (n, k) ,W(α) (n, k) of first and second kind respectively, when w(α) (n, k) , count the number of permutations of n elements withk cycles such that the element of each cycle can mutate in α ways, except the dominant one and W(α) (n, k) , the partitionsof the set 1, 2, 3, ..., n into k subsets such that each element of each subset can mutate in αways, except the dominant one.

∗speaker

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Recentely, Mangontarum and Dibagulun [8] give some properties of this numbers and give also an application of this numberson the Bernoulli polynomials .

This work is devoted to give an other applications of this numbers on the Bernoulli polynomials and the generalizedBernoulli polynomials.

References[1] H. Belbachir, I. E. Bousbaa, Translated Whitney and r -Whitney numbers, A combinatorial approach, J. Integer Seq. 16

(2013) Article 13.8.6.

[2] M. Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996) 13–33

[3] M. Benoumhani, On some numbers related to Whitney numbers of Dowling lattices, Adv. Appl. Math. 19 (1997) 106–116.

[4] M. Benoumhani, Log-concavity of Whitney numbers of Dowling lattices, Adv. Appl. Math. 22 (1999) 186–189.

[5] A. Z. Broder, The r-Stirling numbers. Discrete Math., 49 (1984) 241-259.

[6] G.-S. Cheon, J.-H. Jung, The r-Whitney numbers of Dowling lattices, Discrete Math., 312 (15) (2012) 2337-2348.

[7] T.A. Dowling, A class of gemometric lattices bases on finite groups, J. Combin. Theory Ser. B, 14 (1973) 61-86. ErratumJ. Combin. Theory Ser. B, 15 (1973) 211.

[8] M. Mangontarum A.Dibagulun On the Translated Whitney Numbers and their Combinatorial Properties,BJAST, 11(5),2015; Article no.BJAST.19973

[9] I. Mezo, A new formula for the Bernoulli polynomials, Results Math., 58 (2010) 329-335.

204


b-Domatic number of the join graphs

Mohammed Benatallah1∗ and Noureddine Ikhlef Eschouf2

1 University Ziane Achour of Djelfa, Faculty of Sciences and TechnologyDjelfa, Algeria.

m [email protected] university Dr. Yahia Fares of Medea, Faculty of Sciences and Technology

Medea, Algeria.nour [email protected]

Abstract

A partition of V (G), all of whose classes are dominating sets in G, is called a b-domatic partition of G. Theminimal number of classes of a b-domatic partition of G is called the b-domatic number of G. In this paper weexplore the bounds for the b-domatic numbers of the join of two graphs. The bonds are the best possible in thesense that there exist examples for which equalities are attained. .

Key words: domatic number, b-domatic number, join graph.

1. Introduction

Let G = (V,E) be a finite, simple and undirected graph with vertex-set V and edge-set E. We call |V | the order of Gand denote it by n. For any nonempty subset A ⊂ V , let G[A] denote the subgraph of G induced by A. For any vertex vof G, the neighborhood of v is the set NG(v) = u ∈ V (G) | (u, v) ∈ E and the closed neighborhood of v is the setNG[v] = NG(v) ∪ v. Let ∆(G) (respectively, δ(G) be the maximum (respectively, minimum) degree in G. Through thispaper, the notations Pn, Cn, and Kn always denote a path, a cycle, and a complete graph of order n, respectively. For furtherterminology on graphs we refer to the book by Berge [1].

A set S ⊆ V is called a dominating set if every vertex in VS is adjacent to some vertex in S. The minimum cardinalityof a dominating set is called the domination number and is denoted by γ(G). A dominating set S is indivisible if it cannotbe partitioned into two disjoint dominating sets. By analogy to the concept of chromatic partition, Cockayne and Hedetniemi[2] introduced the concept of domatic partition of a graph. A partition P in which each of its classes is a dominating sets iscalled a domatic partition of G. The domatic number d(G) is defined as the largest number of sets in a domatic partition. Theauthors of [2] showed that

d(G) ≤ min n

γ(G), δ(G) + 1. (1)

The b-domatic spectrum of a graph G is the set of the cardinalities of all the b-maximal domatic partitions of G.As defined in [3] , a domatic partition P is b-maximal if no larger domatic partition P ′ can be obtained by gathering

subsets of some classes of P to form a new class. Formally, it can be defined as follows,

∗speaker205

Definition 1. [3]A partition P = C1, C1, ..., Ck is b-maximal if there do not exist k subsets C′1, . . . , C′

k with C′i ⊂ Ci

(i = 1, . . . , k) such that C′1 ∪ ... ∪ C′

k 6= ∅ and the partition P ′ = C1C′1, C2C′

2, ..., CkC′k, C′

1 ∪ ... ∪ C′k is domatic

Definition 2. [3]The minimum cardinality of a b-maximal domatic partition of G is called the b-domatic number and isdenoted by bd(G).

The concept of b-domatic number has been recently introduced by Favaron [3] . The same author has observed that ifδ(G) = 0, then V (G) is the unique domatic partition and so bd(G) = d(G) = 1. For this, all graphs considered inthis paper are without isolated vertices. Many other properties of domatic and b-domatic partitions were given in [3] . Inparticular, it was observed that for any graph G with degree minimum δ(G) ≥ 2,

2 ≤ bd(G) ≤ d(G). (2)

The joint of two graphG1 andG2 is the graphG1⊕G2 defined as the disjoint union of graphG1 andG2 with additionaledges linking each vertex of G1 with each vertex of G2.

Standard notation is applied for the particular types of graphs, too, such as Kn (the complete graph on n vertices), Pn(the path on n vertices), Cn (the cycle on n vertices), Km,n (the complete bipartite graph), Sn (the star with n leaves).

2. Mains results

It immediate seen that

Proposition 3. For any graph G of order n ≥ 1.

1. bd(G) ≤ nγ(G) .

2. If H is a spanning subgraph of G, then bd(H) ≤ bd(G).

Theorem 4. For any two graph G1, G2 we have

bd(G1) + bd(G2) ≤ bd(G1 ⊕G2) ≤ min δ(G1) + |V (G2)| , δ(G2) + |V (G1)|+ 1.

Proposition 5. Let Pn, Pm two paths and n,m ≥ 2, then bd(Pn ⊕ Pm) = 4.

Proposition 6. Let Cn, Cm two cycle, such that n,m ≥ 4, then

bd(Cn ⊕ Cm) =

5 if Cn, Cm are odd.4 if not.

.

Corollary 7. For m,n ≥ 2, bd(Sn ⊕ Sm) = 4.The b-domatic spectrum of Sn ⊕ Sm is 4, 2 + min n,m.

Corollary 8. For n,m, p, q ≥ 2, bd(Kp,q ⊕Kn,m) = 4.The b-domatic spectrum of Kp,q ⊕Kn,m is 4,min p, q+ min n,m.

Corollary 9. 1. For m,n ≥ 2, bd(Pn ⊕Km) = 2 +m.

2. For n ≥ 1,m ≥ 2, bd(Sn ⊕Km) = 2 +m.

3. For n ≥ 3,m ≥ 2, bd(Cn ⊕Km) = 2 +m.

4. For m,n ≥ 2, bd(Kn ⊕Km) = m+ n.206

References

[1] C. Berge. Graphs. North Holland, 1985.

[2] E.J. Cockayne and S.T. Hedetniemi, Towards a theory of domination in graphs, Networks 7 (1977) 247–261.

[3] O. Favaron. The b-domatic number, Discussiones Mathematicae, Graph Theory 33(2013)747–757.

207


On the Diophantine equation x2 + 13k = yn

Abdelkader Hamtat1∗, Djilali Behloul2,

1 USTHB, ATN LaboratoryBP 32 El Alia 16111, Bab Ezzouar 16111, Algiers, Algeria

[email protected] USTHB, ATN Laboratory

BP 32 El Alia 16111, Bab Ezzouar 16111, Algiers, [email protected]

AbstractThe object of this paper is to find all solutions of the diophantine equation x2 + 13k = yn, in positive

integers x, y with n ≥ 3.

Key words: Diophantine equation, Primitive divisor theorem, Algebraic number theory.

2010 Mathematics Subject Classification:11D65.

1. IntroductionDiophantine equation is any equation in two or more unknowns, such that only the integer solutions are sought or studied.The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made astudy of such equations and was one of the first mathematicians to introduce symbolism into algebra.The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.In 1637, Pierre de Fermat scribbled on the margin of his copy of Arithmetica: ”It is impossible to separate a cube into twocubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. Andthen he wrote, intriguingly: ”Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet” whichmeans : I have discovered a truly remarkable proof of this theorem which this margin is too small to contain. It was provenby the British mathematician Andrew Wiles in 1995.In 1657, Fermat attempted to solve the Diophantine equation 61 ∗ x2 + 1 = y2. The equation was eventually solved by Eulerin the early 18th century, who also solved a number of other Diophantine equations.

Many special cases of the Diophantine equation x2 + qk = yn, where q is a prime and x, y, k and n are positive integershave been studied in the last few years. When q = 2 and k is odd integer, Cohn has proved that this equation have threefamilies of solution, when q = 3 and m is an odd integer[11], Arif and Abu Muriefah [1] proved that this equation has onefamily of solutions and Luca [5] has shown the existence of exactly one family of solutions when q = 3 and m is an eveninteger. Arif and Abu Muriefah has proved that the equation x + q2k+1 = yn when q is odd prime, q ≡ 7mod8 and n is anodd integer ≥ 5, n is not a multiple of 3 and (n, h) = 1, where h is the class number of the field Q(

√−q) has exactly twofamilies of solutions [1]. In this paper, we use this results to prove that the Diophantine equation

x2 + 13k = yn (1)

where n ≥ 3, has exactly one family of solutions in positive integers.∗Speaker

208


1.1. Main results

To prove our results, we need the following lemmas [6] ,[7]

Lemma 1. (Nagell) let n be an odd integer ≥ 3, and let A ∈ N, be an odd and square free. If the class number of the fieldQ(√−A) is not divisible by n, then the Diophantine equation Ax2 + 1 = yn, has no solution in positive integers x and y for

y odd.

Catalan’s conjecture (or Mihailescu’s theorem) is a theorem in number theory that was conjectured by the mathematicianEugne Charles Catalan in 1844 and proven in 2002 by Preda Mihailescu. 23 and 32 are two powers of natural numbers, whosevalues 8 and 9 respectively are consecutive. The theorem states that :

Lemma 2. (Catalan) the Diophantine equation xm − yn = 1 has only one solution in positive integers (x, y,m, n) =(3, 2, 2, 3).

Lemma 3. The Diophantine equation

3x2 + 1 = 13y (2)

when x, y > 0, has only one solution (x, y) = (2, 1).

Theorem 4. The Diophantine equation

x2 + 132m = yn (3)

when n ≥ 3,m > 0, has no solution in positive integers .

Theorem 5. The Diophantine equation

x2 + 132m+1 = yn (4)

when n ≥ 3 has exactly one family of solution (x, y,m, n) = (70.133k, 17.132k, 3k, 3).

References[1] S.A. Arif and Fadwa S. Abu Muriefah : On the Diophantine equation x2 + q2k+1 = yn; The Arabian J. for Sci. and

Engineering, 26(1A), 53-62, 2001.

[2] S.A. Arif and F.S. Abu Muriefah, On the Diophantine equation x2 + q2k = yn; Journal of Number Theory, 95(1),95-100, 2002.

[3] Y.F. Bilu, G. Hanrot and P.M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew.Math., 539, 75-122, 2001.

[4] Cohn, J.H.E., The Diophantine equation x2 + C = yn, Acta Arith. 65 (1993), no.4, 367-381

[5] Yuri Bilu, Guillaume Hanrot, Paul M. Voutier. Existence of Primitive Divisors of Lucas and Lehmer Numbers. [ResearchReport] RR-3792, 1999, pp.41. <inria-00072867> H

[6] Nagell, T., Contributions to the theory of a category of Diophantine equations of the second degree with two unknown,Nova Acta Reg. Soc. Upsal.Ser. 4 (1955), no.16, 1-38.

[7] J.H.E. Cohn, The Diophantine equation x2 + C = yn, Acta Arith., 65(4), 367-381, 1995.

[8] F. Luca. “On the Equation x2 + 2a3b = yn.” Int. J. Math. Math. Sci. 29.4 (2002): 239–244

[9] Luca, F., Togbe, A., On the Diophantine equation x2 + 72k = yn, The Fibonacci Quarterly (2008), preprint

209


[10] Goins, E., Luca, F., Togbe, A.,On the Diophantine equation x2 + 2α5β13γ = yn, ANTS VIII Proceedings: A. J. vander Poorten and A. Stein (eds.), ANTS VIII, Lecture Notes in Computer Science 5011 (2008), 430-442.

[11] Explicit Methods for Solving Diophantine Equations; Henri Cohen . Tucson, Arizona Winter School, 2006

[12] Florian Luca , EFFECTIVE METHODS FOR DIOPHANTINE EQUATIONS ; Universidad Nacional Aut onoma de Mexico.

[13] Muriefah, F.S. Abu, Luca, F., Togbe, A., On the Diophantine equation x2 + 5a13b = yn, Glasgow Math. J. 50 (2008),175-181.

[14] Hui Lin Zhu , Mao Hua Lec; On some generalized Lebesgue–Nagell equations. Journal of Number Theory 131 (2011)458–469

[15] A. Berczes, I. Pink, On the Diophantine equation x2 + p2k = yn, Arch. Math. 91 (2008) 505–517.

[16] L. Tao, On the Diophantine equation x2 + 5m = yn, Ramanujan J. 19 (2009) 325–338.

[17] S. Siksek, J.E. Cremona, On the Diophantine equation x2 + 7 = ym, Acta Arith. 109 (2) (2003) 143–149.

[18] Yann Bugeaud, Florian Luca, Maurice Mignotte, Samir Siksek ; Almost Powers in the Lucas Sequence. Journal deTheorie des Nombres de Bordeaux.

[19] Fadwa S. Abu Muriefah, F. Luca, A.A. Togbe´ , On the Diophantine equation x2 + 5a.13b = yn, Glasg. Math. J. 50(2008) 75–181

210


An MX/G/1retrial queue with breakdown, repair, loss and twoTYPES of customers

Tassadit LACHEMOT 1∗, Hafida SAGGOU2

1 USTHB, Faculty of MathematicsRIIMA Laboratory


2 USTHB, Faculty of MathematicsRIIMA Laboratory

BP 32, El Alia 16111 Bab Ezzouar, Algiers, Algeriasaggou [email protected]

Abstract

This paper deals with the steady behaviour of an MX/G/1 queue with general retrial time, two typesof customers, loss customers and bernoulli vacation schedule for an unreliable server, which consists of abreakdown period and delay period. here we assume that customers arrive according to compound poissonprocesses. While the server is working with primary customers it may breakdown at any instant and server willbe down for short interval of time. Further concept of the delay time is also introduced. the primary customerfinding the server busy, down are queued in orbit with probability p in accordance with FCFS (first come firstserved) retrial policy or leave the system with probability 1−p.after the completion of a service, the server eithergoes for a vacation of random length with probability 1− r or may continue to serve for the next customer withprobability r. The rest of the paper is organized as follows: In the next section, we give a description of themathematical model. the steady state distribution of the server state is studied in section3 by using the conceptof supplementary variable method and the generating function. in section4, some performance measures of thesystem are developed. Some particular cases are given and finally the stochastic decomposition property aredeveloped.

Key words: Transit and recurrent customers; General retrial times; Breakdown; General delay times; repair times; Batcharrival; Bernoulli vacation.

References[1] J.R.Artalejo,I.Atencia, On the single server Retrial queue with Batch Arrivals,Ind.J.Statist 2004, Vol 66, Part1, p140–

158.

[2] I. Atencia, I. Fortes, S. Nishimura, S. Sanchez, A discrete–time retrial queueing system with recurrent customers,Comp. Oper. Res 37 (2010) 1167–1173.

[3] M.Baruah, K.C.Madan, T.Eldabi,A two stage Batch arrival queue with reneging during vacation and breakdown periods,Americ. J. Oper. Res. 2013, 3, 570–580.

∗Speaker211


[4] W.Chan,R.Bartoszynski,D.Pearl, Queues with breakdowns and customer discouragement, Probab.Mathemat.Statist, Vol14. Fasc1(1993),p 77–87.

[5] G. Choudhury, L. Tadj, An M/G/1 queue with two phases of service subject to the server breakdown and delayedrepair, Appl. Mathemat. Model 33 (2009) 2699–2709.

[6] G. Choudhury, J–C. Ke, A batch arrival retrial queue with general retrial times under Bernoulli vacation schedule forunreliable server and delaying repair, Appl. Mathemat. Model 36(2012)255–269.

[7] K.Farahmand, Single line queue with recurrent repeated demands, Queueing Systems 22 (1996) 425–435.

212


Asymptotic normality of high risk estimate for functional data

Boubaker MECHAB1

1 Djillali Liabes University, Laboratory of Statistic and Processus StochasticsBP 89 University Djillali Liabes, Sidi Bel Abbes, Algeria

[email protected]

Abstract

We investigate the asymptotic normality of the high risk estimate. The infill increasing setting is considered,that is when the covariates take values in some abstract function space. Our approach is based on the Doob’stechnique. It is shown that, under the concentration property on small balls of the probability measure of thefunctional estimator and some regularity conditions, the kernel estimate of the three parameters (conditionaldensity, conditional distribution and conditional hazard) are asymptotically normally distributed.

Key words: Asymptotic normality, conditional hazard function, functional data, strong mixing.

2010 Mathematics Subject Classification: 62G05, 62G20, 62N05, 62M09.

1. IntroductionThe conditional hazard rate plays an important role in the statistics. In recent years, the functional estimate has attracted alot of attention in the statistical literature. Functional data arise in a variety of fields including econometrics, epidemiology,environmental science and many others. The kernel type estimation of some characteristics of the conditional cumulativedistribution function and the successive derivatives of the conditional density have been introduced by Ferraty et al. (2006).Some asymptotic properties have been established with a particular application to the conditional mode and conditionalquantile. The only work about the conditional hazard rate in infinite dimensional space for functional covariates is of Ferratyet al. (2007), where they introduce a kernel estimator and they prove some asymptotic properties (with rates) in varioussituations including censored and/or dependent variables.In our setting, we study the asymptotic normality of the kernel estimator of the high risk conditional sup hx defines in themodel.Let (ζn, n ∈ Z) be a sequence of random variables defined on some probabilistic space (Ω,A,P) and taking values insome space (Ω

′,A′). Let us denote, for −∞ ≤ j ≤ k ≤ +∞, by σkj the σ-algebra generated by the random variables

(ζs, j ≤ s ≤ k). The strong mixing coefficients are defined to be the following quantities:

α(n) = supk∈Z,A∈σk

−∞,B∈σ∞n+k|IP(A ∩B)− IP(A)IP(B)| . (1)

The sequence (ζn, n ∈ Z) is said to be α−mixing (or strong mixing), if

limn→∞

α(n) = 0.

213


2. Model and notationsConsider Zi = (Xi, Yi)i∈N be a F ×R-valued measurable random variables (r.v’s), defined on a probability space (Ω,A,P),where (F , d) is a semi-metric space. In most practical applications, F is a normed space (e.g. Hilbert or banach space) withnorm ||.|| so that d(x, x′) = ||x− x′||. In the following x will be a fixed point in F , Nx will denote a fixed neighborhood ofx and S will be fixed compact subset of R.We intend to estimate the conditional hazard function hx using n dependent observations (Xi, Yi)i∈N draw from a randomvariables with the same distribution with Z := (X,Y ) where the regular version F x of the conditional distribution functionof Y given X = x exists for any x ∈ Nx. Moreover we suppose that F x has a continuous density fx with respect to (w.r.t)Lebesgue’s measure over R. we define the function hazard hx, for y ∈ R and F x(y) < 1, by

hx(y) =fx(y)

1− F x(y),

To this aim, we first introduce the kernel type estimator F x of F x defined by

F x(y) =∑ni=1K(h−1

K d(x,Xi))H(h−1H (y − Yi))∑n

i=1K(h−1K d(x,Xi))

, ∀y ∈ R

where K is the kernel, H is a given distribution function and hK = hK,n (resp. hH = hH,n) is a sequence of positive realnumbers.We define the kernel estimator fx of fx by

fx(y) =h−1H

∑ni=1K(h−1

K d(x,Xi))H ′(h−1H (y − Yi))∑n

i=1K(h−1K d(x,Xi))

, ∀y ∈ R

Where H ′ is the derivative of H .We estimate the hazard function hx by

hx(y) =fx(y)

1− F x(y). ∀y ∈ R.

we get easily

hx(y)− hx(y) =1

1− F x(y)

[fx(y)− fx(y)

]+

hx(y)

1− F x(y)

[F x(y)− F x(y)

]. (2)

We try to estimate the point of high risk in S, denoted by θ(x), defined by

hx(θ(x)) = maxy∈S

hx(y). (3)

This model has a great interest in statistics, this is mainly due to its application in various disciplines. Indeed, this is the toolused in seismic risk analysis (see Quintela-del-Ro (2006)). In our context functional, we assume that there is a single pointθ(x) in S satisfying (3). The estimator natural of θ(x), denoted by θ(x), is as:

hx(θ(x)) = maxy∈S

hx(y). (4)

In general, this estimator is not unique. thus, throughout the rest of this article widehat theta(x) mean any random variablesatisfying (4). We keep the assumptions of the previous section to state precisely the rate of convergence of the estimatorθ(x), and we assume that hx is 2-times continuously differentiable around y with:

hx′(θ(x)) = 0 and hx

′′(θ(x)) < 0. (5)

214


3. Hypothesiswe denote by C1, C2, ... any generic positive constant.For h > 0, let B(x, h) := x′ ∈ F/d(x′, x) < h be the ball of center x and radius h.To establish the asymptotic normality of the estimator hx we need to include the following assumptions:

(H1) P(X ∈ B(x, h)) = φx(h) > 0.

(H2) 0 < supi 6=j P [(Xi, Xj) ∈ B(x, h)×B(x, h)] ≤ C(φx(h))(a+1)/a, with 1 < a < δ.

(H3) Let K be the kernel function with support [0, 1] such that, K(0) > 0 and K(1) > 0.

(H4) H’(.) is bounded,∫

RH′2/r(u)du <∞. For some r such that 1− δ < r < 1,

(H5) ∀(x1, x2) ∈ N2x , ∀(y1, y2) ∈ S2

|F x1(y1)− F x2(y2)| ≤ C(db1(x1, x2) + |y1 − y2|b2

), b1 > 0, b2 > 0

and|fx1(y1)− fx2(y2)| ≤ C

(db3(x1, x2) + |y1 − y2|b4

), b3 > 0, b4 > 0.

(H6) (i) nhHφx(hK)→∞. (ii) nh2kH φx(hK)3 → 0.

(H7) There exist sequences of positive integers pn and qn increasing to infinity such that pn + qn ≤ n

qn = o((nhkHφx(hK))1/2),∞∑

k=1

kδα(k)1−r <∞,

snqn = o((nhkHφx(hK))1/2), such that Sn(pn + qn) ≤ n, pn = (nhkHφx(hK))1/2/sn,(n

hkHφx(hK)

)1/2

α(qn)→ 0 and sn

(n

hkHφx(hK)

)1/2

α(qn)→ 0.

(H1’) There exists a function βxr (.) such that

∀s ∈ [0, 1], limh→0

φx(sh)φx(h)

= βxr (s).

Our result will be expressed using the following constants:

aj(x) = Kj(1)−∫ 1

0

(Kj)′(s)βx0 (s)ds for j = 1, 2.

The function βxr (.) is increasing for all fixed r, the limit limr→0

βxr (s) = βx0 (s) plays a fundamental role in all asymptotic, inparticular for the variance term.

4. Main resultsTheorem 1. Under the conditions of conditions (H1)-(H7), if (5) is holds, we have

θ(x)− θ(x) = O(hb1/2K + h

b2/2H

)+O

((log n

nhHφx(hK)

) 14), p.co.

Theorem 2. under hypotheses (H1)-(H6) and (H1’), we have,(nhHφx(hK)

σ2h(y)

)1/2

(θx(y)− θx(y)) D−→ N (0, 1) as n→∞ (6)

where D−→ denote the convergence in distribution, σ2h(y) =

a2(x)hx(y)a21(x)(1− F x(y))

215


References[1] Bosq D., Linear Processes in Function Spaces: Theory and applications, Lecture Notes in Statistics, Springer., 149,

2000.

[2] Ferraty F., Rabhi A., Vieu P., Estimation non-parametrique de la fonction de hasard avec variable explicative fonction-nelle,Rev. Roumaine Math. Pures Appl., 2008, 53, 1–18.

[3] Ferraty F., Vieu P., Nonparametric functional data analysis,Springer Series in Statistics, Springer New York., 2006.

[4] Quintela-del-Rıo A., Nonparametric estimation of the maximum hazard under dependence conditions, Statist. Probab.Lett.., 2006, 76, 1117–1124.

[5] Quintela-del-Rıo A., Hazard function given a functional variable: Non-parametric estimation under strong mixing con-ditions, J. Nonparametr. Stat., 2008, 20, 413–430.

[6] Ramsay J. O., Silverman B. W., Applied functional data analysis; Methods and case studies, Springer-Verlag, NewYork., 2002.

[7] Roussas G., Hazard rate estimation under dependence conditions, J. Statist. Plann. Inference., 1989, 22, 81–93.

[8] Watson G. S., Leadbetter M. R., Hazard analysis, Sankhyia., 1964, 26, 101–116.

216


The local time under sublinear expectations and applications

S.MERADJI1, H.BOUTABIA2, S.STIHI3 ∗

1 Universite Badji Mokhtar, Laboratoire de Probabilite et Statistique (LAPS)Annaba 23000, Algerie

[email protected] Universite Badji Mokhtar, Laboratoire de Probabilite et Statistique (LAPS)

Annaba 23000, [email protected]


sara [email protected]

Abstract

In this paper, we study the notion of local time and obtain the Tanaka formula for the G-Brownien motion.Moreover an intuitive approach to the local time shows its limits in the frame of the classical stochastic analysisand works out the problems to be studied, as an application, we generalize Ito’s formula with respect to theG-Brownian motion to convex functions.

Key words: G-expectation; Local time; Tanaka formula; The G-Ito’s formula.

2010 Mathematics Subject Classification: Stochastic analysis.

1. IntroductionIn this paper, we study the notion of the local time as well as the Tanaka formula for the G−Brownian motion.

Motivated by uncertainty problems in volatility, Peng has introduced a new notion of nonlinear expectation, the so-calledG−expectation , which is associated with the following nonlinear heat equation

∂u(t,x)

∂t = G(

∂2u(t,x)∂x2

), (t, x) ∈ [0,∞)× R,

u (0, x) = ϕ (x) ,

where, for given parameters 0 ≤ σ ≤ σ, the sublinear function G is defined as follows:

G (α) =12(σ2α+ − σ2α−

), α ∈ R.

The G−expectation represents a special case of general nonlinear expectations E, their importance inside the class ofnonlinear expectations stems from the stochastic analysis which it allows to develop.

∗Underline the speaker

217


Peng also introduced the related G−normal distribution and the G−Brownian motion. The G−Brownian motion isa stochastic process with stationary and independent increments and its quadratic variation process is, unlike the classicalcase of a linear expectation, a non deterministic process. Moreover, an Ito calculus for the G−Brownian motion has beendeveloped recently.

The fundamental notion of the local time for the classical Brownian motion has been introduced by Levy, and its existencewas established by Trotter. In virtue of its various applications in stochastic analysis the notion of local time and the relatedTanaka formula have been well studied by several authors, and it has also been extended to other classes of stochasticprocesses.

It should be expected that the notion of local time and Tanaka formula will have the same importance in theG−stochasticanalysis on sublinear expectation spaces, which is developed by Peng. However, unlike the study of the local time forthe classical Brownian motion, its investigation with respect to the G−Brownian motion meets several difficulties: firstly,in contrast to the classical Brownian motion the G−Brownian motion is not defined on a given probability space but ona sublinear expectation space. The G−expectation E can be represented as the upper expectation of a subset of linearexpectations Ep, P ∈ P , i.e.,

E [.] = supEP [.]P∈P

,

where P runs a large class of probability measures P . Let us point out that this is a very common situation in financialmodels under volatility uncertainty. Secondly, related with the novelty of the theory of G−expectations, there is short of a lotof tools in the G−stochastic analysis, among them there is, for instance, the dominated convergence theorem. The definitionof the local time for the G−Brownian motion will be given independently of the probability measures P ∈ P , and the limitsof the classical stochastic analysis in the study of this local time will be indicated.

In this paper we introduce the necessary notations and preliminaries and we give a short recall of some elements of theG−stochastic analysis, we define the notion of the local time for the G−Brownian motion and prove the related Tanakaformula, which non-trivially generalizes the classical one.

References[1] Q.Lin, Local time and Tanaka formula for the G−brownian motion, J. arXiv: [math.PR], 2012, .0912.1515v3.

[2] X. Li, S. Peng, Stopping times and related Ito’s calculus with G−Brownian motion, J. Stochastic Processes and theirApplications., 2011, 121, 1492–1508

[3] H. Boutabia and I. Grabsia,CHAOTIC EXPANSION IN THE G−EXPECTATION SPACE, J. Opuscula Math., 2013,33, no. 4 , 647–666

218


Stochastic calculus for matrix fractional brownian motion andapplications

S.STIHI1, H.BOUTABIA2, S.MERADJI3 ∗


sara [email protected] Universite Badji Mokhtar, Laboratoire de Probabilite et Statistique (LAPS)

Annaba 23000, [email protected]


[email protected]

Abstract

We investigate the process of eigenvalues of a symmetric matrix-valued process which upper diagonal en-tries are independent one-dimensional Holder continuous Gaussian processes of order γ ∈ (1/2; 1). Using thestochastic calculus we show that these eigenvalues do not collide at any time with probability one. When thematrix process has entries that are fractional Brownian motions with Hurst parameter H ∈ (1/2, 1).

Key words: Eigenvalues, Brownian motions with Hurst parameter, Gaussian processes, Matrix-valued process.

2010 Mathematics Subject Classification: Stochastic analysis.

1. IntroductionIn a pioneering work in 1962, the nuclear physicist Freeman Dyson studied the stochastic process of eigenvalues of anHermitian matrix Brownian motion. The case of the (real) symmetric matrix Brownian motion was first considered by McKean in 1969. In both cases the corresponding processes of eigenvalues are called Dyson Brownian motion and are governedby a noncolliding system of Ito Stochastic Differential Equations (SDEs).

More specifically, for the symmetric case, let B (t)t≥0 = bjk (t)t≥0 be a d× d symmetric matrix Brownian motion.

That is, (bjj (t))dj=1, (bjk (t))j<k; is a set of d (d+ 1) /2 independent one-dimensional Brownian motions with parameter

(1 + δjk) t. For each t > 0, B (t) is a Gaussian Orthogonal (GO) random matrix with parameter t. Let λ1 (t) ≥ λ2 (t) ≥... ≥ λd (t) , t ≥ 0; be the d−dimensional stochastic process of eigenvalues ofB. If the eigenvalues start at different positionsλ1 (0) > λ2 (0) > ... > λd (0), then they never meet at any time and furthermore they form a diffusion process satisfying theIto SDE

λi (t) =√

2W it + 2

∑

i6=j

∫ t

0

ds

λi (s)− λj (s), 1 ≤ i ≤ d, t ≥ 0 (1.1)

∗Underline the speaker219


where W 1t , ...,W

dt are independent one-dimensional standard Brownian motions.

In this paper, we study the Dyson process of a symmetric matrix Gaussian process G (t)t≥0 which entries in the upperdiagonal part are independent one-dimensional zero mean Gaussian processes with Holder continuous paths of order

γ ∈ (1\2, 1). In this caseG (t) is still aGO random matrix of parameter t , for each t > 0 ; but G (t)t≥0 is not a matrixprocess with independent increments nor a matrix diffusion. We prove that the corresponding eigenvalues do not collide atany time with probability one.

We study in detail the case when the matrix process has entries that are fractional Brownian motions with Hurst parameterH ∈ (1\2, 1) . For this matrix process we find a stochastic differential equation for the Dyson process, similar to equation(1.1), where instead of

√2W i

t we obtain processes Y it which are expressed as Skorohod indefinite integrals. This equation is

derived applying a generalized version of the Ito formula in the Skorohod sense for the multidimensional fractional Brownianmotion, which has its own interest. Each process Y i has the same H-self-similarity and 1\H-variation properties as a one-dimensional fractional Brownian motion, although there is no reason for them to be fractional Brownian motions. In our casewe conjecture that each Y i is not a one-dimensional fractional Brownian motion, but at this moment we are not able to givea proof of this fact.

In this paper we introduce the necessary notations and preliminaries on Malliavin calculus for the fractional Brownianmotion, we give a generalized version of the Ito formula for the multidimensional fractional Brownian motion and the SDEof the eigenvalues of a matrix fractional Brownian notion.

References[1] T.W. Anderson, An Introduction to Multivariate Statistical Analysis, Third Edition. Wiley., 2013.

[2] F.J. Dyson, A Brownian-motion model for the eigenvalues of a random matrix, J. Math. Phys.,1962, 3.

[3] D. J. Grabiner, Brownian motion in a Weyl chamber, non-colliding particles, and random matrices, Ann. Inst. HenriPoincare, J. Probab. Statist.., 1999,35, 177-204

220


Approximation of multiserver retrial queues by value extrapolationtechnique

Zidani Nesrine1∗, Djellab Natalia2, Belhamra Thara3,

1 University of Badji Mokhtar Annaba , Laboratory LaPSDepartment of Mathematics, BP 12, Annaba, Algeria

[email protected] University of Badji Mokhtar Annaba, Laboratory LaPS

Department of Mathematics, BP 12, Annaba, [email protected]

3 University of Badji Mokhtar Annaba , Laboratory LaPSDepartment of Mathematics, BP 12, Annaba, Algeria

[email protected]

Abstract

This paper discusses multiserver retrial queueing system with C servers. An arriving customer finding afree server enters into service immediately, otherwise the customer either enters into an orbit to try again aftera random amount of time or leave the system without service. As there are not closed-form solutions to thesesystems, approximate methods are required. To the best of our knowledge, all the existing techniques are basedon computing the steady states probabilities. We consider another approach, Value Extrapolation (VE), basedon the relative state values which appear in the Howard equations. Also, we presented some numerical resultswith different parameter for VE .

Key words: Retrial systems, Extrapolation, Blocking probability.

2010 Mathematics Subject Classification: Operation Research.

1. IntroductionAll known approaches rely on the numerical solution of the steady-state Kolmogorov equations of the continuous time Markovchain that describes the system under consideration. Very recently, however, an alternative approach for evaluating infinitestate space Markov processes was introduced by Leino et al. [1]. The new technique, value extrapolation (VE), does notrely on solving the global balance equations but considers the system in its Markov decision process setting. We apply theVE technique to solve a multiserver retrial system. In our model, users arrive according to a Poisson process with rate λcontending for access to a system with C servers. User requests require an exponentially distributed service time with rate µ.When a new request finds all servers occupied, it joins the retrial orbit. After an exponentially distributed time of rate θ, thisuser retries. The retrial is successful if it finds a free server. We assume the retrial orbit to be of infinite capacity. This modelcan be represented as a bidimensional continuous Markov chain with state space S := s = (k,m) : k ≤ C;m ∈ Z+,where k is the number of users being served, and m is the number of users in the retrial orbit. In a Markov decision process

∗Speaker221


framework, after performing an action in state s, the system collects a revenue for that action (r(s)). As the number oftransitions increases, the mean revenue collected converges to r, which is the mean revenue rate of the process. We choosethe revenue function so that r yields the performance metric we want to compute. The relative state value (v(s)) tells howmuch greater the expected cumulative revenue over an infinite time horizon is when the system starts from the initial state sin comparison with r.

ν (s)=E [∫∞

t=0r (S (t)-r)dt|S (0)=s]

Revenues, relative state values, and transition rates from state s to s’ (qss′ ) are related by the Howard equations :

r (s)-r+∑

s′qss′ (ν (s’)-ν (s))=0 ∀ s.

The number of states is infinite, as m can take any value in Z+; thus, we need to truncate the state space to ST := s= (k,m):k≤ C;m ≤ Q. Unlike traditional truncation, that sets qss′=0 ∀ s’∈ ST, VE performs a more efficient approximation byconsidering the relative state values outside ST that appear in the Howard equations (ν(C, Q+1)) as an extrapolation of somerelative state values inside ST. It is important to choose an extrapolating function that makes the Howard equations remain aclosed system of linear equations. The most common extrapolating functions that accomplish this are polynomials.To extrapolate νQ+1, we have used an (n-1)th degree polynomial that interpolates the n points in

(i, νi) | νi = ν(C, i), Q− n ≤ i ≤ Q .

In general, using the Lagrange basis to reduce the complexity, and after some algebra, we obtain a simple closed-formexpression for the extrapolated value:

ν(n)Q+1 =

∑n−1k=0(−1)kCk+1

n νQ−k

As we are interested in the performance characteristics, the revenue function was selected from the states of each character-istic. Then we can compute performance characteristics solving the linear system of equations consisting of the |ST| Howardequations. In this work we calculated performance characteristics for a M/M/C retrial model to analyze the effects of differentparameter on model performance.

References[1] Leino J., Penttinen A;, Virtamo J., Flow level performance analysis of wireless data networks: A case study, Proceedings

of IEEE ICC., 2006, 3, 961–966

[2] Domenech-Benllach M.J., Gimenez-Guzman J.M., Pla, V., Martinez-Bauset, J., Casares-Giner, V., On the efficientsolution of a multiserver system with two reattempt orbits, Mathematical and Computer Modelling., 2010, 51, 1082–1092

[3] Gimenez-Guzman J., Domenech-Benlloch M., Pla, V., Casares-Giner, V., Martinez-Bauset, J., Value ExtrapolationTechnique to Solve Retrial Queues: A Comparative Perspective, ETRI Journal., June 2008, 30(3), 492–494

[4] Puterman M.L., Markov Decision Processes : Discrete stochastic Dynamie programming, Wiley series in Probabilityand statistics., 2008

222


Processus semi-markoviens a temps discret. Theorie et applications.

Asma Benchekor1, Abderrahmane Yousfate2

1 2 Laboratory of Mathematics, University of Sidi Bel-Abbes,P.O Box 89, Sidi Bel-Abbes 22000, Algeria

[email protected] University, Laboratory

Professional address, City, [email protected]

Abstract

On considere un processus stochastique a temps discret (Xn)n∈N a valeurs dans (E, E). Dans l’estimationde duree de processus dans un etat e ∈ E, les modeles de risque ont souvent ete etudies quand #E = 2 telque l’etat etudie soit transient et l’autre absorbant. Dans ce travail, on generalise le modele a un processus aplusieurs etats recurrents dont on etudie les durees de visite respectives par un modele semi-markovien.

Key words: Processus a temps discret,2010 Mathematics Subject Classification: .

1. IntroductionOn considere un processus aleatoire a temps discret (Xn)n∈N a valeurs dans (E, E) avec E = 1, ..., s.

On noteME l’ensemble des matrices reelles sur E × E etME(N) l’ensemble des matrices definies sur N et a valeursdansME . On note IE la matrice identite.

En considerant T0, T1, . . . Tn, . . . les instants successifs de changement d’etat verifiant 0 = T0 < T1 < . . . < Tn < . . .,on note Un = Tn−Tn−1 le temps de sejour dans l’etat apres l’instant du (n− 1)ieme changement d’etat (T1 etant le premierl’instant du 1ier changement d’etat). On considere que (XTn

)n∈N est une chaıne de Markov. En considerant que les Un etUn+1 sont independants sachant XTn

, le couple (XTn, Un)n∈N caracterise la dynamique semi-markovienne du processus

(Xn)n∈N.L’evolution du systeme dans le temps est decrite par:

Definition 1. Une suite de matrices q = (qij(k)) ∈ ME(N) est appelee noyau semi-Markovien a temps discret si ellesatisfait les trois proprietes suivantes:

1. 0 ≤ qij(k), i, j ∈ E, t ∈ N,

2. qij(0) = 0, i, j ∈ E,3.∑∞

k=0

∑j∈E qij(k) = 1, i ∈ E.

Si (X,T ) est une chaıne de renouvellement Markovien (homogene), on peut voir facilement que (XTn)n∈N est une

chaıne de Markov homogene. On note P = (pij)i,j∈E ∈ ME la matrice de transition de (XTn)n∈N. Elle est definie par

pij := P(XTn+1 = j/XTn = i), i, j ∈ E,n ∈ N. Notons aussi que, pour tout i, j ∈ E, pij peut etre exprime dans le noyau

223


semi-Markovien par pij =∑∞

t=0 qij(k). On introduit le noyau semi-Markovien cumule Q = (Q(k); t ∈ N) ∈ ME definipour tout i, j ∈ E et k ∈ N par:

Qij(k) := P(XTn+1 = j, Un+1 ≤ k/XTn= i)

Pour etudier les modeles de durees, on introduit les distributions de temps de sejour dans un etat donne et les distributionsconditionnelles en fonction de l’etat suivant a visiter. Dans ce cas, pour tout i, j ∈ E,

1. la distribution conditionnelle discrete s’ecrit :

fij(k) = P(Un+1 = k/XTn = i,XTn+1 = j), t ∈ N. (1)

2. la fonction de repartition conditionnelle associee s’ecrit :

Fij(k) = P(Un+1 ≤ k/XTn = i,XTn+1 = j) =k∑

l=0

fij(l)), t ∈ N. (2)

evidemment, pour tout i, j ∈ E et pour tout k ∈ N, on a

fij(k) =

qij(k)

pijsi pij ≥ 0

0 sinon

Pour etudier les modeles de durees sans tenir compte de l’etat suivant a visiter, on introduit aussi les distributions detemps de sejour dans un etat donne

Notons hi la distribution du temps de sejour dans l’etat i :

1. hi(k) := P(Un+1 = k/XTn= i) =

∑j∈E qij(k), t ∈ N.

2. Hi(k) := P(Un+1 ≤ k/XTn = i) =∑k

j∈E hi(l), k ∈ N.

Ainsi la fonction de survie s’ecrit Si(k) = 1−Hi(k) = P(T > k).Le modele semi-markovien constitue une generalisation du modele markovien exploitant ainsi la memoire du processus

entre deux changements d’etat par la loi de la duree dans le cas continu. Pour le cas markovien, la duree est exprimee par laloi exponentielle qui est une loi sans memoire.

Des que la fonction de survie est caracterisee, on peut determiner la mesure du processus, la fonction de risque (avec sesdifferentes approches), etc. Le modele est exploitable principalement en medecine, en fiabilite des systemes, en finances eten demographie.

2. ResultatsDans ce travail, on exploite les proprietes spectrales de la matrice de transition pour estimer la vitesse de convergence vers lastationnarite au cas ou le processus est asymptotiquement ergodique. Une etude de la periodicite du processus semi-markovienest egalement presentee.

References[1] Georgiadis, S. & Limnios, N. (2014) Interval reliability for semi-Markov systems in discrete time. Journal de la Societe

Franaise de Statistique.

[2] Korolyuk, V.S., Swishchuk, A. (1995). Random Evolution for Semi- Markov Systems. Kluwer Academic, Singapore..

[3] Pyke, R. (1961a) Markov renewal processes with finitely maNew York states. Ann. of Math. Statist., 32, 1243-1259.

224


DIMACOS-226


List of Authors

Aïssani, Djamil, 85

Abchir, Hamid, 106Abi Aad, Maria, 95Ahmia, Moussa, 143Aidene, Mohamed, 113Aliat, Billel, 128Amin, El Sahili, 95Amrouche, Karim, 111Amrouche, Said, 61Attia, Nehar, 116Attouch, Mohammed Kadi, 197

Belabbaci, Ahlem, 152Belabbaci, Amel, 156Belbachir, Hacène, 9, 58, 61, 64, 101, 120, 124, 145, 167,

181, 189Belhamra, Thara, 222Belkhir, Amine, 120, 167, 189Bellaouar, Slimane, 116Ben Amira, Aymen, 19Benatallah, Mohammed, 206Benchekor, Asma, 224Benmedjdoub, Brahim, 73Benmezai, Athmane, 58Benoumhani, Moussa, 52Benrabia, Imène, 124Benzid, Redha, 48Berkane, Djamel, 39Blouhi, Tayeb, 192Bouabbache, Zahia, 113Bouabsa, Wahiba, 197Boualem, Mohamed, 85Bouchaala, Houcine, 44Bouchemakh, Isma, 22, 73Boudabbous, Imed, 29, 41Boudaoui, Ahmed, 200Boudhar, Mourad, 81, 88, 91, 111Bouroubi, Sadek, 6Bousbaa, Imad Eddine, 12, 167Boussaha, Nadia, 134Boutabia, H, 218, 220Boutiche, Mohamed Amine, 97Bouyakoub, Abdelkader, 36, 58Boyadzhiev, Khristo N, 173

Caraballo, T., 200Chaari, Bechir, 19Chafi, Boudekhil, 138Chemli, Zakaria, 141

Cherfaoui, Mouloud, 85Cherroun, Hadda, 116, 152, 159Chouria, Ali, 12

Dammak, Jamel, 19, 41, 44Dil, Ayhan, 173Djebbar, Bachir, 156Djellab, Natalia, 222Dusart, Pierre, 39

El Sahili, Amin, 103

Fahssi, Nour-Eddine, 148Fariza, Krim, 145

Ganie, Hilal A., 108Guellouma, Younes, 152Guerbienne, Hafida, 132

Hamamache, Kheddouci, 103Hamdi, Fayçal, 126, 128, 132, 134Hamtat, Abdelkader, 209Haned, Amina, 91Harik, Hakim, 101Hemis, Rokia, 132

Igueroufa, Oussama, 64

Kara Terki, Nesrine, 136Khadir, Omar, 79

László, Németh, 1László, Szalay, 55, 143Lachemot, Tassadit, 212Laiche, Daouya, 22

Maamra, Mohammed Said, 164Maidoun, Mortada , 103Masrour, Samia, 36Mechab, Boubaker, 214Medjerredine, Assia, 181Mehdaoui, Abdelghani, 189Mekkia, Kouider, 103Merabet, Nour El Houda Asma, 48Meradji, S, 220Meradji, Selma, 218Messoudi, Fatma, 77Meziani, Nadjat, 88Mihoubi, Miloud, 77, 164, 170, 185, 204Mourid, Tahar, 5, 136

Nadia,Amri, 29


Ouahab, A., 200Ouahab, Abdelghani, 176, 192Oudrar, Djamila, 69Oulad-Naoui, Slimane, 159Oulamara, Ammar, 88

Pirzada, Sariefuddin, 108

Rahim, Asmaa, 185Rahmani, Mourad, 123Rim,Ben Hamadou, 29

Saggou, Hafida, 212Saidi, Yamina, 170Salem, Rahma, 44Si Kaddour, Hamza, 19

Sopena, Eric, 22, 73Souyah, Amina, 32Stihi, Sara, 218, 220

Tchier, Fairouz, 26Tebtoub, Assia Fettouma, 9Tellache, Nour El Houda, 81Tiachachat, Meriem, 204

Yagouni, Mohammed, 16Yahi, Zahra, 6Yaich, Mouna, 41Yalaoui, Farouk, 111Yousfate, Abderrahmane, 224

Zidani, Nesrine, 222

DAFx-06 Proceedings - Algeria, Sidi Bel Abbès

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