Contentsjromai/romaijournal/arhiva/2006/1/RJv2n1.pdf · Contents vii 23 The generalized algorithm for solving the fractional multiobjective transportation problem 197 Alexandra I

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Contents

1Motivation for multicriteria strategies in distributed system resource management 1Costel Aldea

2Some new properties of Lie triple systems 7Ilie Burdujan

3Some problems of the theory of compactifications of topological spaces 25Laurentiu I. Calmutchi, Mitrofan M. Choban

4New iterative chain classes of pseudo-boolean functions 53Vadim Cebotari, Mefodie Rata

5Applications to Lie algebras 57Camelia Ciobanu

6Perturbed bifurcation in a demand-offer model 65Elena Codeci

7Inverse problems of transitive closure 71Adrian Deaconu

8Further study of a microconvection model using the direct method 77Ioana Dragomirescu, Adelina Georgescu

9An optimization method for free surface steady flows 87Tiberiu Ioana, Titus Petrila

10Approximate inertial manifolds for an advection-diffusion problem 91Anca-Veronica Ion

v

vi

11Finite state machine testing from an OR state refinement design 109Florentin Ipate, Tudor Balanescu

12Sharpening of critical spectral parameter by pseudoperturbation method 119Boris V. Loginov*, Olga V. Makeeva**, Elena V. Foliadova**

13Existence results for a class of nonlinear Volterra

integrodifferential problem127

Rodica Luca–Tudorache

14Computer realization of the branching equation construction 135Oleg V. Makeev

15On a sparse matrix memorizing method 149Vlad Monescu, Silviu Dumitrescu, Bogdana Pop

16Regenerative process with a rare event 153Bogdan-Gh. Munteanu

17Stability of limit cycles in a calcium oscillations dynamics model 157Ileana Rodica Nicola

18Estimation of Hausdorff and packing linear measure and dimensions for

some plane sets of points169

Antonio Nuica

19Error handling in software systems: modeling and

testing with Finite State Machines175

Radu Oprisa

20Using stochastic Petri nets in performance analyze of distributed systems 181Norocel Petrache

21Solving method for multiple objective fuzzy constraints mixed variables

optimization189

Bogdana Pop, Silviu Dumitrescu, Vlad Monescu

22On the applicability of the Lindstedt-Poincare method in the two

dimensional case193

Nicolae - Doru Stanescu

Contents vii

23The generalized algorithm for solving the fractional multiobjective

transportation problem197

Alexandra I. Tkacenko

24LISC – A MATLAB package for linear differential problems 203Damian Trif

25Optimal control and stochastic uniform observability 209Viorica Mariela Ungureanu

26Mathematical models of a pipeline - pressure sensor mechanical system 219Petr A. Velmisov, Yuliya V. Pokladova

27Extending object-oriented databases for fuzzy information modeling 225Cristina-Maria Vladarean


MOTIVATION FOR MULTICRITERIASTRATEGIES IN DISTRIBUTED SYSTEMRESOURCE MANAGEMENT PROBLEMS

ROMAI J., 2, 1(2006),1–7

Costel AldeaUniversity ”Transilvania” of [email protected]

Abstract In the resource management systems the following aspects must be consid-ered: the virtual organization policies and the application requirements and,in some cases, the user preferences. According to these multicriteria strate-gies for distributed resource management should be formulated and consid-ered. Some available multicriteriea optimization techniques and methods arediscussed. The dynamic behaviour, uncertainty and incomplete informationare important aspects of the distributed system nature that gives the core ofthe resource management process. Some aspects of the resource managementprocess are: complete information about the accesible resource, applicationsrequirements and user preferences, association of the tasks with the resource inthe best possible ways, supporting the changes in the distributed system withrespect to distributed application control.

1. INTRODUCTIONIn order to describe the resource management process in a distributed sys-

tem it is important to know the participants to this process. There are alsomany aspects in modelling their preferences in order to obtain a good asso-ciation between the tasks and the resources. Nevertheless, one must observeother techniques for satisfying the requirements given by the dynamic behav-iour of the tasks related to the resources as well as for the uncertainty andincomplete information.

Some basic definition and notions related to multicriteria approach are in-troduced. The aim is to motivate the importance of multicriteria strategies indistributed systems resource management.

The performance and the high-throughput of the resource managementimplies the following comparation between two main resource managementstrategies: application level scheduling and job scheduling. It is obvious thatthe combination between those two strategies can bring together all the ben-efits of the both strategies using multicriteria strategies approach and thesuport of artificial inteligence techniques. An analysis of possible criteria thatcan be used in distributed system resource management process is presented

1

2 Costel Aldea

and how the preference models can be exploited in order to assign resourcesto the tasks in the most appropiate way is shown.

2. MULTICRITERIA APPROACHA formulation for the multicriteria decision problem is made and some basic

definition and various ways of modelling decisions maker’s preferences aredescribed.

Multicriteria decision problem. If there are several different ways to achievea goal, a decision problem needs to be solved. More then that, to reach thegoal, optimized choices of the best method must be used. These differentways are decision actions. They are also called the compromise solutions orschedules.

In general the main goal of the multicriteria decision making process is tobuild a global model of the decision makers’ preferences and exploit them inorder to find the solution that is accepted by a decision maker with respectto the values on all criteria. In order to accomplish this, preferential informa-tion concerning the importance of particular criteria must be obtained from adecisionmaker.

If these preferences are not available, then the only objective information isthe dominance relation in the set of decision actions.

Basic definitions. For the choice of the compromise solution the conceptsof non-dominated and Pareto optimal are used. There are two importantspaces to consider: decision variable space and the criteria space. The formercontains possible solutions along with values of their attributes. Let us denoteby D the set of solutions. The set D can be easily mapped into its imagein the criteria space, which creates the sets of points denoted by C(D). Thefunctions φ(x) represent values of criteria and their number is denoted by k.Definition 2.1 (Pareto dominance). The point z ∈ C (D) dominates z

′ ∈C (D) , denoted as z Â z

′, if and only if (∀) j ∈ 1, · · · , k , zj ≥ z

′j ∧ (∃) i ∈

1, · · · , k : zi > z′i (z

′is partially less than z). Thus, one point dominates

another if it is not worse with respect to all criteria and it is better for at leastone of them.Definition 2.2 (Pareto optimality). The point z

′ ∈ C (D) is said to be non-dominated with respect to the set C (D) if there is no z ∈ C (D) that dominatesz′. A solution is x Pareto optimal (efficient) if its image in the criteria space

is non-dominated.Definition 2.3 (Pareto-optimal set). The set of P ∗ of all Pareto-optimalsolutions is called the Pareto optimal set. Thus, it is defined asP ∗ =

x ∈ D

∣∣∣¬ (∃) x′ ∈ D : z

′ Â z

, where z′and z ∈ C (D)

Definition 2.4 (Pareto Front). For a given Pareto optimal set P ∗ , Paretofront (PF ∗) is defined as PF ∗ = z = f1 = z1, · · · , fk (x) = zk |x ∈ P ∗ . A

Motivation for multicriteria strategies in distributed system resource management 3

Pareto front is also called a non-dominated set because it contains all non-dominated points.

Preference models. A preference model defines a preference structure in theset of decision actions. Preference models can aggregate evaluations criteriainto:-functions (e.g weighted sums);-statements (e.g. binary relation: action a is at least as good as action b);-logical statements (e.g. decision rules: if condition on criteria then decision,where if condition on criteria is the conditional part of the rule and thendecision is the decision part of the rule).

3. MOTIVATION FOR MULTIPLE CRITERIAMulticriteria approaches focus on a compromise solution (in this case com-

promise schedule). In this way one can increase the level of satisfaction ofmany stakeholders of the decision making process and try to combine variouspoints of view, rather than provide solutions that are very good from only onespecific perspective as is curently common in other distributed system resourcemanagement approaches (grids). Such specific perspective result in different,often contradictory, criteria (along with preferences ) and make the process ofmapping jobs to resources difficult or even impossible. Consequently all thedifferent perspectives and preferences must be somehow aggregated to pleaseto all participants of the distributed system resource management process.

Various stakeholders and their preferences. Distributed systems schedulingand resource management potentially involves the interaction of many humanplayers (although possibly indirectly). These players can be devided into threeclasses:- end users making use of distributed system applications and portals;- resource administrators and owners;- virtual organization (VO) administrators and VO policy makers.

One goal of the resource management process is to automate the schedulingand resource management process in order to maximize stakeholders partici-pations in the entire process. This is why the final decision is often delegatedto such systems. By decision maker notion one means a scheduler and all hu-man players listed above are stakeholders of a decision making process. Thismodel assumes a single artificial decision maker in a VO. However, in realdistributed systems, multiple stakeholders (agents) may often act accordingto certain strategies, and the decision may be made by means of negotiationsbetween these agents. Such an approach requires distributed decision makingmodels with multiagent interactions [1], which will not be discussed here.

One refers to the assignment of resources to tasks as a solution or a sched-ule. Since one considers many contradictory criteria and objective functions,

4 Costel Aldea

finding the optimal solution is not possible. The solution that is satisfactoryfrom all the stakeholders points of view and that takes into consideration allcriteria is called the compromise solution.

The need for formulation of distributed systems resource management as amulticriteria problem follows from the characteristics of the distributed systemenvironment itself. Such an environment has to meet requirements of differentgroups of stakeholders listed at the beginning of this section. Thus, variouspoints of view and policies must be taken into consideration, and results needto address different criteria for the evaluation of schedules. Different stake-holders have different, often contradictory, preferences that must be somehowaggregated to please all stakeholders. For instance, administrators require ro-bust and reliable work of the set resources controlled by them, while end usersoften want to run their applications on every available resource. Additionally,site administrators often want to maintain a stable load of their machinesand thus load-balancing criteria become important. Resource owners want toachieve maximal throughput (to maximize work done in a certain amount oftime), while end users expect good performance of their application or, some-times, good throughput of their set of tasks. Finally, VO administrators andpolicy makers are interested in maximizing the whole performance of the VOin the way that satisfies both end users and administrators.

Different stakeholders are not the only reason for multiple criteria. Usersmay evaluate schedules simultaneously with respect to multiple criteria. Forexample, one set of end users may want their applications to complete as soonas possible, whereas another one try to pay the minimum cost for the resourcesto be used. Furthermore, the stakeholders may have different preferences eveninside one group. For example, a user may consider time of execution moreimportant than the cost (which does not mean that the cost is ignored).

Job Scheduling involves the use of one central scheduler that is responsiblefor assigning the distributed system resources to applications across multipleadministrative domains. There are three main levels: a set of applications,a central scheduler, and the distributed system resources. In principle, thisapproach is in common use and can be found today in many commercial andpublic-domain job-scheduling systems. A detai1ed analysis and their compar-ative study can be found in [2].

Note that the dynamic behaviour seen in distributed systems environmentsis most often the result of competing jobs executed on the resources. Byhaving a control over all the applications, as happens in the job schedulingapproach, a central scheduler is able to focus on factors that cause variable,dynamic behavior, whereas application-level schedulers have to cope with thiseffects.

Application -Level Scheduling. With application-level scheduling, applica-tions make scheduling decisions themselves, adapting to the availability of

Motivation for multicriteria strategies in distributed system resource management 5

resources by using additional mechanisms and optimizing their own perfor-mance. That is, applications try to control their behavior on the distributedsystem resources independently. Note that all resource requirements are inte-grated within applications.

Such an assumption causes many ambiguities in terms of the classic schedul-ing definitions. When one considers classic scheduling problems, the main goalis to effectively assign all the available task requirements to available resourcesso that they are satisfied in terms of particular objectives. In general, this def-inition fits more the job scheduling approach, where job scheduling of systemstake care of all the available applications and map them to resources in asuitable way. In both cases application-level scheduling differs from the job-scheduling approach in the way applications compete for the resources. Inapplication-level scheduling, an application schedule have no knowledge of theother applications, and in particular of any scheduling decisions made by an-other application. Hence, if the number of applications increases (which is acommon situation in large and widely distributed systems like a Grid), oneobserve worse scheduling decisions from the system perspective, especially interms of its overall throughput. Moreover, since each application must have itsown scheduling mechanisms, information processes may be unnecessarily du-plicated (because of the lack of knowledge about other applications’ schedulingmechanisms).

Hence, a natural question arises: why does one need the application levelscheduling approach? In practice, job scheduling systems do not know aboutmany specific internal application mechanisms, properties, and behaviors. Ob-viously, many of them can be represented in the form of before mentionedhard constraints, as applications requirements, but still it is often impossibleto specify all of them in advance. More and more, end users are interestedin new Grid-aware application scenarios. New scenarios assume that dynamicchanges in application behavior can happen during both launch-time and run-time. Consequently, Grid-aware applications adapt on-the-fly to the dynam-ically changing Grid environment if a pure job scheduling approach is used.The adaptation techniques, which in fact vary with classes of applications, in-clude different application performance models, and the ability to checkpointand migrate.

Consequently, job scheduling-systems are efficient from high-throughputperspective (throughput based criteria) where as application-level schedulersmiss this important criterion, even though internal scheduling mechanisms cansignificantly improve particular application performance (application perfor-mance based criteria).

Hard and soft constraints. By extending present job-scheduling strategies(especially to the war application requirements are expressed), one is able todeal with both throughput and application-centric needs. As noted in the

6 Costel Aldea

previous section, a central scheduler receives many requests from different endusers to execute and control their applications. Requests are represented inthe form of hard constraints, a set of applications and resources requirementsthat must be fulfilled in order to run an application or begin any other actionon particular resources. These constraints are usually specified by a resourcedescription language such as RSL in the Globus Toolkit, ClassAds in Con-dor, or JDL in DataGrid. Simply stated, a resource description languageallows the definition of various bald constraints, such as the operating systemtype, environment variables, memory and disk size, and number and perfor-mance of processors, and generally varies with classes of applications on thedistributed system. Many extensions to the resource description languagesare needed to represent soft constraints, the criteria for resource utilization,deadlines, response time, and so forth needed when many stakeholders areinvolved. Unfortunately, soft constraints are rarely expressed in the resourcedescription language semantics, and consequently they are omitted in manyresource management strategies.

The main difference between hard and soft constraints is that hard con-straints must be obeyed, where as, soft constraints should be taken into con-sideration in order to satisfy all the stakeholders as far as possible. In otherwords, one is able to consider soft constraints if and only if all hard con-straints are met. Yet, soft constraints are actually criteria in the light ofdecision making process, and they are tightly related to the preferences thatthe stakeholders of the process want to consider.We have indicated that resource management in a distributed environmentrequires a multicriteria approach. However, proposition of a method copyingwith multiple criteria during the selection of the compromise schedule doesnot solve the resource management problem entirely.

References[1] Th. W. Sandholm, Distributed rational decision making, Multiagent Systems: a modern

Approach to Distributed Artificial Intelligence, MIT Press, 1999, 201-258.

[2] G. Kris, El-Ghazawi Tarek, N. Alexandridis, F. Vroman, J. R. Radzikowski, PreeyapongSamipagdi, S. A. Suboh, An empirical comparatie study of job management systems.Concurrency: Practice and Experience, to appear, 2003.

[3] G. Allen, D. Angulo, T. Goodale, T. Kielmann, A. Merzky, J. Nabrzyski, J. Pukacki,M. Russel, T. Radke, E. Seidel, J. Schaf, I. Taylor, GridLab:Enabling application onthe grid, in Proceedings of the Third International Workshop on Grid Computing(Grid2002), November 2002.

[4] I. Foster, C. Kesselman, The Globus Project: a status report, Proceedings of the SeventhHeterogenous Computing Workshop, 1998.

[5] C. Aldea, Measuring the performance for parallel matrix multiplication algorithm, Pro-ceedings of the International Conference on Theory and Applications in Mathematicsand Informatics, Alba-Iulia, Romania, 2005.

[6] European DataGrid Project, http://www.eu-datagrid.org.

SOME NEW PROPERTIES OF LIE TRIPLESYSTEMS

ROMAI J., 2, 1(2006),7–23

Ilie BurdujanDept. of Math., Univ. of Agricultural Sciences and Veterinary Medicine”Ion Ionescu de la Brad” Iasiburdujan [email protected]

Abstract The categories LTS of Lie triple systems (briefly, Lts) and LTALG of LT -algebras are considered. There exists a covariant functor LT : LTALG → LTS

that assigns to each LT -algebra A a specific Lts denoted by LT(A). LT has aleft adjoint functor, i.e., there exists a functor U : LTS → LTALG having theproperty HomLts(T, LT(A)) ≈ HomAlg(U(T ), A). The existence of U givesrise to the construction of the universal enveloping LT -algebra. For any LtsT a universal enveloping algebra U(T ) was constructed [4] as being the LT -algebra obtained by factorizing the nonassociative tensor algebra of T by anappropriate ideal. The usual results concerning such a universal envelopingwere proved in a similar way as the corresponding results on the universalenveloping algebra of a Lie algebra. A similar result as that stated in Poincare-Birkhoff-Witt Theorem is proved, namely: the cosets of 1 and the standardmonomials for a basis of Lts T form a basis for the universal enveloping U(T ).

Keywords: Lie triple system, universal enveloping algebra.

2000 MSC: 17A40, 17B35

1. INTRODUCTIONLie triple systems (briefly, Ltss) are subspaces of any Lie algebra L([., .])

which are closed under the ternary composition [x, y, z] = [[x, y], z]. Theywere first noted by E. Cartan in his study on totally geodesic submanifolds[5]. More exactly, if G is the Lie algebra of the Lie group G, then Ltss in G

are connected with the totally geodesic subspaces of G in the same way thatLie subalgebras of G are related to analytic subgroups of G. The Ltss arosealso in the studies on symmetric spaces [5], [20], [22]. They were also usedin differential geometry by P. I. Kovaljov in the study of certain manifoldsendowed with affine connections [17], [18]. It must be also remarked that theyarose naturally in the investigation by H. Freudenthal (1954) on the geometriesof exceptional simple Lie groups [8]. Recently, N. Kamiya and S. Okubo [16]connect Ltss with the study of the Yang-Baxter equation.

From the algebraic point of view, Ltss were studied by N. Jacobson [14],[15], W. G. Lister [19], K. Yamaguti [25], [26], [27], B. Harris [9], N. Hopkins

7

8 Ilie Burdujan

[11], [12], [13], T. Hodge [10] et all. Actually, they were introduced by N. Ja-cobson [14] as being the abstract algebraic structures describing the subspacesof an associative algebra that are closed relative to the ternary operations[[x, y], z], where [x, y] = xy − yx. A crucial moment in the development ofLts structural theory was the result, proved in 1951 by N. Jacobson [14], sta-ting that every Lts (T, [·, ·, ·]) can be embedded into a Lie algebra (L, [·, ·])such that its ternary operation is a superposition of Lie brackets of L (i.e.,[x, y, z] = [[x, y], z],∀ x, y, z ∈ T ). This embedding suggests that Ltss are thetangent algebras of particular homogeneous spaces.

Another important class of Ltss is connected with a class of commuta-tive algebras satisfying an identity of degree 4 , the so called LT -algebras[4]. This connection is yielded by means of the ternary operation [x, y, z] =x · (y · z)− y · (x · z). Moreover, such a commutative algebra can be associatedwith every nonassociative algebra, that allows us to give an embedding resultfor Ltss similar to Jacobson’s embedding result. It was proved [4] that a uni-versal enveloping LT -algebra can be associated with every Lts. More exactly,for any Lts T , the universal enveloping LT -algebra U(T ) is a nonassociativealgebra having the property that any Lts homomorphism of T into the Ltsassociated with a LT -algebra A extends to an algebra isomorphism of U(T )into A. The algebra U(T ) is a quotient of the nonassociative tensor algebraTT by a suitable two sided ideal and it is a filtered algebra.

In this paper we prove, for U(T ), a similar result to that stated in thePoincare-Birkhoff-Witt theorem for universal enveloping of a Lie algebra, namely:the cosets of 1 and the standard monomials for a basis of Lts T form a basisfor the universal enveloping U(T ).

2. PRELIMINARIESDefinition 2.1 A Lie triple system (briefly, Lts) is a vector space T over thefield K, with a ternary composition [·, ·, ·] : T × T × T → T, (a, b, c) → [a, b, c],which is threelinear and satisfies the following axioms:

(Lts.1) [x, x, y] = 0,(Lts.2) [x, y, z] + [y, z, x] + [z, x, y] = 0,(Lts.3) [x, y, [u, v, w]] = [[x, y, u], v, w] + [u, [x, y, v], w] + [u, v, [x, y, w]],

∀ x, y, z, u, v, w ∈ T .

We consider, for every x, y ∈ T , the vector space endomorphism D(x,y) :T → T defined by D(x,y)(z) = [x, y, z]; (Lts. 3) assures us that D(x,y)

is a Lts-derivation which is called a inner derivation of T. Moreover, D =SpanKD(x,y)| x, y ∈ T has a Lie algebra structure; D is the so-called innerderivation algebra and it is also denoted by InnDer(T ).

A Lts is called Abelian if [x, y, z] = 0 for any triple (x, y, z). Of course,

Some new properties of Lie triple systems 9

every K-module T is an Abelian Lts by defining the Lts-bracket as [x, y, z] =0,∀x, y, z ∈ T .

Let LTSK denote the category of Lie triple systems over K with the Ltshomomorphisms as morphisms.

If L is a Lie algebra with product [a, b], then the ternary composition[a, b, c] = [[a, b], c] satisfies the above identities. In 1951 N. Jacobson showed[15] that any Lts may be considered as a subspace of a Lie algebra L in sucha way that [a, b, c] = [[a, b], c]; this Lie algebra is called the standard envelop-ing Lie algebra of T , and it is denoted by Ls(T ) or merely Ls; the naturalembeding map of T into Ls(T ) is 1-1. It was also constructed a universal en-veloping Lie algebra Lu(T ), with the property that any homomorphism of Tinto a Lie algebra extends to a homomorphism of Lu(T ). Further, a universalassociative algebra (with identity) U(T ) with the same property relative to thehomomorphisms of T into associative algebras was constructed. U(T ) is justthe universal associative algebra of Lu(T ); the natural maps of T into U(T )and Lu(T ) are 1-1. In the Abelian case, Ls(T ) identifies with the AbelianLie algebra having underlying vector space T . Also, Lu(T ) = T ⊕ Λ2T , with[x, y] = x ∧ y for x, y ∈ T and with all other products vanishing.

The existence of a new kind of universal enveloping algebra for Ltss wasproved in [4]. This is neither a Lie algebra nor an associative one, but it isa commutative algebra satisfying a weak associative law, the so-called LT -algebra. Recall the definition of the LT -algebra.

Definition 2.2 Any commutative algebra A(·) (with or without identity ele-ment) satisfying the identity

2((y · x) · x) · x + y · (x2 · x) = 3(y · x2) · x, ∀x, y ∈ A (1)

is called a LT -algebra.

By linearizing (2.1) we get its equivalent form

f(x, y, z, u) = 0 (2)

where

f(x, y, z, u) = x·(y·(z·u)−y·(x·(z·u))−(x·(y·z))·u++(y·(x·z))·u−z·(x·(y·u))+z·(y·(x·u)).(3)

As examples we consider: 1 LT-algebras obtained by factorizing a commu-tative algebra A(·) by the ideal generated by f(x, y, z, w);x, y, z, w ∈ A [4],2 Jordan algebras satisfying the conditions from Corollary of Theorem 2 [23]are LT-algebras.

Let us denote by LTALGK the category of LT -algebras over K with thealgebra homomorphisms as morphisms.

10 Ilie Burdujan

Every LT -algebra A(·) becomes a Lts relative to the composition

[x, y, z] = x · (y · z)− y · (x · z), ∀x, y, z ∈ A; (4)

this Lts is denoted by LT(A). Consequently, there exists a covariant functorLT : LTALGK → LTSK , carrying every LT -algebra A(·) in LT(A). It mustbe remarked that every inner derivation for LT(A) is a derivation for A(·).Moreover, the following identities hold in any LT-algebra

x1 · (x2 · (a1 · (a2 · ... · (am−1 · am)...)) = (5)

= x2 · (x1 · (a1 · (a2 · ... · (am−1 · am)...))+

+D(x1,x2)(a1 · (a2 · ... · (am−1 · am)...)), m ≥ 2,

a1 · (a2 · ... · (ai−1 · (ai · (ai+1 · (ai+2 · ... · (am−1 · am)...)) = (6)

= a1 · (a2 · ... · (ai−1 · (ai+1 · (ai · (ai+2 · ... · (am−1 · am)...))+

a1 · (a2 · ... · (ai−1 · (D(ai,ai+1)(ai+2 · ... · (am−1 · am)...)).

Notations. For the later use we introduce the notations

[x, y, z]⊗ = x⊗ (y ⊗ z)− y ⊗ (x⊗ z).

f⊗(x, y, z, w) = x⊗ (y ⊗ (z ⊗ w))− y ⊗ (x⊗ (z ⊗ w))− z ⊗ (x⊗ (y ⊗ w)+

+z ⊗ (y ⊗ (x⊗ w))− (x⊗ (y ⊗ z))⊗ w + (y ⊗ (x⊗ z))⊗ w.

They are obtained by substituting everywhere in the expressions (2.4), (2.3)of [x, y, z], f(x, y, z, u), respectively, the binary operation symbol ”· ” by ”⊗”. In [4] the existence of a left adjoint for LT was proved; this is equivalentwith the construction of a universal enveloping algebra U(T ) for any Lts T .U(T ) is obtained by a factorization of the nonassociative tensor algebra TTby an apropriate ideal. Recall that

TT = T⊗0 ⊕ T⊗1 ⊕ T⊗2 ⊕ ...⊕ T⊗n ⊕ ....

where, T⊗0 = K1, T⊗1 = T and, for n ≥ 2,

T⊗n =n−1∑

i=1

T⊗i ⊗ T⊗n−i;

the product of v ∈ T⊗i and w ∈ T⊗j is defined as v · w = v ⊗ w. Thealgebra TT has the following universal property: for any K-algebra A andany K-linear map f : T → A there exists a unique algebra homomorphismf0 : TT → A extending f . In other words, the functor T is left adjoint to


the underlying functor to K-vector space which forgets the algebra structure.This assertion is easily proved by noting that f0([v1⊗ ...⊗vn]s) may be definedby [f(v1) · f(v2) · ... · f(vn)]s (here [v1 ⊗ ... ⊗ vn]s denotes the tensor productof the ordered set (v1, ..., vn) realized in pairs delimited by a fixed selection sof parenthesis, and [f(v1) · f(v2) · ... · f(vn)]s is the element obtained in A bycomposing the elements f(vi) and preserving the selection s of parenthesis.

Definition 2.3 Let T be a Lts over K. The pair (U, i), where U is a LT -algebra over K and i is a Lts-homomorphism from A to LT(U) (i.e. it is alinear map such that

i([x, y, z]) = i(x) · [i(y) · i(z)]− i(y) · [i(x) · i(z)], ∀ x, y, z ∈ T ), (7)

is called a universal enveloping LT -algebra of T if the following propertyholds: for any LT -algebra B and any Lts-homomorphism j from T to LT(B),there exists a unique homomorphisms of LT -algebras Φ : U → B such thatΦ i = j.

The uniqueness of the universal enveloping LT -algebra (U, i) of T is easilyprovable in a standard way. Further, the LT -algebra U = U(T ) is generatedby i(T ). In order to prove the existence of a suitable pair (U, i) we considerTT and factorize it by the two sided ideal J generated by x ⊗ y − y ⊗x, x⊗ (y⊗ z)− y⊗ (x⊗ z)− [x, y, z], f⊗(x, y, z, w)| x, y, z, w ∈ T, where T isconsidered naturally imbedded in TT. We put U = U(T ) = TT/J , denoteby π : T(T ) → U(T ) the canonical homomorphism, and define i : T → U(T )to be the restriction of π to T . Notice that J ⊂ ⊗∞i=1T

⊗i(T ), so π mapsT isomorphically into T ⊂ TT, and consequently, i is a 1-1 map. Then,(U(T ), i) is a universal enveloping algebra of T . In [4] the following result wasproved.

Theorem 2.1 1. Let T1 and T2 be two Ltss, (U1, i1), (U2, i2)-their corre-sponding universal enveloping LT -algebras, and α : T1 → T2 be a LT -algebrahomomorphism. Then, there exists a unique LT -homomorphism α′ : U1 → U2

such that α′ i1 = i2 α.2. Let B be a two-sided ideal of T and let R be the two-sided ideal of U gen-erated by i(B). If a ∈ T , then j : a + B → i(a) + R is a Lts-homomorphismof T/B and LT(V), where V = U/R; further (V, j) is the universal envelopingLT -algebra for T/B.3. If D is a derivation for the Lts T , then there exists a uniquely definedderivation D′ for the LT -algebra U such that D′ i = i D.

3. STANDARD POLYNOMIALSIn what follows, we consider the Lts T having the basis X = xjj∈J

where J is an ordered set (possible, a finite one). In particular, we can con-sider X = x1, x2, ..., xn, ... or X = x1, x2, ..., xn. Let us consider the set

12 Ilie Burdujan

S`[X] ⊂ TX recurrently defined by

1 . X ⊂ S`[X],2 . xi1 ⊗ (xi2 ⊗ ..⊗ (xim−1 ⊗ xim)..)) ∈ S`[X], if i1 ≤ i2 ≤ .. ≤ in, ∀m ≥ 2.Every element of S`[X] is called a left standard nonassociative monomial.

Now, we can define the set S[X] ⊂ TX by

1 . S`[X] ∈ S[X],2 . if xj ∈ X, v, w ∈ S[X] \X, then, (v)⊗ xj , (v)⊗ (w) ∈ S[X],3 . if xj ∈ X,u ∈ S[X] \ S`[X], then xj ⊗ (u) ∈ S[X].

The elements of S[X] are called the standard nonassociative monomialsover X. Any linear combination of standard monomials is called a standardpolynomial over X.

4. THE ANALOG OFPOINCARE-BIRKHOFF-WITT THEOREM

Let T be a K-Lts and U(T ) be its universal enveloping. Also, let X =x1, x2, ..., xn, ... be a basis for T and tijkm ∈ K be its structure constants.Firstly, we shall prove the following result.

Lemma 4.1 Let T be a Lts and TT be its nonassociative tensor algebra.Then, the following identities(i) ui ⊗ uj = uj ⊗ ui(modJ),(ii) ui ⊗ (uj ⊗ uk) = uj ⊗ (ui ⊗ uk) + [ui, uj , uk](modJ),(iii) ui ⊗ (uj ⊗ (uk ⊗ u`)) = uj ⊗ (ui ⊗ (uk ⊗ u`)) + uk ⊗ [ui, uj , u`] ++ [ui, uj , uk]⊗ u`(modJ),(iv) uj1 ⊗ (uj2 ⊗ (uj3 ⊗ ...⊗ (ujm−1 ⊗ ujm)...)) ==uj2 ⊗ (uj1 ⊗ (uj3 ⊗ ...⊗ (ujm−1 ⊗ ujm)...)) ++D⊗

(uj1,uj2

)(uj3⊗...⊗(ujm−1⊗ujm)...) =uj2⊗(uj1⊗(uj3⊗...⊗(ujm−1⊗ujm)...))++ D(uj1

,uj2)(uj3 ⊗ ...⊗ (ujm−1 ⊗ ujm)...)(modJ),m ≥ 5,

hold for all ui, uj , uk, u`, uj1 , uj2 , ..., ujm ∈ T ; here D⊗(x,y) and D(x,y) are de-

fined, respectively, by

D⊗(x,y)([a1, a2, ..., an]s) = (8)

=n∑

i=1

[a1, a2, ..., ai−1, [x, y, ai]⊗, ai+1, ..., an]s,

D(x,y)([a1, a2, ..., an]s) =


=n∑

i=1

[a1, a2, ..., ai−1, [x, y, ai], ai+1, ..., an]s.

Proof. The proof must certainly take account of the generators for J .Obviously, we have ui ⊗ uj = uj ⊗ ui + (ui ⊗ uj − uj ⊗ ui) = uj ⊗ ui(modJ).From the definition of [ui, uj , uk]⊗, we get

ui ⊗ (uj ⊗ uk) = uj ⊗ (ui ⊗ uk) + [ui, uj , uk]⊗ =

= uj ⊗ (ui ⊗ uk) + ([ui, uj , uk]⊗ − [ui, uj , uk]) + [ui, uj , uk] =

= uj ⊗ (ui ⊗ uk) + [ui, uj , uk](modJ).

In order to prove formula (iii) we make use of the identity

x1 · (x2 · (x3 · x4)) = (9)

= x2 · (x1 · (x3 · x4)) + [x1, x2, x3] · x4 + x3 · [x1, x2, x4] + f(x1, x2, x3, x4)

which is satisfied for every binary algebra A(·) with [x, y, z] and f(x, y, z, w)defined by (2.4) and (2.3), respectively. Replacing everywhere ”· ” by ”⊗” weget

ui ⊗ (uj ⊗ (uk ⊗ u`)) = uj ⊗ (ui ⊗ (uk ⊗ u`)) + uk ⊗ [ui, uj , u`]⊗ ++ [ui, uj , uk]⊗ ⊗ u` + f⊗(ui, uj , uk, u`) = uj ⊗ (ui ⊗ (uk ⊗ u`)) ++ uk ⊗ ([ui, uj , u`]⊗ − [ui, uj , u`]) + uk ⊗ [ui, uj , u`] ++ ([ui, uj , uk]⊗ − [ui, uj , uk])⊗ u` + [ui, uj , uk]⊗ u` + f⊗(ui, uj , uk, u`) == uj ⊗ (ui ⊗ (uk ⊗ u`)) + uk ⊗ [ui, uj , u`] + [ui, uj , uk]⊗ u`(modJ).

We shall prove now, recurrently, formula (iv). Formula (iii) shows us that (iv)is valid in the case m = 4. Suppose that (iv) is valid for certain m. We put itin the form

uj1 ⊗ (uj2 ⊗ (uj3 ⊗ ...⊗ (ujm−1 ⊗ ujm)...)) = (10)

= uj2 ⊗ (uj1 ⊗ (uj3 ⊗ ...⊗ (ujm−1 ⊗ ujm)...))+

+m−1∑

i=3

uj3⊗(uj4⊗...⊗(uji−1⊗([uj1 , uj2 , uji ]⊗(uji+1⊗...⊗(ujm−1⊗ujm)...))...)+

+uj3 ⊗ (uj4 ⊗ ...⊗ (ujm−1 ⊗ [uj1 , uj2 , ujm ])...)(modJ), m ≥ 5.

Replacing in this formula ujm by ujm ⊗ ujm+1 and taking into account that[x, y, z]⊗ is congruent mod J with [x, y, z], the previous formula can be writtenin the form

uj1 ⊗ (uj2 ⊗ (uj3 ⊗ ...⊗ (ujm−1 ⊗ (ujm ⊗ ujm+1))...)) =

14 Ilie Burdujan

= uj2 ⊗ (uj1 ⊗ (uj3 ⊗ ...⊗ (ujm−1 ⊗ (ujm ⊗ ujm+1))...))+

+m−1∑

i=3

uj3⊗..⊗(uji−1⊗([uj1 , uj2 , uji ]⊗(uji+1⊗..⊗(ujm−1⊗(ujm⊗ujm+1))..)))..)+

+uj3 ⊗ (uj4 ⊗ ...⊗ (ujm−1 ⊗ [uj1 , uj2 , ujm ⊗ ujm+1 ]⊗)...)(modJ).

To conclude, it is enough to remark the readily provable identity

[uj1 , uj2 , ujm ⊗ ujm+1 ]⊗ = [uj1 , uj2 , ujm ]⊗ ⊗ ujm+1+

+ujm ⊗ [uj1 , uj2 , ujm+1 ]⊗ + f(uj1 , uj2 , ujm , ujm+1).

This completes the proof of Lemma.Formulae (ii)-(iv) give the opportunity to make the following remarks:

1) since [ui, uj , uk] ∈ T it follows that it is a standard polynomial of degree 1,2) since [ui, uj , uk] ∈ T it follows that uk ⊗ [ui, uj , u`] + [ui, uj , uk] ⊗ u` iscongruent modJ with a standard polynomial of degree 2,3) the element

D⊗(uj1

,uj2)(uj3 ⊗ ...⊗ (ujm−1 ⊗ ujm)...)

is congruent modJ with a standard polynomial of degree m− 2.

Then, we can restate Lemma 4.1 in the form.

Lemma 4.2 (Switching Lemma) Let T be Lts and TT be its nonasso-ciative tensor algebra. Then, the following identities hold:(1’) ui ⊗ uj = uj ⊗ ui(modJ),(2’) ui ⊗ (uj ⊗ uk) = uj ⊗ (ui ⊗ uk) + P⊗

1 (x1, ..., xn, ...)(modJ),(3’) ui ⊗ (uj ⊗ (uk ⊗ u`)) = uj ⊗ (ui ⊗ (uk ⊗ u`)) + P⊗

2 (x1, ..., xn, ...)(modJ),(4’) uj1 ⊗ (uj2 ⊗ (uj3 ⊗ ...⊗ (ujm−1 ⊗ (ujm)...)) =

= uj2⊗(uj1⊗(uj3⊗...⊗(ujm−1⊗(ujm)...))+P⊗m−2(x1, ..., xn, ...)(modJ), m ≥ 5,

where P⊗k x1, ..., xn, ...)(k = 1, 2, ...) are left standard polynomials of degree k

(i.e. they are finite linear combinations of left standard monomials of S[X]).

It must be remarked that P⊗k (x1, ..., xn, ...) can depends on those elements from

X which occur in the linear combinations expressing the elements defining theanalyzed monomial as well as those introduced by the expressions of [ui, uj , uk]in the basis X.

Corollary 4.1 The following formula holds

uj1 ⊗ (uj2 ⊗ ...⊗ (uji−1 ⊗ (uji ⊗ (uji+1 ⊗ ...⊗ (ujm−1 ⊗ (ujm)...)) = (11)

= uj1 ⊗ (uj2 ⊗ ...⊗ (uji−1 ⊗ (uji+1 ⊗ (uji ⊗ (uji+2 ⊗ ...⊗ (ujm−1 ⊗ (ujm)...))+


+P⊗m−i−1(x1, ..., xn, ...)(modJ), m ≥ 3,

where P⊗m−i−1(x1, ..., xn, ...) depends on those elements from X which occur in

connection of uji , uji+1 , ..., ujm.

Proof . The assertion follows from Lemma 4.2 applied to uji ⊗ (uji+1 ⊗ ... ⊗(ujm−1 ⊗ (ujm)...) and taking into account that uj1 ⊗ (uj2 ⊗ ...⊗ (uji−1 is notinvolved in the computations.

In what follows, we need to consider the subspace T`T of TT, namedthe left tensor algebra of T . It is defined as

T`T = T⊗0` ⊕ T⊗1

` ⊕ T⊗2` ⊕ ...⊕ T⊗n

` ⊕ ...

where T⊗0` = K1, T⊗1

` = T, and T⊗n` = T ⊗ T⊗n−1

` for n ≥ 2.

Lemma 4.3 Every element of T`T is congruent mod J to a K-linear com-bination of 1 and left standard monomials.

Proof. The assertion is clear for K1 and monomials of degree 1. Let X =x1, x2, ..., xn, ... be a basis for T . It is sufficient to apply the results of Lemma4.2 to monomials in X. For any left monomial of degree n ≥ 2, we shall provethe following formula

xj1 ⊗ (xj2 ⊗ (xj3 ⊗ ...⊗ (xjn−1 ⊗ (xjn))...) = (12)

= xi1 ⊗ (xi2 ⊗ (xi3 ⊗ ..⊗ (xin−1 ⊗ (xin))..) + P⊗n−2(x1, x2.., xk, ...)(modJ),

where (i1, i2, .., in) is the set, obtained by putting in natural order the elementsof the set (j1, j2, .., jn), and P⊗

n−2(x1, x2..., xk, ...) is a left standard polynomialof degree n − 2, i.e. a (finite) linear combination of elements from S`[X] . Ifthe set (j1, j2, .., jn) has k inversions, then we get (4.12) after k times of usingLemma 4.2. Of course, P⊗

n−2(x1, x2..., xk, ...) can depend (not necessarily) onxj1 , xj2 , ..., xjn as well as on those xj occurring through the usual formulaedefining the structure constants [xj1 , xj2 , xj3 ] =

∑tij1j2j3

xi of the Lts T .

Lemma 4.4 Every element of TT is congruent mod J to a K-linear com-bination of 1 and left standard monomials.

Proof. Since every monomial of degree n in X is uniquely obtained as sometensorial products of elements from T`T, the proof is the result of usingLemma 4.4 for every element of T`T which is factor of the analyzed mono-mial.

We prove now that the classes mod J of 1 and standard monomials arelinearly independent and, consequently, they represent a basis in U(T ). To

16 Ilie Burdujan

this aim we consider the vector spaces Bn spanned by all standard monomialsof degree n and the vector space B = K1 ⊕ B1 ⊕ ... ⊕ Bn ⊕ .... Their linearindependence follows from the following statement.

Lemma 4.5 There exists a linear mapping σ between TT and B such that

σ(1) = 1, (13)

σ([xj1 ⊗ xj2 ⊗ ...⊗ xjn ]s) = (14)

= [xj1 · xj2 · ... · xjn ]s ∈ S[X], ∀[xj1 ⊗ xj2 ⊗ ...⊗ xjn ]s ∈ S[X]

σ([xj1 ⊗ xj2 ⊗ ...⊗ xji−1 ⊗ [xk1 , xk2 , xk3 ]⊗ ⊗ xji+1 ⊗ ...⊗ xjn−2 ]s) = (15)

= σ([xj1 ⊗ xj2 ⊗ ...⊗ xji−1 ⊗ [xk1 , xk2 , xk3 ]⊗ xji+1 ⊗ ...⊗ xjn−2 ]s),

i ∈ 1, 2, ..., n, n ≥ 3

σ([xj1 ⊗ xj2 ⊗ ..⊗ xji−1 ⊗ f⊗(xk1 , xk2 , xk3 , xk4)⊗ xji+1 ⊗ ..⊗ xjn−3 ]s) (16)

= 0, i ∈ 1, 2, ..., n, n ≥ 4

Proof. This definition is connected with the number of the inversions that arenecessary to put the set of label in the natural order. We define it such thatto take as the value on a monomial the corresponding standard polynomial.Actually, the existence of σ is proved if the standard polynomial correspondingto a monomial is unique. Certainly, it is the case of monomials of degree 2.That is why we set

σ(1) = 1, (17)

σ(xj) = xj , ∀xj ∈ X. (18)

Taking into account Lemma 4.1 (1) we define

σ(xi ⊗ xj) = xi · xj , if i ≤ j, (19)

σ(xi ⊗ xj) = xj · xi, if i > j.

A monomial of degree 3 can have 0, 1, 2 or 3 inversions. According to thesecases we define

σ(xi ⊗ (xj ⊗ xk)) = xi · (xj · xk)), if i < j < k (20)

σ(xi ⊗ (xj ⊗ xk)) = xi · (xk · xj)), if i < k < j

σ(xi ⊗ (xj ⊗ xk)) = xj · (xi · xk)) + [xi, xj , xk], if j < i < k

σ(xi ⊗ (xj ⊗ xk)) = xj · (xk · xi)) + [xi, xj , xk], if j < k < i

σ(xi ⊗ (xj ⊗ xk)) = xk · (xi · xj)) + [xi, xk, xj ], if k < i < j


σ(xi ⊗ (xj ⊗ xk)) = xk · (xj · xi)) + [xi, xk, xj ], if k < j < i

The monomial has no inversion iff it is a standard monomial. In this casewe necessarily define σ by (4.201). If the labels of a monomial have only oneinversion, then, by Lemma 4.1, (1), (2), the monomial is congruent modJwith a unique standard monomial; this supports the definitions (4.202) and(4.203). The formulae (4.204) and (4.205) are also correct because, even if themonomials present exactly two inversions, the order of making acting theseinversions is necessarily unique. Finally, (4.206) occurs by applying Lemma4.1 (i) and (ii) to realize the inversions (j, k), (i, k) and (i, j). If we shouldmake the three inversions in the order (i, j), (i, k) and (j, k), then we obtain

σ(xi ⊗ (xj ⊗ xk)) = xk · (xj · xi) + [xj , xk, xi] + [xi, xj , xk].

By subtracting the two expressions for σ(xj ⊗ xk)) we get

[xi, xk, xj ]− [xj , xk, xi]− [xi, xj , xk] = 0,

i.e., (4.206) is well-defined. Moreover, as it can easily be checked, the formulae(20) imply

σ([xi, xj , xk]⊗ − [xi, xj , xk]) = 0.

As f⊗(x1, x2, x3, x4) is a generator for J , we put

σ(f⊗(x1, x2, x3, x4)) = 0, ∀x1, x2, x3, x4 ∈ X. (21)

Then, we shall prove the assertion of Lemma by a complete induction on thedegree n > 3 of the monomials and on the number of inversions of labelsoccurring in these monomials; the number of inversions of labels occurringin a monomial is called its index. We denote by T⊗n,j the subspace of T⊗n

spanned by the monomials of degree n and index ≤ j. In order to ascertainthe affiliation to a subspace T⊗n,j , we associate with the monomial xj1⊗(xj2⊗(xj3 ⊗ ...⊗ (xjn−1 ⊗ (xjn))...), with n > 3, a permutation

τ :(

1 2 ... ni1 i2 ... in

),

where i1 is the ordinal number giving the position of j1 in the totally orderedset j1, j2, ..., jn etc. It is well-known that every permutation τ is the com-position of a finite number of transpositions. In their turn, each transpositionis the product of inversions.

Unfortunately, a representation of any permutation as a product of all itsinversions is unique up to their ordering. Certainly, there exists a naturalbijection between the inversions in the labels of monomial xj1 ⊗ (xj2 ⊗ (xj3 ⊗

18 Ilie Burdujan

... ⊗ (xjn−1 ⊗ (xjn))...) and the inversions of the associated permutation τ .Suppose that a linear mapping σ satisfying to conditions stated in Lemma 4.2has already been defined for T⊗0 ⊕ T⊗1 ⊕ ... ⊕ T⊗n−1 for the monomials inT`T, only. In order to extend linearly to T⊗0

` ⊕ T⊗1` ⊕ ... ⊕ T⊗n−1

` ⊕ T⊗n,0`

we put

σ(xj1 ⊗ (xj2 ⊗ (xj3 ⊗ ..⊗ (xjn−1 ⊗ (xjn))..) = xj1 · (xj2 · (xj3 · .. · (xjn−1 · (xjn))..),

if j1 ≤ j2 ≤ ... ≤ jn, i.e. we have solved the problem in the case when theleft monomials of degree n have no inversion. Consider now the left mono-mials having a single inversion. The most representative such monomial isx2 ⊗ (x1 ⊗ (x3 ⊗ ... ⊗ (xn−1 ⊗ xn)...)). Then, by making use of the identity(2.5) and the induction hypothesis, we give the following definition

σ(x2 ⊗ (x1 ⊗ (x3 ⊗ ...⊗ (xn−1 ⊗ xn)...))) =

= x1 · (x2 · (x3 · ... · (xn−1 · xn)...)))+

+σ(D(x2,x1)(x3 ⊗ ...⊗ (xn−1 ⊗ xn)...)))).

Of course, this definition is correct. Moreover, the identity (2.6) suggests usto use the following definition for any monomial of degree n and index 1

σ(xj1 ⊗ (xj2 ⊗ ...⊗ (xji−1 ⊗ (xji+1 ⊗ (xji ⊗ (xji+2 ⊗ ...⊗ (xn−1 ⊗ xn)..))))..)) =

= σ(xj1 ⊗ (xj2 ⊗ ...⊗ (xji−1 ⊗ (xji ⊗ (xji+1 ⊗ (xji+2 ⊗ ..⊗ (xn−1⊗ xn)..))))..))+

+σ(xj1 ⊗ (xj2 ⊗ ...⊗ (xji−1 ⊗ (D⊗(xji+1

,xji)(xji+2 ⊗ ...⊗ (xn−1 ⊗ xn)...)))).

Assume that σ has already been defined for T⊗0⊕T⊗1⊕...⊕T⊗n−1⊕T⊗n,i−1

satisfying the assertion of Lemma for monomials belonging to this space andlet xj1 ⊗ (xj2 ⊗ ... ⊗ (xn−1 ⊗ xn)...) be a left monomial of the index i ≥ 1.Suppose jk > jk+1. Then we set

σ(xj1 ⊗ (xj2 ⊗ ..⊗ (xjk−1⊗ (xjk

⊗ (xjk+1⊗ ..⊗ (xn−1 ⊗ xn)..)))) = (22)

= σ(xj1 ⊗ (xj2 ⊗ ...⊗ (xji−1 ⊗ (xjk+1⊗ (xjk

⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(xj1 ⊗ (xj2 ⊗ ...⊗ (xji−1 ⊗ (D⊗

(xjk,xjk+1

)(xjk+2⊗ ...⊗ (xn−1 ⊗ xn)...)))).

This definition makes a sense since the terms on the right-hand side are inT⊗0⊕ T⊗1⊕ ...⊕ T⊗n−1⊕ T⊗n,i−1. First we show that this definition is inde-pendent of the choice of the pair (jk, jk+1), jk > jk+1. Let (j`, j`+1) be anotherpair with j` > j`+1. Essentially there are two cases: I. ` > k + 1, II. ` = k + 1.Since the computations do not involve the part xj1 ⊗ (xj2 ⊗ ...⊗ (xjk−1

of the


monomial and taking into account the existence of a bijection between the in-versions in the labels of a monomial and those of the associated permutation,it follows that it is sufficient to consider the following two types of monomials

I. x2 ⊗ (x1(x3...(xi−1 ⊗ (xi+1 ⊗ (xi ⊗ (xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...))))...)),II. x3 ⊗ (x2 ⊗ (x1 ⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...)))).

I. The associated permutation is τ = (1, 2)(i, i + 1). Then we get the firstevaluation E1 by starting with (1, 2), namely

E1 == σ(x2 ⊗ (x1 ⊗ (x3 ⊗ ..⊗ (xi−1 ⊗ (xi+1 ⊗ (xi ⊗ (xi+2 ⊗ ..⊗ (xn−1 ⊗ xn)..))))..))) == σ(x1 ⊗ (x2 ⊗ (x3 ⊗ ..⊗ (xi−1 ⊗ (xi+1 ⊗ (xi ⊗ (xi+2 ⊗ ..⊗ (xn−1 ⊗ xn)..))))..)))++

∑i−1k=3 σ(x3 ⊗ (x4 ⊗ ...⊗ (xk−1 ⊗ (D(x2,x1)(xk)⊗ (xk+1 ⊗ ...⊗ (xi−2 ⊗ xi−1)⊗

⊗(xi+1 ⊗ (xi ⊗ xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...))))...)))++σ((x3 ⊗ ...⊗ (xi−2 ⊗ (xi−1 ⊗ (D(x2,x1)(xi+1)⊗ (xi ⊗ (xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ((x3 ⊗ ...⊗ (xi−2 ⊗ (xi−1 ⊗ (xi+1 ⊗ (D(x2,x1)(xi)⊗ (xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ((x3 ⊗ ...⊗ (xi−2 ⊗ (xi−1 ⊗ (xi ⊗ (xi+1 ⊗ (D(x2,x1)(xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ((x3 ⊗ ...⊗ (xi−2 ⊗ (xi−1 ⊗ (D⊗

x(i+1,xi)(D⊗

(x2,x1)(xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...)))) == σ(x1 ⊗ (x2 ⊗ (x3 ⊗ ..⊗ (xi−1 ⊗ (xi ⊗ (xi+1 ⊗ (xi+2 ⊗ ..⊗ (xn−1 ⊗ xn)..))))..)))++σ((x1 ⊗ (x2...⊗ (xi−1 ⊗ (D⊗

(xi+1,xi)(xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...))))+

+∑i−1

k=3 σ(x3 ⊗ (x4 ⊗ ...⊗ (xk−1 ⊗ (D(x2,x1)(xk)⊗ (xk+1 ⊗ ...⊗ (xi−2 ⊗ xi−1)...))⊗⊗(xi ⊗ (xi+1 ⊗ xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...))))...)))+

+∑i−1

k=3 σ(x3 ⊗ (x4 ⊗ ...⊗ (xk−1 ⊗ (D(x2,x1)(xk)⊗ (xk+1 ⊗ ...⊗ (xi−2 ⊗ xi−1)...))⊗⊗(D⊗

(xi+1,xi)(xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...))))+

+σ(x3 ⊗ ...⊗ (xi−2 ⊗ (xi−1 ⊗ (D(x2,x1)(xi+1)⊗ (xi ⊗ (xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...))))..))++σ((x3 ⊗ ...⊗ (xi−2 ⊗ (xi−1 ⊗ (xi+1 ⊗ (D(x2,x1)(xi)⊗ (xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ((x3 ⊗ ...⊗ (xi−2 ⊗ (xi−1 ⊗ (xi ⊗ (xi+1 ⊗ (D⊗

(x2,x1)(xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ((x3 ⊗ ...⊗ (xi−2 ⊗ (xi−1 ⊗ (D⊗

x(i+1,xi)(D⊗

(x2,x1)(xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...)))).

If we start with (i + 1, i), we get the second evaluation E2, namely

E2 == σ(x2 ⊗ (x1 ⊗ (x3 ⊗ ..⊗ (xi−1 ⊗ (xi+1 ⊗ (xi ⊗ (xi+2 ⊗ ..⊗ (xn−1 ⊗ xn)..))))..))) == σ(x2 ⊗ (x1 ⊗ (x3 ⊗ ..⊗ (xi−1 ⊗ (xi ⊗ (xi+1 ⊗ (xi+2 ⊗ ..⊗ (xn−1 ⊗ xn)..))))..)))++σ(x2 ⊗ (x1 ⊗ (x3 ⊗ ..⊗ (xi−1 ⊗ (D⊗

(xi+1,xi)(xi+2 ⊗ ..⊗ (xn−1 ⊗ xn)..))))..))) =

= σ(x1 ⊗ (x2 ⊗ (x3 ⊗ ..⊗ (xi−1 ⊗ (xi ⊗ (xi+1 ⊗ (xi+2 ⊗ ..⊗ (xn−1 ⊗ xn)..))))..)))++σ(D(x2,x1)(x3 ⊗ ..⊗ (xi−1 ⊗ (xi ⊗ (xi+1 ⊗ (xi+2 ⊗ ..⊗ (xn−1 ⊗ xn)..))))++σ(x1 ⊗ (x2 ⊗ (x3 ⊗ ..⊗ (xi−1 ⊗ (D⊗

(xi+1,xi)(xi+2 ⊗ ..⊗ (xn−1 ⊗ xn)..))))+

+∑i−1

k=3 σ(x3 ⊗ (x4 ⊗ ...⊗ (xk−1 ⊗ (D(x2,x1)(xk)⊗ (xk+1 ⊗ ...⊗ (xi−1⊗⊗(D⊗

(xi+1,xi)(xi+2 ⊗ (xn−1 ⊗ xn)...))))+

20 Ilie Burdujan

+σ(x3 ⊗ (x4 ⊗ ...⊗ (xi−1 ⊗ (D⊗(x2,x1)(D⊗

(xi+1,xi)(xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...)))...))).

By substracting E1 − E2 it follows

E1 − E2 == σ((x3 ⊗ ...⊗ (xi−2 ⊗ (xi−1 ⊗ (xi+1 ⊗ (D(x2,x1)(xi)⊗ (xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...))))−−σ((x3 ⊗ ..⊗ (xi−2 ⊗ (xi−1 ⊗ (D(x2,x1)(xi)⊗ (xi+1 ⊗ (xi+2 ⊗ ..⊗ (xn−1 ⊗ xn)..))))++σ((x3 ⊗ ..⊗ (xi−2 ⊗ (xi−1 ⊗ (D(x2,x1)(xi+1)⊗ (xi ⊗ (xi+2 ⊗ ..⊗ (xn−1 ⊗ xn)..))))−−σ((x3 ⊗ ..⊗ (xi−2 ⊗ (xi−1 ⊗ (xi ⊗ (D(x2,x1)(xi+1)⊗ (xi+2 ⊗ ..⊗ (xn−1 ⊗ xn)..))))++σ(x3 ⊗ ...⊗ (xi−1 ⊗ (D⊗

(xi+1,xi)(D⊗

(x2,x1)(xi+2 ⊗ ...⊗ (xn−1 ⊗ xn))))))...)−−σ(x3 ⊗ ...⊗ (xi−1 ⊗ (D⊗

(x2,x1)(D⊗(xi+1,xi)

(xi+2 ⊗ ...⊗ (xn−1 ⊗ xn))))))...) == σ((x3 ⊗ ...⊗ (xi−2 ⊗ (xi−1 ⊗ (D⊗

([x2,x1,xi+1],xi)(xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...))))+

+σ((x3 ⊗ ...⊗ (xi−2 ⊗ (xi−1 ⊗ ((D⊗(xi+1,[x2,x1,xi])

(xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(x3 ⊗ ...⊗ (xi−1 ⊗ (D⊗

(xi+1,xi)(D⊗

(x2,x1)(xi+2 ⊗ ...⊗ (xn−1 ⊗ xn))))))...)−−σ(x3 ⊗ ...⊗ (xi−1 ⊗ (D⊗

(x2,x1)(D⊗(xi+1,xi)

(xi+2 ⊗ ...⊗ (xn−1 ⊗ xn))))))...) == σ(x3 ⊗ ...⊗ (xi−1 ⊗ (D⊗

([x2,x1,xi+1],xi)+ D⊗

(xi+1,[x2,x1,xi])+

+D⊗(xi+1,xi)

(D⊗(x2,x1) −D⊗

(x2,x1)D⊗(xi+1,xi)

(xi+2 ⊗ ...⊗ (xn−1 ⊗ xn)...)))...) = 0

By using (Lts.5) and (Ls.6) it follows that the two evaluations coincide.

II. Let us consider the case σ(x3 ⊗ (x2 ⊗ (x1 ⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...)))).Firstly, we use the order of transformations following the pairs (3, 2), (3, 1) and(2, 1) and get the first form E1 of evaluation, namely

E1 == σ(x3 ⊗ (x2 ⊗ (x1 ⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...)))) == σ(x2 ⊗ (x3 ⊗ (x1 ⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(D(x3,x2)(x1)(x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(x1 ⊗ (D(x3,x2) ⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...)))) == σ(x2 ⊗ (x1 ⊗ (x3 ⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(x2 ⊗ (D(x3,x1)(x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(D(x3,x2)(x1)⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(x1 ⊗ (D(x3,x2)(x4 ⊗ ...⊗ (xn−1 ⊗ xn)...)))) == σ(x1 ⊗ (x2 ⊗ (x3 ⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(D(x2,x1)(x3)⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(x3 ⊗ (D(x2,x1)(x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(x2 ⊗ (D(x3,x1)(x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(D(x3,x2)(x1)⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(x1 ⊗ (D(x3,x2)(x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))

If we use the order of transformations following the pairs (2, 1), (3, 1) and(3, 2) we get the second form E2 of evaluation, namely


E2 == σ(x3 ⊗ (x2 ⊗ (x1 ⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...)))) == σ(x3 ⊗ (x1 ⊗ (x2 ⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(x3 ⊗ (D(x2,x1)(x4 ⊗ ...⊗ (xn−1 ⊗ xn)...)))) == σ(x1 ⊗ (x3 ⊗ (x2 ⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(D(x3,x1)(x2 ⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(x3 ⊗ (D(x2,x1)(x4 ⊗ ...⊗ (xn−1 ⊗ xn)...)))) == σ(x1 ⊗ (x2 ⊗ (x3 ⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(x1 ⊗ (D(x3,x2)(x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(D(x3,x1)(x2)⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(x2 ⊗ (D(x3,x1)((x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ(x3 ⊗ (D(x2,x1)(x4 ⊗ ...⊗ (xn−1 ⊗ xn)...)))).

Substracting E1 − E2 one gets

E1 − E2 == σ([x2, x1, x3]⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))++σ([x3, x2, x1]⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...))))+σ([x1, x3, x2]⊗ (x4 ⊗ ...⊗ (xn−1 ⊗ xn)...)))) = 0.

By using (Lts.3) and (Ls.4) it follows that the two evaluations are coincident.Consequently, the definition (19) is independent of the choice of the pair (i +1, i). We now apply this definition for defining σ for the monomials of degreen and index i. Since every monomial of degree n is obtained uniquely as atensorial product (with a specific ordering for parenthesis) [w1⊗w2⊗ ...⊗wp]sand s ≤ n − 1, it follows that σ is naturally extended to the space T⊗n,i.Consequently, we get a mapping on T⊗0⊕T⊗1⊕ ...⊕T⊗n−1⊕T⊗n,i satisfyingthe conditions stated by Lemma. This completes the proof of Lemma. Byusing Lemmas 4.1 and 4.2 we can prove the following analogue of Poincare-Birkhoff-Witt Theorem.

Theorem 4.1 The cosets of 1 and the standard monomials modJ form abasis for U(T ) = TT/J .

Proof. Lemma 4.1 shows that every coset is a linear combination of 1 + Jand the cosets of the standard monomials. Lemma 4.2 gives a linear mappingof TT into B satisfying the stated conditions. Every element of J is a linearcombination of elements of the form

[xj1 ⊗ xj2 ⊗ ...⊗ xji−1 ⊗ (xji , xji+1)⊗ xji+1 ⊗ ...⊗ xjn−2 ]s−−[xj1 ⊗ xj2 ⊗ ...⊗ xji−1 ⊗ (xji+1 , xji)⊗ xji+1 ⊗ ...⊗ xjn−2 ]s,[xj1 ⊗ xj2 ⊗ ...⊗ xji−1 ⊗ [xk1 , xk2 , xk3 ]

⊗ ⊗ xji+1 ⊗ ...⊗ xjn−2 ]s−−[xj1 ⊗ xj2 ⊗ ...⊗ xji−1 ⊗ [xk1 , xk2 , xk3 ]⊗ xji+1 ⊗ ...⊗ xjn−2 ]s,i ∈ 1, 2, ..., n, n ≥ 3,[xj1 ⊗ xj2 ⊗ ...⊗ xji−1 ⊗ f⊗(xk1 , xk2 , xk3 , xk4)⊗ xji+1 ⊗ ...⊗ xjn−3 ]s,i ∈ 1, 2, ..., n, n ≥ 4.

22 Ilie Burdujan

Since σ maps these elements into 0, σ(J) = 0 and so σ induces a linearmapping of U(T ) into B. This mapping carries the cosets of 1 and standardmonomials [xj1 ⊗xj2 ⊗ ...⊗xjn ]s ∈ S[X], into 1 and [xj1 ·xj2 · ... ·xjn ]s ∈ S[X],respectively. Since these images are linearly independent in B, we have thelinear independence in U(T ) of the cosets of 1 and the standard monomials.

Corollary 4.2 The mapping i of T in U(T ) is 1-1 and K1⋂

i(T ) = 0.Proof. If X = x1, x2, ..., xn, ... is a basis for T over K, then 1 = 1 + Jand the cosets i(xj) = xj + J are linearly independent. This implies bothstatements.

References[1] Burdujan, I., Sur un theoreme de K. Yamaguti, Proc. of Inst. of Math. Iasi, Ed. Acad.

Rom. 1974, 23-30.

[2] Burdujan, I., On homogeneous systems associated to a binary algebra, An. St. Univ.Ovidius Constantza, 2 (1994), 31-38.

[3] Burdujan, I., On derivation algebra of a real algebra without nilpotents of order two,Italian J. of Pure and Applied Math., 8 (2000), 137-154.

[4] Burdujan, I., A universal enveloping algebra for a Lie triple system, Proc. of the 3-rdInt. Colloquium ”Mathematics in Engineering and Numerical Physics” Oct. 7-9, 2004,59-70.

[5] Cartan, E., Oeuvres completes, Part 1, 2, 101, 138, Gauthier-Villars, Paris, 1952.

[6] Cartan, E., Sur la structure des groupes de transformations finis et continues, (These)Paris, 1894.

[7] Cartan, E., La geometrie des groupes de transformations, J. Math. Pures et Appliquees,6 (1927), 1 -119.

[8] Freudenthal, H., Beziehungen der E7 und E8 zur oktavenebene, I, II, Proc. Kon. Ned.Akad. Wet. A, 57 (1954), 218-230, 363-368.

[9] Harris, B., Cohomology of Lie triple systems and Lie algebras with involution, Trans.Amer. Math. Soc., 98 (1961), 148-162.

[10] Hodge, L. T., Parshall L.,B., On the representation theory of Lie triple systems, Trans.Amer. Math. Soc., 354, 11 (2002), 4359-4391 (electronic).

[11] Hopkins, N. C., Some structure theory for a class of triple systems, Trans. Amer. Math.Soc., 289, 1 (1985), 203-212.

[12] Hopkins, N. C., Decomposition of Lie module triple systems, Algebras, Groups andGeometries, 3 (1986), 399-413.

[13] Hopkins, N. C., On the derivation algebras of Lie module triple systems, J. of Algebra,111 (1987), 520-527.

[14] Jacobson, N., Lie and Jordan triple systems, Amer. J. Math., 71 (1948), 148-170.

[15] Jacobson, N., General representation theory of Jordan algebras, Trans. Amer. Math.Soc., 70 (1951), 509-530.

[16] Kamiya, N., Okubo, S., On Lie triple systems and Yang-Baxter equations, Proc. of the7th Int. Colloquium on Differential Equations (Plovdiv, 1996), VSP, Utrecht, 1997,189-196.


[17] Kovaljov, P.I., Lie triple systems and spaces with affine connections,(in Russian), Mat.Zametki, 14, 1 (1973), 107-112.

[18] Kovaljov, P.I., On a class of Lie triple systems,(in Russian), Ukr. Geom. Sb., 14 (1973),24-28.

[19] Lister, W.G., A structure theory of Lie triple systems, Trans. Amer. Math. Soc., 72(1952), 217-242.

[20] Loos, O., Symmetric spaces, vol. I, II, W. A. Benjamin, New York, 1969.

[21] Meng, D. J., Shi, Y. Q., The uniqueness of decomposition of Lie triple systems, Adv.Math. (China), 33, 1 (2004), 52-56.

[22] Mykytyuk, I. V., The Lie triple system of the symmetric space F4/SPIN(9), Asian J.Math., 6, 4 (2002), 713-718.

[23] Osborn, J. M., Commutative algebras satisfying an identity of degree four, Proc. A. M.S., 16, 5 (1965), 1114-1120.

[24] Shestakov, I. P., Every Akivis algebra is linear, Geometrie dedicata, 77 (1999), 215-223.

[25] Yamaguti, K., A note on a theorem of N. Jacobson, J. Sci. of Hiroshima Univ., Ser. A,22, 3 (1958), 187-190.

[26] Yamaguti, K., On the Lie triple system and its generalization, J. Sci. of HiroshimaUniv., Ser. A, 21 (1958), 155-160.

[27] Yamaguti, K., On algebras of totally geodesic spaces (Lie triple systems), J. Sci. ofHiroshima Univ., Ser. A, 21 (1957-58), 107-113.

[28] Yamaguti, K., On weak representations of Lie triple systems, Kumamoto J. Sci. Ser.A, 8, 3 (1968), 107-114.

[29] Yamaguti, K., On cohomology groups of general Lie triple systems, Kumamoto J. Sci.Ser. A, 8, 4 (1969), 135-146.

[30] Zhevlakov, K. A., Slinko, A. M., Shjestakov, I. P., Shirshov, A. I., Rings that are nearlyassociative, Academic, New York, 1982.


SOME PROBLEMS OF THE THEORY OFCOMPACTIFICATIONS OF TOPOLOGICALSPACES

ROMAI J., 2, 1(2006),25–51

Laurentiu I. Calmutchi, Mitrofan M. ChobanTiraspol State University, Republic of Moldova

Abstract If bX and cX are two compactifications of a space X, then a mapping ϕ : bX →cX is canonical if ϕ is continuous and ϕ(x) = ϕ−1(x) = x for any x ∈ X. TheWallman compactification wX may be constructed for an arbitrary T0-spaceX. A compactification cX of a space X is a ρ-compactification if there exits acanonical mapping ϕ : ωX → cX onto cX. If ϕ is closed, then we say that cXis an ωα-compactification.

Keywords: compactification, g-compactification, ωα-proximity, perfect compactification,

WS -compactification.

2000 MSC: 54C25, 54D35, 54D40, 54D60, 54E15, 54F15.

1. INTRODUCTIONCompactness is one of the most important notions and the problems con-

nected with special embeddings of spaces in compact spaces were studied inmany works [7, 9, 10, 13, 30, 40].

The general problems of the theory of compactifications are the following.Problem 1. To find the methods of construction and study of the com-

pactifications with concrete properties of a given space.Problem 2. To construct, for every space X from a given class K of spaces

and a class M of continuous mappings, a class L(X) of compactifications ofX such that:

– in the class L(X) there exists a maximal element mLX for any X ∈ K;– if X, Y ∈ K and f : X → Y is a mapping from M , then there exists a

continuous extension g : mLX → mLY of the mapping f .One of the solutions of the Problem 2 for a class K of completely regular

spaces is the class L(X) of Hausdorff compactifications. The class of Hausdorffcompactifications was profoundly studied (see[3, 8, 9, 11, 12, 13, 15, 16, 23,30, 34, 35, 38, 40]).

Since the class of all T1-compactifications of a given infinite T1-space is nota set, there exist many obstacles for solution of these problems in the classesof T1-spaces and T0-spaces.

25

26 Laurentiu I. Calmutchi, Mitrofan M. Choban

Significant moments in the general theory of T1-compactifications of T1-spaces were:

– the construction by L. Wallman of the Wallman compactification wX ofa T1-space X (see[1, 7, 13, 41]);

– the Shanin’s generalization of the Wallman method of constructions ofT1-compactifications (see[7, 10, 17, 20, 21, 22, 24, 27, 28, 29, 32, 33, 36, 37]);

– the construction of the theory of ωα-compactifications and of ωα-proximities(see[25, 26]);

– the existence of a Hausdorff compactification bX for some discrete spaceX which is not of the Wallman-Shanin type (see[33, 39]).

Various methods of construction of extensions of spaces were proposed in[3, 5, 7, 10, 16, 18, 20].

The aim of the present work is to investigate the class of T0-compac-tifications of T0-spaces.

In Section 1 we discuss the general notions of compactifications and g-compactifications.

Sections 2-6 are devoted to the investigation of the Wallman-Shanin methodof construction of T0-compactifications of T0-spaces.

In Section 7 we study the Hausdorff g-compactifications of spaces.In Section 8 the notion of ρ-compactification of a T0-space is introduced.In Section 9–10 the ωα-compactifications of T0-spaces are studied.In Section 11 we consider the locally compact-small spaces (see [19]).In Section 12 we define the notion of the ωα-proximity for T0-spaces.In Sections 13 and 14 the notions of perfect and punctiform compacti-

fications are studied for T0-spaces.We use the terminology from [13].By clXA or clA it is denoted the closure of a set A in a space X. By |A|

we denote the cardinality of a set A. The weight of a space X is denoted byw(X). If X is a space, then c(X) = x ∈ X : the set x is closed in X andz(X) = X \ c(X).

By exp(X) we denote the cardinality of a set of all subsets of a set X.Every space is considered to be a non–empty T0-space. Let N = 1, 2, . . ..If τ is a cardinal number, then we put exp(τ) = 2τ .

1. COMPACTIFICATIONS OF SPACES

1.1 Definition. A g-compactification of a space X is called a pair (Y, f),where Y is a compact space, f : X → Y is a continuous mapping and theset f(X) is dense in Y . If f is an embedding of X into Y , then (Y, f) is acompactification of X.

If (Y, f) is a compactification of a space X, then we identify x ∈ X withf(x) ∈ Y and we consider that X is a dense subspace of Y , i.e X = f(X) ⊆ Y .

Some problems of the theory of compactifications of topological spaces 27

Denote by GC(X) the class of all g-compactifications and by C(X) the classof all compactifications of a space X.

Let (Y, f) and (Z, g) be g-compactifications of a space X. We considerthat (Z, g) ≤ (Y, f) if there exists a continuous mapping ϕ : Y → Z suchthat g = ϕf . If ϕ is a homeomorphism of Y onto Z, then we consider that(Y, f) = (Z, g). We identify the equal g-compactifications. If (Z, g) ≤ (Y, f)and (Y, f) ≤ (Z, g), then the g-compactifications (Y, f) and (Z, g) are calledequivalent and we denote (Y, f) ∼ (Z, g).

For every space X and every g-compactification (Y, f) of X the class (Z, g) ∈GC(X) : (Z, g) ∼ (Y, f) is not a set. In particular, GC(X) and C(Y ) are notsets (see [7], Proposition 1.1.7).

1.2 Definition. (see [21] for T1-compactifications). A g-compactification(Y, f) of a space X is called:

– a weakly correct g-compactification if clY f(A) : A ⊆ X is a closed baseof Y ;

– a correct g-compactification if it is weakly correct and for every y ∈Y \ f(X) the set y is closed in Y ;

– a strongly correct g-compactification if it is correct and if Φ is a closedsubset of Y , F is a closed subset of a subspace f(X) and y ∈ Φ ∩ (clY F \ F ),then there exists a filter η of closed subsets of f(X) such that F ∈ η andy ∈ ∩clY H : H ∈ η ⊆ Φ.

Denote by wKGC(X) the class of all weakly correct g-compactifications,by KGC(X) the class of all correct g-compactifications and by sKGC(X) theclass of all strongly g-compactifications of a space X.

1.3. Proposition. The class wKGC(X) of all weakly correct g-compactificationsof a space X is a set.

Proof. If (Y, f) ∈ wKGC(X), then w(Y ) ≤ exp(X) and |Y | ≤ exp(exp(X)).The proof is complete.

1.4. Proposition. If (Y, f) and (Z, g) are equivalent correct g-compacti-fications of a space X, then (Y, f) = (Z, g).

Proof. Let ϕ : Y → Z and ψ : Z → Y be continuous mappings for whichg = ϕ f and f = ψ g. We put X1 = f(X) and h(y) = ψ(ϕ(y)) for anyy ∈ Y . It is obvious that y = h(y) for any y ∈ X1. Suppose that y ∈ Y andy1 = h(y) 6= y. It is clear that y ∈ Y 8X1 and y is a closed subset of Y .There exists a closed subset F of X1 such that y, y1 ∩ clY F = y. Since his a continuous mapping, h(clY F ) ⊆ clY h(F ). By construction, y1 ∈ h(clY F )and F = h(F ), a contradiction. Thus h(y) = y for any y ∈ Y and ψ = ϕ−1.The proof is complete.

1.5. Operation of correction. Let (Y, f) be a g-compactification ofa space X. On Y we consider the topology T ′ generated by the closed base∩clY f(A) : i ≤ n : A1,A2, . . . , An ⊆ X, n ∈ N. If T is the initial topologyon Y , then T ′ ⊆ T . For any y ∈ Y we put i(y) = ∩U \ V : U, V ∈ T ′, y ∈


U \ V . There exists a mapping ϕ : Y → Z, where ϕ−1(ϕ(y)) = i(y) for anyy ∈ Y . On Z we consider the topology H ⊆ Z : ϕ−1(H) ∈ T ′. Then Z isa T0-space and (Z, g = ϕ f) is a g-compactification of X. By construction,ϕ | f(X) : f(X) → ϕ(f(X)) is a homeomorphism and clZg(A) : A ⊆ X isa closed base of the space Z. Thus (Z, g) is a weakly correct compactificationof X. If (Y, f) is a compactification of X, then (Z, g) is a compactificationof X too. The g-compactification (Z, g) is called the correction of the g-compactification (Y, f). If (S, h) is a correct g-compactification of X and(S, h) ≤ (Y, f), then (S, h) ≤ (Z, g). Thus (Z, g) is the maximal weaklycorrect g-compactification from the set (S, h) ∈ KGC(X) : (S, h) ≤ (Y, f).

1.6 Definition. We say that a g-compactification (Y, f) of a space X :– is weakly incompressible if a compact subset Φ of Y such that Y 6= Φ and

f(X) ⊆ Φ does not exist;–is incompressible if for every subset A of f(X) a compact subset Φ of Y

such that Φ 6= clY f(A) and f(X) ∩ clY f(A) ⊆ Φ ⊆ clY f(A) does not exist.1.7. Example. Let D be an infinite discrete space, A and B be two infinite

subsets of D for which A ∩ B = ∅ and D = A ∪ B. Let a, b be two distinctpoints, D ∩ a, b = ∅ and cD = D ∪ a, b. On cD we consider the topologygenerated by the open base x; x ∈ D ∪ (D \ F ) ∪ a : F is a finitesubset of D ∪ (B \ F ) ∪ b : F is a finite subset of D. The space cD is acompactification of D with the properties:

– cD is not weakly correct ;– cD is not incompressible;– cD is not weakly incompressible;– if L ⊆ D and the set L ∩ B is infinite, then the set L ∪ a is compact

and clcDL = L ∪ a, b.1.8. Example. Let A,B, C be infinite subsets of a discrete space D and

A ∩ B = A ∩ C = B ∩ C = ∅, D = A ∪ B ∪ C. On bD = D ∪ a, b,where a 6= b and D ∩ a, b = ∅, consider the topology generated by theopen base x : x ∈ D ∪ a ∪ ((A ∪ B) \ F ) : F is a finite subset ofD ∪ b ∪ ((B ∪ C) \ F ) : F is a finite subset of D. The space bD is acompactification of D with the proprieties:

– if the set L ⊆ B is infinite, then the sets L ∪ a, L ∪ b are compactand clbDL = L ∪ a, b;

– bD is not weakly correct;–bD is not incompressible;–bD is weakly incompressible.1.9. Example. Let Y = [0, 1] with the topology T = X,∅∪[0, x) : 0 <

x < 1. Fix a ∈ Y and set X = Y \ a. Then Y is a weakly correct compact-ification of X. The compactification Y is incompressible if and only if a = 1.For a < 1 the space X is compact and Y is not a correct compactification.The compactification Y is strongly correct if and only if a = 1.


2. WS-COMPACTIFICATIONSA family L of subsets of a space X is called a ring of subsets of X if X,∅ ∈ L

and A ∩B, A ∪B ∈ L for all A,B ∈ L.If L is a ring of subsets of a space X and any set A ∈ L is closed in X, then

we say that L is a closed ring of X.If a ring is a closed base of a space X, then L is called a basic ring of X.Fix a ring L of closed subsets of X. For any x ∈ X we put ξ(x,L) = F ∈

L : x ∈ F. A family ξ of subsets of X is called an L-filter if: ξ ⊆ L; ∅ /∈ξ; A ∩ B ∈ ξ for any A,B ∈ ξ; if A ⊆ B, A ∈ ξ and B ∈ L, then B ∈ ξ. Amaximal L-filter is called an L-ultrafilter. A filter ξ is a free L-filter if ∩ ξ = ∅.We put ωLX = ξ(x,L) : x ∈ X ∪ ξ ⊆ L : ξ is a free L-ultrafilter.

Consider the mapping ωL : X → ωLX, where ωL(x) = ξ(x,L) for anyx ∈ X. For every A ∈ L we put cL(A) = ξ ∈ ωLX : A ∈ ξ.

2.1. Lemma. For every ring L of closed subsets of a space X we have:1. cL(∅) = ∅ and cL(X) = X;2. cL(A ∪B) = cL(A) ∪ cL(B) for all A, B ∈ L;3. cL(A ∩B) = cL(A) ∩ cL(B) for all A, B ∈ L;4. c(L) = cL(A) : A ∈ L is a closed base for some T0–topology on ωLX;5. c(L) is a ring of subsets;6. ω−1

L (cL(A)) = A for any A ∈ L.Proof. The assertions 1 and 2 are obvious. If ξ ∈ ωLX and A,B ∈ L, then

A ∪ B ∈ ξ if and only if ξ ∩ A, B 6= ∅. Thus the assertion 3 is true. Hencec(L) is a ring of subsets of ωLX. In particular, c(L) is a closed base of someT0–topology on ωLX. The assertions 4 and 5 are proved. The assertion 6 isobvious. The proof is complete.

On ωLX we consider the topology generated by the closed base c(L).2.2. Theorem. Let L be a closed ring of subsets of a space X. Then

(ωLX,ωL) is a correct weakly incompressible g-compactification of the spaceX. Moreover, if Y = ωLX, then:

1. clY ωL(A) = cL(A) for any A ∈ L;2. if ξ ∈ ωLX \ ωL(X), then the set ξ is closed in ωLX;3. if A ∈ L, Φ is a compact subset of Y and ωL(A) ⊆ Φ, then clY ωL(A) ⊆

Φ.Proof. It is well-known that ωLX is a compact space. Fix A ∈ L. By

definition of the topology on ωLX, we have clY ωL(A) ⊆ cL(A). Supposethat ξ ∈ cL(A)\ clY ωL(A). Then ξ is a free L-ultrafilter, A ∈ ξ, ωL(H) ∩clY ωL(A) ⊇ ωL(H ∩ A) 6= ∅ for any H ∈ ξ and ∩cL(H) : H ∈ ξ = ξ.Thus ∩clY ωL(A) ∩ cL(H) : H ∈ ξ = ∅, a contradiction with the conditionof compactness of the set clY ωL(A). The assertions 1 and 2 are proved. Theproof of the assertion 3 is similar with the proof of the assertion 1. The proofis complete.


2.3. Corollary. If L is a closed ring base of a space X, then ωLX is acorrect weakly incompressible compactification of the space X.

2.4. Definition. If L is a ring of closed subsets of X, then (ωLX, ωL) iscalled the Wallman-Shanin g-compactification or the WS-g-compactificationgenerated by the closed ring L.

2.5. Definition. A g-compactification (Y, f) of a space X is called aWS - g-compactification of X if (Y, f) = (ωLX,ωL) for some closed ring L ofX.

2.6. Definition. If L is the ring of all closed subsets of X, then (ωX, ωX) =(ωLX,ωL) is called the Wallman compactification of the space X.

2.7. Corollary. The Wallman compactification ωX of a space X is correct,incompressible and the set ξ is closed in ωX for any point ξ ∈ ωX \X.

2.8. Remark. The compactifications bD and cD of the discrete space Dfrom Examples 1.7 and 1.8 are not WS-compactifications. If a = 1 in theExample 1.9, then Y = ωX.

2.9. Definition. A mapping g : X → Y of a space X into a space Yis weakly perfect if g is a closed continuous mapping and the set g−1(y) iscompact provided y ∈ z(Y ).

Thus, every continuous closed mapping g : X → Y of a T0-space X into aT1-space Y is weakly perfect.

2.10. Theorem. Let g : X → Y be a weakly perfect mapping of a space Xinto a space Y . Then:

1. there exists a weakly perfect mapping ϕ : ωX → ωY such that g = ϕ | X;2. if Ψ : ωX → ωY is a closed continuous mapping and g = Ψ | X, then

Ψ = ϕ.Proof. We consider that Y = g(X). If ξ is an ultrafilter of closed subsets

of X, then g(ξ) = g(H) : H ∈ ξ is an ultrafilter of closed subsets of Y .In particular, if ξ ∈ ωX \ X, then g(ξ) ∈ ωY , g(ξ) is an ultrafilter of closedsubsets of Y and the set g(ξ) is closed in ωX.

We put ϕ(x) = g(x) for x ∈ X and ϕ(ξ) = g(ξ) for ξ ∈ ωX \X. If y ∈ z(Y )then the set g−1(y) is compact and ξ /∈ g−1(y) for any ultrafilter ξ ∈ ωX. Thusϕ−1(y) = g−1(y). By construction, ϕ−1(clωY F ) = clωXg−1(F ) for any closedsubset F of Y . Thus ϕ is a continuous mapping and the set ϕ−1(y) is compactfor any y ∈ ωX. If F is a closed subset of X, then ϕ(clωXF ) = clωY g(F ).Thus the mapping ϕ is closed. The assertion 1 is proved. The assertion 2 isobvious. The proof is complete.

2.11. Corollary. Let (Y, f) be a g-compactification of a space X, Z =f(X), ϕ : ωX → Y be a closed continuous mapping and f = ϕ | X : X → Zbe a weakly perfect mapping.

1. Then there exists a unique weakly perfect mapping Ψ : ωX → ωZ anda continuous closed mapping h : ωZ → Y such that f = Ψ | X = h Ψ andh(z) = h−1(z) = z for any z ∈ Z.


2. (Y, f) is a correct g-compactification of the space X and a correct com-pactification of the space Z.

3. (Y, f) is an incompressible g-compactification of the space X.4. If F is a closed subset of Z and y ∈ clY F \ F , then there exists an

ultrafilter η of closed subsets of Z such that F ∈ η and ∩clY H : H ∈ η = y.

3. CONSTRUCTION OF ALL STRONGLYCORRECT G-COMPACTIFICATIONS WITH AFINITE REMAINDER

3.1. Theorem. Let (Y, f) be a strongly correct g-compactification of aspace X, Y \ f(X) ⊆ c(Y ) and the set Y \ f(X) is finite: Then:

1. (Y, f) is a weakly incompressible g-compactification of X.2. (Y, f) is a WS - g-compactification of the space X.Proof. Let Z = f(X). The assertions are obvious if Z = Y . Suppose that

Y \ Z is a non-empty finite set. Let Y \ Z = y1,y2,...yn, where the pointsy1,y2,...yn are distinct. If R = y1,y2,...yn, then every subset L ⊆ R is closedin Y . Since the sets L ⊆ R are closed and clY A : A ⊆ Z is a closed base ofY , then there exist the closed subsets F1, F2, . . . , Fn of Z such that :

1. clY Fi = Fi ∪ yi for any i ≤ n;2. Fi ∩ Fj = ∅ for all 1 ≤ i < j ≤ n.Let L0 = H ⊆ Z : H is closed in Y . It is obvious that ∅ ∈ L0 and

A ∪B, A ∩B ∈ L0 for any A,B ∈ L0.If Φ is a closed compact subset of Z and Φ /∈ L0, then the set Φ is not

closed in Y and the g-compactification Y is not incompressible. Suppose thatΦ is a compact subset of Y and Z ⊆ Φ 6= Y . We may assume that y1 /∈ Φ.Since y1 is a closed subset of Y and y1 ∈ clY F1, there exists a filter η ofclosed subsets of Z such that F1 ∈ η and ∩clY H : H ∈ η = y1. ThenH ∩ Φ = ∅ for some H ∈ η and H ⊆ Z \ Φ, a contradiction. Thus Y is aweakly incompressible g-compactification.

For every i ≤ n there exists a free ultrafilter ηi of closed subsets of Z suchthat Fi ∈ ηi. In this case ∩clY H : H ∈ ηi = yi for all i ≤ n. LetLn = ηn. For any i ≤ n− 1 we put Li = H0 ∪H1 ∪ . . . ∪Hi : H0 ∈ L0 andHj ∈ L ∈ ηj : L ⊆ Fj for 1 ≤ j ≤ i. Let L∗ = L0 ∪ L1 ∪ . . . ∪ Ln. Then L′is a closed base and a ring of subsets of the space Z. In this case ωLZ = Y .If M = f−1(H) : H ∈ L, then ωMX = ωLZ. The proof is complete.

3.2. Definition. Let L be a ring of closed subsets of X. The familyL0 = L \ H ∈ L : H ∈ ξ for some free L-filter ξ is called the nucleus of thering L.

3.3. Corollary. If L is a ring of closed subsets of a space X, then L0 =H ∈ L : ωL(H) is closed in ωLX.


3.4. Definition. Let L be a ring of closed subsets of X and n ∈ N .We say that the rank r(L) ≤ n if there exists F1, F2, . . . , Fn ∈ L such thatξ ∩ F1, F2, . . . , Fn 6= ∅ for any free L-ultrafilter ξ.

We put ||A|| = |A| if the set A is finite and ||A|| = ∞ if the set A is infinite.3.5. Corollary. r(L) = ||ωLX \ ωL(X)|| for any ring L of closed subsets

of a space X.3.6. Construction. Let L0, ξ1,ξ2, . . . , ξn be families of closed subsets of

X and F1, F2, . . . , Fn be closed subsets of X such that:1. Fi ∩ Fj = ∅ for 1 ≤ i < j ≤ n;2. Fi ∈ ξi for any i ≤ n;3. ∩ξi = ∅ for any i ≤ n;4. if A,B ∈ L0, then A ∪B, A ∩B ∈ L0;5. if A,B ∈ ξi and 1 ≤ i ≤ n, then A ∪B, A ∩B ∈ ξi and ∅ /∈ ξi;6. if 1 ≤ i ≤ n and A ∈ L0, then A ∩H = ∅ for some H ∈ ξi.We put Ln = X∪A0∪(A1∩F1)∪. . .∪(An−1∩Fn−1)∪An, A0 ∈ L0, A1 ∈

ξ1, . . . , An−1 ∈ ξn−1, An ∈ ξn and Li = A0∪(A1∩F1)∪ . . .∪(Ai∩Fj) : A0 ∈L0, A1 ∈ ξ1, . . . , Ai ∈ ξj for any 1 ≤ i ≤ n − 1. Let L = L0 ∪ L1 ∪ . . . ∪ Ln.Then L is a ring of closed subsets of X, L0 is the nucleus of L and r(L) = n.The pair (ωLX, ωL) is a WS - g-compactification and |ωLX \ ωL(X)| = n.Every WS - g-compactification with the finite remainder may be obtained inthis way.

3.7. Remark. The one-point correct T1-compactifications of T1-spaceswere constructed by P. C. Osmatescu [28].

3.8. Construction. Let L0 be a family of compact closed subsets of aspace X such that A∪B, A∩F ∈ L0 for all A,B ∈ L0 and any closed subsetF of X. Fix a free ultrafilter ξ of closed subsets of X. Then L0 ∪ ξ = L is abase ring of closed subsets of X and ωLX is a one-point compactification ofthe space X. We say that ωLX is generated by the nucleus L0.

If aX and bX are two one-point WS-compactifications of the space X, thenL1 = F ⊆ X : F is closed in aX is the nucleus of aX, L2 = F ⊆ X : F isclosed in bX is the nucleus of bX and bX = aX if and only if L1 = L2.

If aX is a one-point compactification of X and every compact closed subsetof X is closed in aX, then aX is called the one-point Alexandroff compactifi-cation of the space X.

4. EXISTENCE OF A WS-COMPACTIFICATIONWHICH IS NOT A CLOSED IMAGE OF THEWALLMAN COMPACTIFICATION

Let X be a T0-space and Φ = an : n ∈ N be an infinite set of X suchthat every subset L ⊆ Φ is closed in X. We put B = bn = a3n+1 : n ∈ N,C = cn = a3n+2 : n ∈ N and A = Φ \ (A ∪ B). There exists the free


ultrafilters λ, µ, η of closed subsets of X such that A ∈ λ, B ∈ µ, C ∈ η. Fixa family L0 of closed compact subsets of X such that A ∪B, A ∩ F ∈ L0 forall A,B ∈ L0 and a closed subset F of X. We put L2 = η, L1 = H0 ∪ (H1 ∩A) ∪ (H2 ∩ B) : H0 ∈ L0, H1 ∈ λ and H2 ∈ B and L = L0 ∪ L1 ∪ L2. ThenL is a base ring of closed subsets of X. Suppose that ϕ : ωX → ωLX is amapping of ωX onto ωLX and ϕ−1(x) = x for any x ∈ X.

Case 1. X is not an open subset of ωX. In this case the mapping ϕ is notcontinuous: X is an open subset of ωLX and ϕ−1(X) = X.

Case 2. There exists a closed compact subset F of X such that F /∈ L0.In this case F is not closed in ωLX and ϕ(F ) = F . Thus in this case themapping ϕ is not closed.

Case 3. X is a normal locally compact space and for X a Hausdorffcompactification does not exist, bX for which bX \X is a finite set and |bX \X| ≥ 2 does not exist. Since |ωLX \X| = 2, then ϕ is not a closed mapping.In particular, if ωX \X is a connected space, then ϕ is not a closed mapping.

5. EXISTENCE OF COMPACTIFICATIONS OFTHE GIVEN WEIGHT

5.1. Theorem. For a space X and a cardinal number τ the followingassertions are equivalent :

1. w(X) ≤ τ ;2. there exists a compactification bX of X such that w(bX) ≤ τ ;3. there exists a WS-compactification ωLX such that w(ωLX) ≤ τ .Proof. The implications 2 → 3 → 1 are obvious. If w(X) ≤ τ , then there

exists an open base B of X such that |B| ≤ τ . Let L be the minimal ring ofclosed subsets of X for which X \ U : U ∈ B ⊆ L. Then |L| ≤ τ and ωLXis the desired compactification. The proof is complete.

5.2. Definition. A cα-network on X is a family B of closed subsets suchthat for every point x ∈ X and any neighbourhood U of X there exists anelement H ∈ B for which x ∈ H ⊆ U .

The cα-network weight of a space X is defined as the smallest cardinalnumber |B|, where B is a cα-network of X and this cardinal number is denotedby cαw(X).

The cardinal number cαw(X) is defined if and only if X is a T1-space. Iffor X the cardinal number cαw(X) is not defined, then we put cαw(X) = ∞.

5.3. Theorem. For a space X and a ring L of closed subsets of X thefollowing assertions are equivalent:

1. ωLX is a T1-compactification of the space X;2. L is a closed base and a cα-network of the space X;3. for every x ∈ X the family ξ(x,L) is an L-ultrafilter ;4. L is a closed base of X and c(L) is a cα–network of ωLX.


Proof. 1. The ring L is a closed base of X if and only if ωLX is acompactification of X. In this case we assume that X ⊆ ωLX.

2. ωLX is a T1-compactification of X if and only if L is a closed base andc(L) is a cα–network of ωLX. The implications 4 → 1 → 4 → 2 are proved.

3. If ωLX is a T1-compactification of X, then every family ξ(x,L) is an L-ultrafilter. A point ξ ∈ ωLX is closed in ωLX if and only if ξ is an L-ultrafilter.The implication 1 → 3 is proved.

4. Let L be a closed base and a cα–network of X. Fix x ∈ X. If F ∈ L andx /∈ F , then there exists H ∈ L such that x ∈ H ⊆ X \ F . Thus F /∈ ξ(x,L)and ξ(x,L) is an L-ultrafilter. The implication 2 → 3 is proved. The proof iscomplete.

5.4. Theorem. (A.V.Arhangel’skii [4]). For a T1-space X and an infinitecardinal number τ the following assertions are equivalent:

1. there exists a T1-compactification bX of X such that w(bX) ≤ τ ;2. there exists a WS-compactification bX of X such that w(bX) ≤ τ ;3. w(X) + cαw(X) ≤ τ .Proof. The implications 1 → 3 → 1 are proved by A.V.Arhanghel’skii

[4]. The implication 2 → 1 is obvious. If L1 is a closed base of X, L2 is acα-network of X, |L1|+ |L2| ≤ τ and L is the minimal ring of closed subsetsof X such that L1 ∪L2 ⊆ L, then |L| ≤ τ, ωLX is a T1-compactification andw(ωLX) ≤ |L| ≤ τ . The proof is complete.

5.5. Example. (see A. V. Arhanghel’skii [4]). Let X = [0, 1], Q be the setof rational numbers of X and U(x, ε) = x ∪ y ∈ X \Q : x− ε < y < x + εfor every x ∈ X and ε > 0. On X we consider the topology generated bythe open base U(x, ε) : x ∈ X and ε > 0. It is obvious that the set X \ Qis open in X and X is a Hausdorff H-closed space with a countable base.The cα-weight of X is not countable. Thus for X a T1-compactification ofcountable weight does not exist.

6. THE MAXIMALITY OF A RING OF CLOSEDSETS

6.1. Theorem. For a g-compactification (Y, f) of a space X and a ring L

of closed subsets of X the following assertions are equivalent:1. (Y, f) = (ωLX, ωL);2. a family b(L) = b(H) = clY f(H) : H ∈ L is a closed base of the space

Y , b(F )∩ f(X) = f(F ), F = f−1(f(F )) and b(F ∩H) = b(F )∩ b(H) for allF,H ∈ L.

Proof. The implication 1 → 2 follows from Lemma 2.1.Let Z = f(X), b(L) be a closed base of Y , b(F )∩Z = f(F ) and b(F ∩H) =

b(F ) ∩ b(H) for all F, H ∈ L. For every x ∈ X we put ϕ(ξ(x,L)) = f(x) andϕ(ξ) = ∩b(H) : H ∈ ξ for any ξ ∈ ωLX \ ωL(X). The mapping ϕ is one-to-


one and ϕ(cL(H)) = b(H) for any H ∈ L. Thus ϕ is a homeomorphism. Theproof is complete.

6.2. Definition. A ring L of closed subsets of X is called a maximal ringif (ωLX,ωL) 6= (ωMX, ωM) for every ring M of closed subsets of X for whichL ⊆ M and L 6= M.

From Theorem 6.1 it follows6.3. Corollary. For a ring L of closed subsets of a space X the following

assertions are equivalent:1. the ring L is maximal;2. for every closed subset Φ of X, for which Φ /∈ L, there exists F ∈ L such

that clY (ωL(F ∩ Φ)) 6= clY ωLF ∩ clY ωLΦ.6.4. Corollary. Let (Lα : α ∈ A) be a family of rings of closed subsets

of X, L = ∪Lα : α ∈ A and for every α, β ∈ A, F ∈ Lα and Φ ∈ Lβ

there exists µ ∈ A such that F, Φ ∈ Lµ. If (ωLαX, ωLα) = (ωLβX, ωLβ)for all α, β ∈ A, then L is a ring of closed subsets of X and (ωLX,ωL) =(ωLαX, ωLα) for any α ∈ A.

6.5. Corollary. For every ring L of closed subsets of X there exists amaximal ring M of closed subsets of X such that L ⊆ M and (ωLX, ωL) =(ωMX,ωM).

6.6. Example. Let X be a space and ξ, µ be two distinct free ultrafil-tres of closed subsets of X. We put λ = F ⊆ X : F is a compact closedsubset of X, L = λ ∪ ξ and M = µ ∪ λ. Then L, M are maximal rings ofclosed subsets of X, L 6= M and (ωLX, ωL) = (ωMX, ωM) is the one-pointAlexandroff compactification aX of the space X.

6.7. Remark. From Corollary 6.3 it follows that the proof of Proposition4.10 from [19] is not correct. By virtue of Example from [39] it follows thatthe Proposition 4.10 and Theorem 4.11 from [19] are not true.

7. HAUSDORFF G-COMPACTIFICATIONSLet HGC(X) be the family of all Hausdorff g-compactifications of a space

X and HC(X) = C(X) ∩HGC(X). A space X is completely regular if andonly if HC(X) 6= ∅.

It is obvious that every Hausdorff g-compactification is strongly correct andincompressible.

7.1. Definition. A class B of g-compactifications of a space X is called:– a complete upper semilattice if B is a non-empty set and for every non-

empty set M ⊆ B there exists a g-compactification (∨M) ∩B;– a complete lattice if B is a complete upper semilattice and (∧B)∩B 6= ∅.If B is a complete upper semilattice of g-compactification of a space X ,

then in B there exists some maximal element (βBX,βB).


It is obvious that HGC(X) is a complete lattice of g-compactifications ofa space X and the maximal Hausdorff g-compactification (βX, βX) is unique.The g-compactification (βX, βX) is called the Stone-Cech g-compactificationof the space X.

7.2. Theorem. (see [13], Theorem 3.6.21 for T1-spaces). If f : X → Y isa continuous mapping of a space X into a Hausdorff compact space Y , thenthere exists a unique perfect mapping ϕ : ωX → Y such that f = ϕ | X. Iff(X) is dense in Y , then (Y, f) ∈ HGC(X).

Proof. Suppose that the subspace Z = f(X) is dense in Y . For x ∈ Xwe put ϕ(x) = f(x). If ξ ∈ ωX \X, then there exists a unique point y ∈ Ysuch that y ∈ ∩clY f(H) : H ∈ ξ and we put ϕ(ξ) = y. By construction,ϕ : ωX → Y is a single-valued mapping. Fix a closed subset F of Z. LetΦ = f−1(F ). If x ∈ clωXΦ, then ϕ(x) ∈ clY F . Suppose that y ∈ clY F andη = clZ(U ∩ Z) : U is open in Y and y ∈ U. Then y = ∩clY (H ∩ F ) :H ∈ η. There exists an ultrafilter ξ of closed subsets of X such that Φ ∈ ξand f−1(H) ∈ ξ for any H ∈ η. Then ξ ∈ clωXΦ and y = ϕ(ξ). ThusclωXΦ = ϕ−1(clY F ) and the mapping ϕ is continuous. Since Y is a Hausdorffspace, then ϕ is a perfect mapping. The proof is complete.

7.3. Corollary. There exists a unique continuous mapping ϕ : ωX → βXsuch that βX = ϕ | X.

7.4. Corollary. For every continuous mapping ϕ : X → Y there exists aunique continuous mapping βf : βX → βY such that βf βX = βY f . Inparticular, the mapping βf is perfect.

8. RELAXED COMPACTIFICATIONS8.1. Definition. A g-compactification (Y, f) is called a relaxed g-compactification

or a ρ - g-compactification of a space X if there exists a continuous mappingϕ : ωX → Y of ωX onto Y such that f = ϕ | X.

Let ρGC(X) be the class of all ρ - g-compactifications of a space X andρC(X) = C(X) ∩ ρGC(X).

It is obvious that ρGC(X) is a set and ρC(X) 6= ∅. Moreover, HGC(X) ⊆ρGC(X). The compactification ωX is the maximal element of the classesρGC(X) and ρC(X).

We say that the g-compactification (Y, f) is a ρ∗ - g-compactification if (Y, f) ∈ρGC(X) and for every two continuous mappings ϕ, ψ : ωX → Y for whichϕ | X = ψ | X = f we have ϕ = ψ. Let ρ∗GC(X) be the class of allρ∗ - g-compactifications of X and ρ∗C(X) = C(X) ∩ ρ∗GC(X).

8.2. Example. Let D be a discrete space and cD = D ∪ a, b be thecompactification constructed in Example 1.7. We put ϕ(x) = ψ(x) = x forany x ∈ D, ϕ−1(a) = clωDA \ A, ϕ−1(b) = clωDB \ B, ψ−1(b) = ∅ and


ψ−1(a) = ωD \ D. The mappings ϕ, ψ are continuous and ϕ 6= ψ. ThuscD ∈ ρC(X) \ ρ∗C(X).

8.3. Proposition. Let (Yα,fα) : α ∈ A be a non-empty set of g-compactifications of a space X, ϕα : ωX → Yα : α ∈ A be a family ofcontinuous mappings, fα = ϕα | X for any α ∈ A, ϕ(x) = (ϕα(x) : α ∈ A) ∈uYα : α ∈ A for every x ∈ ωX, Y = ϕ(ωX) and f = ϕ | X. Then (Y, f) isa ρ - g-compactification of the space X and (Yα,fα) ≤ (Y, f) for any α ∈ A. If(Yα,fα) ∈ ρ∗GC(X) for every α ∈ A, then (Y, f) = ∨(Yα,fα) : α ∈ A and(Y, f) ∈ ρ∗GC(X).

Proof. It is obvious that (Y, f) ∈ ρGC(X) and (Yα, fα) ≤ (Y, f) for everyα ∈ A.

Suppose that (Yα,fα) ∈ ρ∗GC(X) for every α ∈ A. Let (Z, g) ∈ ρGC(X)and (Yα,fα) ∈ (Z, g) for any α ∈ A.

Fix a continuous mapping h : ωX → Z, where g = h | X. For every α ∈ Afix a continuous mapping hα : Z → Yα,where fα(x) = hα(h(ω)) for everyx ∈ X.

Since (Yα,fα) ∈ ρ∗GC(X), ϕα = hα h for any α ∈ A. Thus ψ(z) =(hα(z) : α ∈ A) = (ϕα(h−1(z)) : α ∈ A) is a continuous mapping of Z into Yand f = g ψ. The proof is complete.

8.4. Corollary. For every space X:– the set ρ∗GC(X) is a complete lattice with a maximal element (ωX, ωX)

and a minimal element (mX, mX), where mX is the singleton space;– the set ρ∗C(X) is a complete upper semilattice with the maximal ele-

ment(ωX, ωX).8.5. Theorem. Let (Y, f) be a g-compactification and ϕ : ωX → Y be a

continuous closed mapping for which f = ϕ | X. Then:1. (Y, f) ∈ ρ∗GC(X);2. if F is a closed subset of the space Z = f(X) and y ∈ clY F \ F ,

then there exists an ultrafilter ξ of closed subsets of Z such that F ∈ ξ and∩clY H : H ∈ ξ = y.

Proof. Let F be a closed subset of Z and y ∈ clY F \ F . Then y ∈ Y \ Z,ϕ−1(y) is a compact closed subset of ωX and ϕ−1(y) ⊆ ωX \X. It is obviousthat clωXf−1(F ) ∩ ϕ−1(y) 6= ∅. Let η ∈ ϕ−1(y) ∩ clωXf−1(F ). Then η isan ultrafilter of closed subsets of X and f−1(F ) ∈ η. Hence ξ = f(η) is thedesired ultrafilter. The assertion 2 is proved.

Suppose that ψ : ωX → Y is a continuous mapping, ψ |X = f and ϕ 6=ψ. There exists a point η ∈ ωX \ X such that ϕ(η) 6= ψ(η). Since ϕ is aclosed mapping, the set ϕ(η) is closed and ψ(η) = ∩clY f(H) : H ∈ η.There exists F ∈ η such that ψ(η) /∈ clY f(F ). Thus η ∈ clωXF and ψ(η) /∈clY f(F ) = clY (F ), a contradiction. Thus (Y, f) ∈ ρ∗GC(X). The proof iscomplete.


8.6. Example. Let X be a locally compact non-compact T1-space andbX = X ∪ b be the one-point compactification of X with the nucleus L0 =F ⊆ X : F is a finite set. Then bX ∈ ρ∗C(X). If the space X is not discrete,then does not exist a closed mapping ϕ : ωX → bX such that ϕ(x) = x forany x ∈ X does not exist.

8.7. Remark. HGC(X) ⊆ ρ∗GC(X) for any space X.

9. ωα-COMPACTIFICATIONS9.1. Definition. A g-compactification (Y, f) of a space X is called:– an ωα-compactification of the space X if there exists a continuous closed

mapping ϕ : ωX → Y such that f = ϕ | X, the mapping f : X → f(X) isclosed and the set f−1(y) is compact for any point y ∈ z(f(X));

– an α-compactification if there exists a closed continuous mapping ϕ :ωX → Y such that f = ϕ |X.

9.2. Lemma. Let (Y, f) be a g-compactification of a space, f : X → f(X)be a closed mapping, ϕ : ωX → Y be a closed continuous mapping and f =ϕ |X. The following assertions are equivalent:

1. the mapping ϕ is perfect;2. ϕ−1(y) is a compact subset of Y for any y ∈ z(Y );3. if Z = f(X), then the subset f−1(y) is compact for any y ∈ z(Z);4. ϕ−1(y) = f−1(y) and the subset f−1(y) is compact for any y ∈ z(Z);Proof. By definition, c(Y ) = y ∈ Y : the set y is closed inY and

z(Y ) = Y \ c(Y ). The implications 1 → 2 and 4 → 3 are obvious.If ξ ∈ ωX \X, then the set ϕ(ξ) is closed in Y and ϕ(ξ) ∈ c(Y ). Thus

ϕ−1(y) = f−1(y) for any y ∈ z(Y ). Since z(Z) ⊆ z(Y ), then the implications2 → 3 → 4 are proved. If y ∈ c(Y ), then ϕ−1(y) is a closed compact subset ofωX. The implication 4 → 1 is proved. The proof is complete.

Let ΩACG(X) be the set of all ωα - g-compactifications of a space X andΩAC(X) = C(X) ∩ ΩAGC(X).

From Theorem 8.5 it follows9.3. Corollary. For every space X:1. ΩAGC(X) ⊆ ρ∗GC(X);2. ΩAC(X) ⊆ ρ∗C(X);3. ΩAGC(X) and ΩAC(X) are sets of g-compactifications with the maximal

element (ωX, ωX) and (mX,mX) is a minimal element of the set ΩAGC(X).9.4. Proposition. For a space X the following assertions are equivalent:1. X is a T1-space;2. ωX is a T1-space;3. some compactification of X is a T1-space;4. every ωα-compactification of X is a T1-space;5. every ωα-g-compactification of X is a T1-space.


Proof. Every subspace of a T1-space is a T1-space and the closed image ofthe T1-space is a T1-space. If Y is a ωα-compactification of X, then z(Y ) =z(X). The proof is complete.

9.5. Theorem. Let (Y, f) be a g-compactification of a space X, ϕ : ωX →Y be a continuous mapping, f = ϕ | X be a closed mapping, c(f(X)) ⊆ c(Y )and the set f−1(y) be compact for any y ∈ z(f(X)). The following assertionsare equivalent:

1. (Y, f) is a ωα-g-compactification of the space X;2. the mapping ϕ is closed ;3. (Y, f) is a strongly correct and incompressible g-compactification;4. (Y, f) is an incompressible g-compactification.Proof. The implication 2 → 1 → 2 and 3 → 4 are obvious. The im-

plication 1 → 3 follows from Corollary 2.11. Suppose now that (Y, f) isan incompressible g-compactification. Since f(ωX) is a compact subset andZ = f(X) ⊆ f(ωX), f(ωX) = Y . Since ϕ is continuous, ϕ−1(y) is a closedcompact subset for any y ∈ i(Y ). Let F be a closed subset of X. ThenΦ = f(F ) = ϕ(F ) is a closed subset of Z. The set ϕ(clωXF ) is compact andΦ ⊆ ϕ(clωXF ) ⊆ clY Φ. Since (Y, f) is incompressible, clY Φ = ϕ(clωXF ) andϕ(clωXF ) is a closed subset of Y . Let ξ ∈ ωX8X. Then ξ is an ultrafilterof closed subsets of X and η = f(ξ) is an ultrafilter of closed subsets of Z.If y ∈ ∩η, then y = ∩η and y ∈ c(Z). Thus ϕ−1(y) = f−1(y) is a com-pact subset of ωX for any y ∈ z(Z). Suppose that y ∈ Y \ Z, z ∈ Y, z 6= yand z ∈ clY y. In this case L = Y \ y is a compact subset of Y andX ⊆ L, a contradiction. Thus, y is a closed subset for any y ∈ Y \ Z.Since c(Z) ⊆ c(Y ), then z(Y ) = z(Z) and ϕ is a compact mapping. If Φ isa closed subset of ωX, then ϕ(Φ) = ∩ϕ(clωXF ) : F is a closed subset of Xand Φ ⊆ clωXF is a closed subset of Y . Thus, ϕ is a perfect mapping. Theimplication 4 → 2 is proved. The proof is complete.

9.6. Example. Let X be a non-compact space, x0 ∈ X and x0 =∩clXU : U is open in X and x0 ∈ U. Let (Y, f) be a ωα-compactificationof X and y0 ∈ Y \ X, where f(X) = X. There exists a perfect mappingϕ : X → Y such that ϕ−1(x) = ϕ(x) = f(x) = x for any x ∈ X. On Y weconsider a new topology T ′ = U ⊆ Y \y0 : U is open in Y ∪U ∪V : U, Vare open in Y and x0 ∈ U. Denote by Z the set Y with the topology T ′.The mapping ϕ : ωX → Y is continuous, the set ϕ−1(x) = x is compactfor any x ∈ X and ϕ(x0) = f(x0) is not closed in Z : clZx0 = x0, y0.Thus (Z, f) is not a ωα-compactification of X. By construction, x0 ∈ c(X) ∩z(Z). The compactification (Z, f) is incompressible. Thus the requirementc(f(X)) ⊆ c(Y ) is essential in the conditions of Theorem 9.4.

9.7. Example. Let X be a non-compact space, U be an open subset of X,clXU 6= U, V be an open subset of ωX and V ∩X = U . Consider the mappingϕ : ωX → Y , where ϕ−1(ϕ(x)) = x for any x ∈ V and ϕ−1(ϕ(x)) = V for


any x ∈ V . In Y we consider the topology H ⊆ Y : ϕ−1(H) is open in ωX.Let f = ϕ | X and z0 = ϕ(V ). Then z0 is an open subset of Y and (Y, f) isa ρ - g-compactification of X. The mappings f and ϕ are continuous. Supposenow that if L ⊆ U and L is closed in X, then L = ∅. In this case V = U andthe mappings ϕ : ωX → Y and f : X → f(X) are closed. If the set V is notcompact, then (Y, f) is not a ωα - g-compactification of X.

9.8. Example. Let X = [0, 1) with the topology T = [0, x) : 0 ≤ x ≤ 1.Let U = [0, 2−1). Then ωX = [0, 1] with the topology ωX ∪ T . In thiscase the g-compactification (Y, f) is not a ωα-g-compactification of X, as inExample 9.6.

9.9. Example. Let X = [0, 1) with the topology [0, x) : 0 ≤ x ≤ 1.Then ωX = X ∪ 1 with the topology ωX ∪ [0, x) : 0 ≤ x ≤ 1 and Xis open in ωX. Consider the space Y = 0, 1 with the topology ∅, 0, Y .Let ϕ(x) = f(x) = 0 for any x ∈ X and ϕ(1) = 1. Then ϕ : ωX → Y isa continuous closed mapping and f : X → f(X) = 0 is a closed mapping.Thus Y, f is an α-compactification and a ρ-compactification of the spaceX. Since the mapping ψ : ωX → Y , where ψ(x) = 0 for any x ∈ ωX, iscontinuous and f = ψ |X, then (Y, f) is not a ρ∗-compactification and anωα-compactification.

10. INTRINSICAL CHARACTERISTICS OFωα - G-COMPACTIFICATIONS

10.1. Definition. Let Z be a subspace of a space Y . A set H ⊆ Y iscalled a quasi-clopen in Y relatively to Z subset or a Z-clopen subset of Y if:

1. H ∩ Z is an open subset of Z;2. if F is a closed subset of Z and F ⊆ H, then clY F ⊆ H.10.2. Definition. Let Z be a subspace of a space Y and L ⊆ H ⊆ Y . A

set H is a Z-clopen neighbourhood of L in Y if H is a Z-clopen subset of Yand there exists an open subset U of Y such that L ⊆ U ⊆ H.

10.3. Definition. Let Z be a subspace of a space Y . A family B of subsetsof Y is a Z-clopen basis of Y if for every open subset V of Y and any pointy ∈ V there exists a Z-clopen neighbourhood H ∈ B of y in Y , such thaty ∈ H ⊆ V .

10.4. Lemma. If Z is a subspace of a space Y and B is a Z-clopen basisof Y , then the set Z is dense in Y .

Proof. It is obvious.10.5. Theorem. Let (Y, f) be a ωα - g-compactification of a space X,

Z = f(X) and BY be the set of all Z-clopen subsets of Y .1. BY is a Z-clopen basis of Y ;2. if A,B ∈ BY , then A ∪B, A ∩B ∈ BY .Proof. It is obvious that A ∩B ∈ BY for all A,B ∈ BY .


Fix a perfect mapping ϕ : ωX → Y such that f = ϕ |X and f−1(y) =ϕ−1(y) for any y ∈ z(Z).

For any open subset U of X we put = ωX \ clωX(X \ U) = ∪V : Vis open in ωX and X ∩ V ⊆ U. The set is X-clopen in ωX. LetH be a X-clopen subset of ωX and U = H ∩ X. The set U is open in X.Suppose that F is a closed subset of X and ξ ∈ clωXF \ F . Then ξ is anultrafilter of closed subsets of X and F ∈ ξ. If F ⊆ U , then X \ U /∈ ξ andξ ∈ ωX \ clωX(X \ U) =. Thus clωXF ⊆. Therefore a setH ⊆ ωX is a X-clopen subset of ωX if and only if the set U = H ∩X is openin X and ⊆ H. Denote by BωX the set of all X-clopen subsets of ωX.By construction, : U is open in X is a Z-clopen base of ωX. ThusBωX is an X-clopen base of ωX.

Since = ∪ < V > for all open subsets U, V of X, thenA ∪ B ∈ BωX for all A,B ∈ BωX . It is obvious that A ∩ B ∈ BωX forA, B ∈ BωX .

Since the mapping ϕ is closed, ϕ−1(BY ) ⊆ BωX .Let H ⊆ Y and L = ϕ−1(H) be an X-clopen subset of ωX. Let F ⊆ Z be

a closed subset of Z and F ⊆ L ∩ Z. The set Φ = f−1(F ) is closed in X andthe set U = X ∩ L = f−1(X ∩ Z) is open in X. Since Φ ⊆ U, clωXΦ ⊆ L.Therefore clY F = ϕ(clωXΦ) ⊆ ϕ(L) = H and H is a Z-clopen subset of Y .Therefore A ∪B ∈ BY for any A,B ∈ BY .

Let V be an open subset of Y and y0 ∈ V . The set ϕ−1(y0) is compact.There exists an open subset U of X such that ϕ−1(y0) ⊆⊆ ϕ−1(V ).Let H = y ∈ Y : ϕ−1(y) ⊆ and L = H ∪ (ϕ() ∩ (Y \ Z)). Theset H is open in Y, y0 ∈ H and L ∩ Z = H ∩ Z. Let F be a closed subset ofZ and F ⊆ L. Then f−1F ⊆ U and clY F \ F ⊆ ϕ(clωX f−1(F )) ∩ (Y \ Z) ⊆(ϕ() ∩ (Y \ Z)) ⊆ L.

Since ϕ−1(H) ⊆, then ϕ−1(H) ⊆< X ∩ϕ−1(H) >⊆ U . By construc-tion, ϕ−1(y0) ⊆< X ∩ ϕ−1(H) >. Thus there exists a closed subset Φ of Xsuch that Φ ⊆ ϕ−1(H) and ϕ−1(y0) ∩ clωXΦ 6= ∅. Then f(Φ) ⊆ H ∩ Z andy0 ∈ clY f(Φ) ⊆ L.

Therefore L ∈ BY and y0 ∈ H ⊆ L. The proof is complete.10.6. Proposition. Let Z be a subspace of a compact space Y , Y \ Z ⊆

c(Y ) and B be a Z-clopen base of the space Y such that A ∪B ∈ B for everyA, B ∈ B.

1. if H ∈ B, then F = Z \H is a closed subset of Z and clY F ∪H = Y ;2. if F is a closed subset of Z and y ∈ clY F \ F , then there exists an

ultrafilter ξ of closed subsets of Z such that F ∈ ξ and y = ∩clY H : H ∈ ξ.Proof. For every H ∈ B we put ω(H) = ∪V : V is open in Y and

V ⊆ H. Then H ∩ Z = Z ∩ ω(H) and if Φ ⊆ Z ∩H is a closed subset of Z,then clY Φ ⊆ H.


Fix H ∈ B. Let F = Z \ H. Suppose that Y \ (clY F ∪ H) 6= ∅. Fixy ∈ Y \ (clY F ∪H). Since clY F is a compact set, there exists A ∈ B such thatclY F ⊆ ω(A) ⊆ A ⊆ Y \y. Then A∪H = B ∈ B, Z ⊆ ω(B) ⊆ B ⊆ Y \yand clY Z ⊆ B, a contradiction. Therefore H ⊆ Y \ clY F . The assertion 1 isproved.

Let F be a closed subset of Z and y ∈ clY F \ F . Since y ∈ Y \ Z, the sety is closed in Y . Let Hα : α ∈ A = H ∈ B : y /∈ H. For every α, β ∈ Athere exists λ = λ(α, β) ∈ A such that Hλ = Hα ∪Hβ. Let Fα = F \Hα forany α ∈ A. We affirm that Fα 6= ∅. If Fα = ∅, then F ⊆ Z ∩Hα ⊆ Hα andclY F ⊆ Hα, a contradiction. Thus Fα 6= ∅ for any α ∈ A. We may considerthat Hα = ∅ and Fα = F for some α ∈ A.

Suppose that α ∈ A and y /∈ clY Fα. Then, for some β ∈ B, we haveclY Fβ ⊆ ω(Hβ) ⊆ Hβ ⊆ Y \ y. If λ = λ(α, β) ∈ A, then F ⊆ Hλ ∩ Zand clY F ⊆ Hλ ⊆ Y \ y, a contradiction. Thus y ∈ clY Fα for any α ∈ A.If x ∈ Y \ y, then there exists α ∈ A such that x ∈ ω(Hα) ⊆ Hα andx 6= clY Fα ⊆ Y \ ω(Hα). Therefore ∩clY Fα : α ∈ A = y. Denote byξ the ultrafilter of closed subsets of Z for which Fα : α ∈ A ⊆ ξ. Since∅ 6= ∩clY H : H ∈ ξ ⊆ ∩clY Fα : α ∈ A = y, the proof is complete.

10.7. Theorem. (First intrinsic characteristic, P. C. Osmatescu [25] forT1-compactifications). Let (Y, f) be a g-compactification of a space X, f :X → f(X) be a closed mapping, z(f(X)) = z(Y ) and the set f−1(y) becompact for any y ∈ z(f(X)). The following assertions are equivalent:

1. (Y, f) is an ωα-g-compactification of the space X;2. if Z = f(X), then there exists a Z-clopen base B of the space Y such

that A ∪B ∈ B for any A,B ∈ B.Proof. The implication 1 → 2 follows from Theorem 10.5.Let B be a Z-clopen base of Y and A ∪B ∈ B for all A,B ∈ B.For any x ∈ X we put ϕ(x) = f(x).Let ξ ∈ ωX \ X. Then f(ξ) is an ultrafilter of closed subsets of Z. We

put ϕ(ξ) = ∩clY f(H) : H ∈ ξ. By construction, ϕ(ξ) is a closed compactsubset of Y . Suppose that y1 ∈ ϕ(ξ) and y2 ∈ Y . If Z ∩ ϕ(ξ) 6= ∅, then| ϕ(ξ) ∩ Z |≤ 1. Hence, we may suppose that y2 ∈ Y \ Z. Since the sety2 is closed in Y , then there exist H ∈ B and an open subset U of Ysuch that y1 ∈ U ⊆ H ⊆ Y \ y2. From Proposition 10.6 it follows thaty2 ∈ clY (Z \ H) ⊆ Y \ U ⊆ Y \ y1. Moreover, if y2 ∈ clY F for everyF ∈ f(ξ), then F \H 6= ∅ for every F ∈ f(ξ). In this case F \H ∈ f(ξ) andy1 /∈ ϕ(ξ), a contradiction. Thus ϕ(ξ) is a singleton set for any ξ ∈ ωX \X.Hence ϕ : ωX → Y is a single-valued mapping.

By construction, ϕ(clωXF ) ⊆ clY f(F ) for any closed subset F of X.Let F be a closed subset of X and y ∈ clY f(F ). If y ∈ f(F ), then y ∈

ϕ(clωXF ). Suppose that y ∈ clY f(F ) \ Z. Then there exists an ultrafilterη of closed subsets of Z such that f(F ) ∈ η and y = ∩clY H : H ∈ η.


Since f is a closed mapping, there exists an ultrafilter ξ of closed subsets ofX such that η = f(ξ) and F ∈ ξ. Then ϕ(ξ) = y and ξ ∈ clωXF . Henceϕ(clωXF ) = clY f(F ) for every closed subset F of X.

Let x ∈ ωX, y = ϕ(x), W be an open subset of Y and y ∈ W . Thereexist H ∈ B and an open subset V of Y such that H ∩ Z = V ∩ Z andy ∈ V ⊆ H ⊆ W . Let U = f−1(V ) and U ′ = ωX \ clωX(X \ U). If x /∈ U ′,then x ∈ clωX(X \ U) and y ∈ clY (Z \ V ) ⊆ Y \ V , a contradiction. Thusx ∈ U ′. By construction, ϕ(U ′ ∩ X) = f(U ′ ∩ X) = V ∩ Z ⊆ V . Letx ∈ (ωX \X) ∩ U ′. Then x is an ultrafilter of closed subsets of X, η = f(x)is an ultrafilter of closed subsets of Z and y′ = ∩clY H : H ∈ η. Byconstruction, Z \ V /∈ η and y′ /∈ (Z \ V ). Since Y = H ∪ clY (Z \ V ), theny′ ∈ H. Thus ϕ(U ′) ⊆ H ⊆ W . Hence ϕ is a continuous mapping. In thiscase ϕ−1(y) is a non-empty closed compact subset of ωX for any y ∈ c(Y ). Ify ∈ z(Z) = z(Y ), then ϕ−1(y) = f−1(y) is a compact subset of X.

Let F be a closed subset of ωX and Hα : α ∈ A = H ⊆ X : F ⊆clωXH,F is a closed subset of X. For every α, β ∈ A we have Hλ ⊆ Hα ∩Hβ

for some λ ∈ A. Since ϕ is a compact mapping and ϕ(clωXHα) is a closedsubset of Y , then ϕ(F ) = ∩ϕ(clωXHα) : α ∈ A is a closed subset of Y . Thusϕ is a perfect mapping. The implication 2 → 1 and the theorem are proved.

10.8. Theorem. Let (Y, f) be an ωα-g-compactification of a space X andZ = f(X). There exists the perfect mappings ϕ : ωX → Y , ψ : ωZ → Y andh : ωX → ωZ such that:

1. f = h | X = ϕ | X;2. ϕ = ψ h;3. if g = ψ | Z, then (Y, g) is an ωα-compactification of the space Z;4. (ωZ, f) is an ωα-g-compactification of the space X.Proof. Since (Y, f) is an ωα-g -compactification, there exists a perfect

mapping ϕ : ωX → Y such that f = ϕ |X.The family B = ωZ \ clωZ(Z \ U) : U is an open subset of Z is a Z-

clopen base of the space ωZ. By virtue of Theorem 10.7, there exists a perfectmapping h : ωX → ωZ such that f = h |X. The assertion 4 is proved. Now weput ψ(z) = ϕ(h−1(z)) for any z ∈ ωZ. The mappings ϕ, ψ, h are constructed.The proof is complete.

10.9. Definition. A g-compactification (Y, f) of a space X is a g-compac-tification with the properties Ωα if:

Ωα1. the mapping f : X → f(X) is closed and f−1(y) is a compact set forany y ∈ z(f(X));

Ωα2. if C, D are closed subsets of Y and C ∩ D = ∅, then there exists aclosed subset F of f(X) such that C ⊆ clY F ⊆ Y \ D and C ∩ clY H = ∅provided H is a closed subset of f(X) and F ∩H = ∅;


Ωα3. if y ∈ Y \f(X), F is a closed subset of f(X) and y ∈ clY F , then thereexists a filter η of closed subsets of f(X) such that F ∈ γ and y = ∩clY H :H ∈ η;

Ωα4. the family clY A : A ⊆ f(X) is a closed base of the space Y .10.10. Lemma. If (Y, f) is a g-compactification of a space X with Prop-

erty Ωα3, then z(Y ) = z(f(X)).Proof. One easily can verify that Y \f(X) ⊆ c(Y ), i.e. the set y is closed

in Y for every y ∈ Y \ f(X). Let y ∈ f(X), the set y be closed in f(X) andy′ ∈ clY y. By assumptions, f(X) ∩ clY y = y. If F = f(X) ∩ clY y =y,y′ 6= y and y′ ∈ clY F , then y′ ∈ f(X) \ X and there exists a filter η ofclosed subsets of f(X) such that F ∈ η and ∩clY H : H ∈ η = (y′). Forsome H ∈ η we have y /∈ H. Thus ∅ = F ∩H ∈ η, a contradiction. The proofis complete.

10.11. Lemma. If (Y, f) is a T1 - g-compactification of a space X withProperty Ω2, then (Y, f) is a g-compactification with Property Ωα4.

Proof. Let C be a closed subset of Y and y /∈ C. Since the set D = y isclosed in Y , then there exists a closed subset F of f(X) such that C ⊆ clY F ⊆Y \ y. The proof is complete.

10.12. Theorem. (Second intrinsic characteristic). For a g-compactifi-cation (Y, f) of a space X the following assertions are equivalent:

1. (Y, f) is an ωα-g-compactification of X;2. (Y, f) is a g-compactification of X with Properties Ωα.Proof. Let Z = f(X), ϕ : ωX → Y be a perfect mapping of X onto Y and

f = ϕ |X : X → Y be a closed mapping for which the set f−1(y) is compactprovided y ∈ z(Z).

Let C and D be two closed subsets of Y and C ∩ D = ∅. There existsa closed subset F1 of X such that ϕ−1(C) ⊆ clωXF1 ⊆ ωX \ ϕ−1(D). LetF = ϕ(F1). If H is closed in Z and H ∩ F = ∅, then C ∩ clY H = ∅.Thus (Y, α) is a g-compactification with Property Ωα2. From Theorem 8.5 itfollows that (Y, f) is a g-compactification with Property Ωα3. It is obviousthat clY A : A ⊆ Z is a closed base of Y . The implication 1 → 2 is proved.

Let (Y, f) be a g-compactification with Properties Ωα.If x ∈ X we put ϕ(x) = f(x).If ξ ∈ ωX \X, then ϕ(ξ) = ∩clY f(H) : H ∈ ξ.It is obvious that | ϕ(ξ)∩Z |≤ 1. Let y ∈ ϕ(ξ), y′ ∈ Y \Z and y′ 6= y. Since

y and y′ are two distinct closed sets, then there exists a closed subsetF of Z such that clY F ∩ y, y′ = y and if H is a closed subset of Z andH ∩ F = ∅, then y /∈ clY H. In this case F ∈ f(ξ) and y′ /∈ ϕ(ξ). Thus ϕ is asingle-valued mapping. By construction, ϕ(clωXF ) ⊆ clY f(F ) for any closedsubset F of X. From Condition Ωα3 it follows that ϕ(clωXF ) = clY f(F ) forany closed subset of X. Thus ϕ−1(clY Φ) = clωXf−1(Φ) for any closed subsetΦ of Z. By that fact and Property Ωα4 it follows that ϕ is a continuous


compact mapping. Since clωXA : A ⊆ X is a closed base of ωX, ϕ isa compact mapping and the set ϕ(clωXA) is closed in Y , the mapping ϕ isperfect. The implication 2 → 1 and Theorem are proved.

10.13. Definition. A g-compactification (Y, f) of a space X is a g-compactification of X with Properties ωα if:

ωα1. the mapping f : X → f(X) is closed and the set f−1(y) is compactfor any y ∈ z(f(X));

ωα2. if C and D are closed subsets of f(X) with clY C ∩ clY D = ∅, thenthere exists a closed subset B of f(X) such that C ⊆ B, clY B ∩ clY D = ∅and clY F ∩ clY C = ∅ provided the set F is closed in f(X) and F ∩B = ∅;

ωα3. if F is a closed subset of f(X) and y ∈ F ∩ (Y \ f(X)), then thereexists a filter η of closed subsets of f(X) such that y = ∩clY H : H ∈ η;

ωα4. for every closed subset Φ of Y there exists a family Fα : α ∈ A ofclosed subsets of f(X) such that Φ = ∩clY Fα : α ∈ A and for every α, β ∈ Athere exists µ ∈ A for which Fµ ⊆ Fα ∩ Fβ;

ωα5. Y \ f(X) ⊆ c(Y ).10.14. Theorem. (Third intrinsic characteristic). For a g-compactifi-

cation (Y, f) of a space X the following assertions are equivalent:1. (Y, f) is an ωα - g-compactification of X;2. (Y, f) is a g-compactification with Properties ωα.Proof. The implication 1 → 2 is obvious. From Properties ωα3 and ωα4

follow the Properties ωα3 and ωα4. Theorem 10.2 complete the proof.10.15. Corollary. Let L be a ring of closed subsets of a space X. The

following assertions are equivalent:1. (ωLX,ωL) is an ωα-g-compactification of X.2. L is a ring with the following properties:2.1. the mapping ωL : X → ωL(X) is continuous, closed and the set ω−1

L (y)is compact for any y ∈ z(Z), where Z = f(X);

2.2. if A, B ∈ L and A ∩ B = ∅, then there exists F ∈ L such thatA ⊆ F ⊆ X\B and if H is a closed subset of ωL(X) such that F∩ω−1

L (H) = ∅,then clY ωL(A) ∩ clY H = ∅, where Y = ωLX;

2.3. if F is a closed subset of Z and y ∈ clY F \ Z, then there exists a filterη of closed subsets of Z such that F∈ η and ∩clY H : H ∈ η = y.

Proof. Implication 1 → 2 follows from Theorem 10.14. It is obvious that(ωLX,ωL) is a g-compactification with Properties ωα1, ωα3, ωα4 and ωα5.

Let C and D be closed subsets of Y and C ∩D = ∅. From Lemma 2.1 itfollows that there exist C ′, D′ ∈ L such that C ′∩D′ = ∅, C ⊆ clY ωL(C ′), D ⊆clY ωL(D′). From Condition 2.2 there exists a closed subset F of Z such thatC ⊆ ωL(C ′) ⊆ F ⊆ Z \D′ and clY ωL(C ′) ∩ clY H = ∅ provided H is a closedsubset of Z and H ∩ F = ∅. Thus (ωLX, ωL) is a g-compactification withProperties ωα. The proof is complete.


10.16. Example. Let X be a non-compact space a ∈ X, b ∈ ωX \X andC /∈ ωX. Suppose that a is a closed subset of X. We put Y = Z = ωX∪c.On Y and Z we consider the following topologies:

– ωX is an open subspace of the spaces Y and Z;– c ∪ U : U is open in ωX and a ∈ U is the base of the space Y at a

point c;– c ∪ (U \ b) : U is open in ωX and b ∈ U is the base of the space Z

at a point c.Thus Y and Z are compactifications of the space X. It is obvious that:– Y is a compactification with Properties Ωα1, Ω2, Ωα4, ωα1, ωα2, ωα4 and

without Properties Ωα3, ωα3;– Z is a compactification with Properties Ωα1, ωα1, ωα2 and without other

properties;– the compactifications ωX, Y, Z are equivalent.

11. LOCALLY COMPACT SPACESA set B of a space X is compact-small if every closed subset F ⊆ B of the

space X is compact (see[19]).A space X is called:– locally compact if for every point x ∈ X there exists an open subset U of

X such that x ∈ U and the set clXU is compact;– weakly locally compact if for every point x ∈ X there exists an open subset

U and a compact subset F such that x ∈ U ⊆ F ;– locally compact-small if for every point x ∈ X there exists an open

compact-small subset U of X such that x ∈ U .For every non-compact space X it is determined the one-point Alexandroff

compactification aX, where aX = X ∪ a, a /∈ X, X is an open subspace ofaX and aX \ F : F is a compact closed subset of X is a base of the spaceaX at a point a.

For a compact space X we put aX = X.11.1. Theorem. A space X is a locally compact-small space if and only if

aX is an ωα-compactification of X.Proof. If U is an open compact-small set, then U is open in ωX. Let

ϕ : ωX → aX be the mapping for which ϕ−1(a) = ωX \ X and ϕ(x) = xfor any x ∈ X. If X is a locally compact-small space, thenX is open in ωXand aX. Thus ϕ is a perfect mapping and aX is an ωα-compactification. Ifϕ is continuous, then X is open in ωX. For every point x ∈ X there exists aclosed subset F of X such that x /∈ F and ωX \X ⊆ clωXF . Then X \ F isan open compact-small subset. The proof is complete.

11.2. Theorem. ([19] for T1–spaces). For a space X the following asser-tions are equivalent:


1. X is a locally compact-small space;2. aX is a ωα-compactification of X;3. X is an open subspace of ωX;4. aX is a ρ-compactification of X;5. X is open in some ωα-compactification;6. X is open in every ωα-compactification.

12. PROXIMITIES ON SPACESLet X be a set, δ be a relation on a family of all subsets of X. We shall

write δ(A, B) = 0 if the subsets A, B ⊆ X are δ-related, otherwise we shallwrite δ(A,B) = 1.

We put δ(x, A) = δ(x,A), δ(A) = x ∈ X : δ(x,A) = 0 and [A]δ =∩B ⊆ X : A ⊆ B = δ(B).We put δ(x, y) = δ(x, y).

12.1. Definition. A relation δ on the family of all subsets of a set X is aproximity on the set X if δ satisfies the following conditions:

A1. δ(x, y) + δ(y, x) = 0 if and only if x = y;A2. δ(A,B ∪ C) = minδ(A,B), δ(A,C);A3. δ(A ∪B, C) = minδ(A,C), δ(B,C);A4. δ(∅, X) = δ(X,∅) = 1;A5. δ(A,B) = δ(A, [B]δ).Conditions (A1)–(A5) imply the following properties of proximities:B1. if δ(A,B) = 0, A ⊆ A1and B ⊆ B1, then δ(A1, B1) = 0;B2. if A ∩B 6= ∅, then δ(A,B) = 0;B3. δ(∅, A) = δ(A,∅) = 1 for any A ⊆ X;B4. T (δ) = X \ [A]δ : A ⊆ X is a T0-topology on the set X.We say that T (δ) is the topology generated by the proximity δ.For every proximity δ on X we put δ(A,B) = minδ(A,B), δ(B, A). The

following properties are obvious:B5. δ(A,B) = δ(B, A);B6. δ is a proximity on X if and only if T (δ) is a T1-topology on X;B7. if δ is a proximity on X, then T (δ) = T (δ).12.2. Definition. We say that the proximity δ on a set X is an ωα-

proximity on X ifA6. if δ(A,B) = 1, then there exists a subset C ⊆ X such that A ⊆

C, δ(A,X \ C) = 1 and δ((X \ C) ∩ [D]δ, B) = 0 provided δ(D, B) = 0.If δ1, δ2 are proximities on X and δ1(A,B) ≤ δ1(A,B) for all subsets A,B

of X, then we put δ1 ≤ δ2.12.3. Remark. The notion of the symmetric ωα-proximity was introduced

by A. V. Arhanghel’skii (see [26]). For a symmetric proximity δ Condition A5follows from Conditions A1, A2, A3, A4 and A6 (see[26], p.194).


12.4. Theorem. Let X be a compact space with the topology T andδmX (A,B) = 1 if and only if A ∩ clXB = ∅. Then:1. δm

X is an ωα-proximity on X and T = T (δmX );

2. if δ is a proximity on X and T = T (δ), then δmX ≤ δ ≤ δm

X .Proof. It is obvious.12.5. Remark. Let δ and δ be proximities on X. Then δ is an ωα-

proximity if and only if δ is an ωα-proximity.12.6. Theorem. Let Y be a compactification of a space X and δY (A,B) =

δmY (A,B) for any subsets A,B of X. Then:1. δY is a proximity on X which generated the topology of the space X;2. if Y is an ωα-compactification, then δY is an ωα-proximity.Proof. The assertion1 it is obvious.By construction, δY (A,B) = 1 if and only if clY A∩ clY B = ∅. Fix

A, B ⊆ X. Suppose that δ(A, B) = 0. By virtue of Theorem 10.7, there existan open subset U of Y and an X-clopen subset C of Y such that clY A ⊆U ⊆ C ⊆ Y \ clY B. Let D be a closed subset of X and δ(D, B) = 0. Weput H = D ∩ (X \ C). Fix y ∈ clY D ∩ clY B. There exists an ultrafilter ξof closed subsets of X such that D ∈ ξ and y = ∩clY E : E ∈ ξ. Byconstruction, U ∩ X = C ∩ X is an open subset of X and δ(A, X \ C) = 1.If X \ C /∈ ξ, then E ∩ (X \ C) = ∅ and y ∈ clY E ⊆ C ⊆ Y \ clY B forsome E ∈ ξ, a contradiction.Thus X \ C ∈ ξ, F = (X \ C) ∩ D ∈ ξ andδ(D ∩ (X \ C), B) = δ(F,B) = 0. The proof is complete.

12.7. Definition. The proximities δ1, δ2 on X are called equivalent ifδ1 = δ2.

If δ is a proximity on X , then the proximities δ, δ are equivalent.12.8. Corollary. If Y, Z are two distinct ωα-compactifications of a space

X, then the ωα-proximities δY , δZ on X are not equivalent.12.9. Theorem. For every ωα-proximity δ on a space X there exists a

unique ωα-compactification Y such that δ and δY are equivalent.Proof. The uniqueness of the ωα-compactification Y follows from Coro-

llary 12.8.Fix an ωα-proximity δ on a space X. Two ultrafilters ξ, µ ∈ ωX are called

δ-equivalent if δ(A, B) = 0 for all A ∈ ξ and B ∈ µ. From Condition A6 itfollows that the δ-equivalence of ultrafilters on X is an equivalence relation.Thus, there exist a set Y and a mapping ϕ : ωX → Y of ωX onto Y suchthat ϕ−1(ϕ(x)) = x for any x ∈ X and ϕ−1(ϕ(ξ)) = µ ∈ ωX \X : ξ and µare δ-equivalent ultrafilters on X for every ξ ∈ ωX \ X. Thus ϕ−1(y) is aclass of δ-equivalent ultrafilters for any y ∈ Y \X, where X = ϕ(x). If F is aclosed subset of X, then c(F ) = F ∪y ∈ Y \X : F ∈ ξ for some ξ ∈ ϕ−1(y).By construction, ϕ−1(c(F )) = clωXF . On Y consider the topology generatedby the closed base c(F ) : F is a closed subset of X. Thus ϕ is a continuous


mapping and ϕ(clωXF ) = c(F ) for every closed subset F of X. If y ∈ Y \Xand ξ ∈ ϕ−1(y), then y = ∩c(F ) : F ∈ ξ is a closed subset of Y . Thusthe mapping ϕ is compact. Hence ϕ is a perfect mapping and Y is an ωα-compactification of X. The proof is complete.

13. PERFECT COMPACTIFICATIONS13.1. Definition. Let Y be a compactification of a space X and sY (A) =

∪V ⊆ Y : V is open in Y and V ∩ X ⊆ A. We say that Y is a perfectcompactification of X if clY (sY (U)) \ sY (U) = clY (clXU \ U) for every opensubset U of X.

A subset B of a space X is connected in X if do not exist two closed subsetsF and H of X such that F ∩H = ∅, B ⊆ F ∩H, B ∩ F 6= ∅, B ∩H 6= ∅.

13.2. Theorem. Let Y be an ωα-compactification of a space X. Thefollowing assertions are equivalent:

1. Y is a perfect compactification of X;2. there exists a closed continuous mapping ϕ : ωX → Y such that ϕ(x) = x

for every x ∈ X and the set ϕ−1(y) is connected in ωX for any y ∈ Y .Proof. For T1-spaces the theorem is proved in [26]. The case of T0-spaces

is similar.We mention, that the case of completely regular spaces was studied by E.

G. Sklearenco [34].13.3. Remark. If X is a space, then:1. ωX is a perfect compactification;2. if X is a completely regular space, then the Stone-Cech compactification

βX of X is perfect;3. if X is a rimcompact space (i.e. the family U : U is open in X and clXU\

U is compact is a base of X), then the Freudenthal-Morita compactificationγX of X (see[6]) is perfect and γX ≤ Y for every perfect ωα-compactificationY of X.

14. PUNCTIFORM COMPACTIFICATIONS14.1. Definition. A compactification Y of a space X is called punctiform

if |B| ≤ 1 for every connected in Y subset B ⊆ Y \X.14.2. Theorem. Let f : X → Y be a perfect mapping of a space X onto

a space Y , bX be a perfect ωα-compactification of X and bY be a punctiformωα-compactification of Y . Then there exists a perfect mapping ϕ : bX → bYsuch that f = ϕ | X.

Proof. There exist two perfect mappings g : ωX → bX and h : ωY → bYsuch that g−1(x) = x and h−1(y) = y for all x ∈ X and y ∈ Y. Since f isperfect, there exists a perfect mapping ψ : ωX → ωY such that ψ−1(y) =


f−1(y) for any y ∈ Y . The mapping ϕ(x) = h(ψ(g−1(x))), x ∈ bX is thedesired one. The proof is complete.

The case of separable metric spaces was examined by Duda, the case ofcompletely regular spaces – by E. G. Skliarenco and the case of T1-spaces –by P. C. Osmatescu (see [26, 34]).

14.3. Corollary. If bX is a perfect ωα-compactification of X and cX is apunctiform ωα-compactification of X, then cX ≤ bX.

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NEW ITERATIVE CHAIN CLASSES OFPSEUDO-BOOLEAN FUNCTIONS

ROMAI J., 2, 1(2006),53–56

Vadim Cebotari, Mefodie RataInstitute of Mathematics and Computer Science of A.S.R.M., [email protected], [email protected]

Abstract We continue the investigations founded in the book [8]. The basic idea con-sists of new approach to the problem of the description of closed classes of thefunctions of simplest non-classical logic. The approach consists of the consid-erations of the notion of chain classes as the most fundamental. We describea new infinite sequence of iterative chain classes of pseudo-boolean functions,and other 8 the same classes.

Keywords: iterative pseudo-boolean function, iterative chain class, First Jaskowski’s Ma-

trix, expressibility, pre-complete system, conserving of predicate (relation).

2000 MSC: 03B20, 03B50, 03B55

E. Post [1, 2, 3] described all closed (with respect to superpositions) classesof boolean functions. Each of these classes possesses a finite basis, and they inall constitutes an algebraic countable lattice. Otherwise is the situation withclosed classes of k-valued logic when k = 3, 4, . . . . Namely, Yu. Yanov and A.Muchnik [3] proved that there exist classes of the functions of 3-valued generallogic with countable bases, and since they constitute in all a continuous set,there are classes in general without basis.

M. Rata [5, 8] proved that all mentioned new phenomena take place even inthe simplest not classical logic. That is the logic of First Jaskowski’s Matrix[6, 9]. We denote this logic as well as the class of its functions by the letterJ . The formulas of the J logic are constructed from variables x, y, z, . . . ,possibly indexed, by means of logical operations &, ∨, ⊃ and ¬, and roundbrackets. We denote the formulas (x&¬x) (constant 0), (x ⊃ x) (constant 1),(x ∨ ¬x) (ternondatia of x) and ((x ⊃ y)&(y ⊃ x)) (equivalence of x and y)by symbols 0, 1, ⊥x and (x ∼ y), respectively. The logic J is intermediatebetween classical and intuitionistic logics. It can be defined as the logic of3-element pseudo-boolean algebra < 0, τ, 1; &,∨,⊃,¬ > by means of therelations: 0 < τ < 1, x&y = min(x, y), x∨ y = max(x, y), x ⊃ y = 1 if x ≤ y,and x ⊃ y = y if x > y, and ¬x = x ⊃ 0.

Assertion 1 [8, 9]. Any function of 3-valued general logic belongs to the Jlogic if and only if it can be expressed by means of some formula.

53

54 Vadim Cebotari, Mefodie Rata

Following A. Kuznetsov [7], we say that the function f(x1, . . . , xn) conservesthe predicate (relation) R(x1, . . . , xm) on the set M if , for every elementsαij ∈ M (i = 1, . . . , m; j = 1, . . . , n), the truth expressions

R(α11, . . . , αm1), . . . , R(α1n, . . . , αmn)

imply the fact that the expression

R(f(α11, . . . , α1n), . . . , f(αm1, . . . , αmn))

is also true. Often it is convenient to use, instead of predicate R, the corre-sponding matrix

(αij) (i = 1, . . . , m; j = 1, . . . , l),

such that R is true on m-element set < α1j , . . . , αmj > if and only if the lastone constitutes some column of matrix.

Assertion 2 [8, 9]. Any function of 3-valued general logic belongs to the logicJ if and only if it conserves both predicates x 6= τ and ¬x = ¬y.

We say that, the formula F is expressible in the logic L by means of a Σsystem of formulas when F may be obtained from the formulas belonging toΣ and from variables by means of a finite number of the applications of weaksubstitution rule and replacement by equivalent in L. A system of Σ formulas(in the language of logic L) is called a complete in L if all the formulas areexpressible in L by means of Σ. The system Σ of functions of class K is saidto be pre-complete in K if, for every function f ∈ K \ Σ, the system Σ ∪ fis complete in K.

It is known [8, 9] that the class J contains 10 pre-complete classes denotedby the symbols J0, J1, . . . , J9 and defined as the subclasses of functions of Jconserving, respectively, the predicates: x = 0, x = 1, x 6= y, x ≤ y and(x ∼ y) = (z ∼ u) on 0, 1, (x 6= τ)&((x&y) 6= τ), (x 6= τ)&((x ∨ y) 6= τ),(x ≤ y)&(¬x = ¬y), ((x ∨ y) 6= τ)&(¬x = ¬y) and (((x ∼ y)&¬¬y) =z)&(¬x = ¬y = ¬z) on 0, τ, 1.

We say that, a system Σ of functions of class K is a chain in K if all closedclasses of K, in which Σ is included, are comparable with respect to inclusion(i.e. they constitute a chain). It is clear that every of classes J0, J1, . . . , J9 isa chain in J . In the monograph [8] all chain classes containing the function(x&¬x) (constant 0) are described .

Let us introduce in analysis, for any µ = 1, 2, . . . , the predicates

R11, R

21, . . . , R

µ1 , . . . (1)

over the set 0, τ, 1, where

Rµ1 (x1, . . . , xn) == (x1 6= τ)& . . .&(xµ 6= τ)&((x1 ∨ · · · ∨ xµ) = 1).

New iterative chain classes of pseudo-boolean functions 55

Remark that R11 defines the class J1 not containing the constant 0. It is

necessary to introduce the following 8 matrices:

M1.1 = (τ), M1.3 =(

0 1τ 1

), M1.4 =

(0 1 1τ τ 1

),

M1.5 =(

0 0 1τ 1 1

),M1.6 =

(0 1 11 τ 1

), M1.7 =

(τ τ 1τ 1 1

),

M1.8 =(

τ 1 11 τ 1

), M1.9 =

τ τ 1 1τ 1 τ 11 τ τ 1

.

We denote the classes of pseudo-boolean functions conserving the introducedpredicates of line (1.1) or matrix M1.1, M1.3,M1.4, . . . ,M1.9 by symbolsJ1

1 , J21 , . . . , Jµ

1 , . . . , J1.1, J1.3, J1.4, . . . , J1.9, respectively.

Theorem 1 (criterion of completeness in J1). In order that a system Σ offunctions of J1 be complete in J1 it is necessary and sufficient that Σ be notincluded in any of the classes

J0 ∩ J1, J3 ∩ J1, J4 ∩ J1, J1.1, J1.3, J1.4, . . . , J1.9. (2)

Consequence 1. The classes of line (2) and only they are pre-complete inthe class J1.

Consequence 2. The classes J1.1, J1.3, J1.4, . . . , J1.9 and only they are itera-tive chains in the class J1.

Let us consider the functionh∗µ = &µ+1

i=1 (x1 ∨ · · · ∨ xi−1 ∨ xi+1 ∨ · · · ∨ xµ+1), µ = 1, 2, . . . ,and its particular cases

h∗1 = x1&x2, h∗2 = (x1&x2) ∨ (x1&x3) ∨ (x2&x3).

Lemma 1. h∗µ ∈ Jµ1 \ Jµ+1

1 , µ = 1, 2, . . . .

This Lemma is dual to Lemma 2.1 of monograph [8].

Consequence 3. The following strong inclusions

Jµ+11 ⊂ Jµ

1 , µ = 1, 2, . . .

hold.

Theorem 2. The classes J11 , J2

1 , . . . constitute a infinite decreasing sequenceof iterative chain classes of pseudo-boolean functions.

56 Vadim Cebotari, Mefodie Rata

References[1] Post, E. L., Introduction to a general theory of elementary propositions, Amer. J. Math.,

43(1921), 163–185.

[2] Post, E. L., Two-valued iterative systems of mathematical logic, Princeton Univ. Press,Princeton, 1941.

[3] Jablonskii, S. V., Gavrilov G. P., Kudreavtsev V. B., The functions of the algebra oflogic and Post classes, Nauka, Moscow, 1974. (Russian).

[4] Yanov, Yu. I, Muchnik A. A., On the existence of k-valued closed classes without finitebasis, Dokl. AN SSSR, 127, 1 (1959), 44–46. (Russian).

[5] Rata, M. T., On functional completeness in the intuitionistic propositional logic, Prob-lemy Kibernetiki, Nauka, Moskow, 39 (1982), 107–150. (Russian).

[6] Jaskowski, R., Recherches sur le systeme de la logique intuitioniste, Actes du CongresIntern. de Philosophie Scientifique, Paris, 1936, Part VI, 58–61.

[7] Kuznetsov, A. V., On functional expressibility in the super-intuitionistic logics, Matem-aticheskie Issledovania, Stiinta, Chisinau, 6, 4(1971), 75–122. (Russian).

[8] Rata, M. T., Iterative chain classes of pseudo-boolean functions, Stiinta, Chisinau,1991, 204 p. (Russian).

[9] Rata, M. T., On a class of 3-valued logic, corresponding to First Jaskowski’s Matrix,Problemy Kibernetiki, Nauka, Moskow, 21(1969), 185–214 (Russian).

APPLICATIONS TO LIE ALGEBRAS

ROMAI J., 2, 1(2006),57–65

Camelia Ciobanu”Mircea cel Batran” Naval Academy, Constanta

Abstract In order to find the Jacobson radical of finite-dimensional associative algebra,polynomial time algorithms are described. An efficient algorithm for computingthe solvable and nilpotent radicals of Lie algebra is presented.

Keywords: Lie algebra, Jacobson radical

2000 MSC: 16N28, 16N40, 17B15.

1. INTRODUCTIONConsider some basic algorithmic problems related to finite dimensional as-

sociative algebras. Our starting point is the structure theory of these alge-bras and we touch upon some applications of the associative decompositionalgorithms, including efficient algorithms for calculating the radicals of Liealgebras.

2. BASIC DEFINITIONS AND THEOREMSA linear space L over the field k is an algebra over k if it is equipped with a

binary, k-bilinear operation (called multiplication). Denote by xy the productof x, y ∈ L . Multiplication is assumed to be associative, i.e. x(yz) = (xy)zfor every x, y, zL. Throughout we assume that dimk L = n < ∞. We say thatL is a commutative algebra if xy = yxf or every x, y ∈ L. A k- subspace Sof L is a subalgebra of L if S is closed under multiplication: if x, y ∈ S thenxy ∈ S. A k subspace I of L is a left ideal of L if yx ∈ L whenever x ∈ I andy ∈ L. A right ideal is defined similarly. An k- subspace I of L is an ideal ofL if I is both left and right ideal of L. If I is an ideal in L, then we can formthe factor algebra L/I. The notions of homomorphism and L-module are usedin the standard way. The algebra L is simple if it has no ideals except (0)and L, and LL 6= 0, where LL is the algebra generated by products ab witha, b ∈ L. We say that L is the direct sum of its ideals L1, ..., Ls (written asL1 ⊕ ...⊕ Ls) if L is the direct sum of these linear subspaces.

Theorem 2.1 (Representation Theorem). Let L be an algebra over thefield k and suppose that dimk L = n. Then L is isomorphic to a subalgebra ofMn+1(k) - the algebra of all n + 1 by n + 1 matrices over k.

57

58 Camelia Ciobanu

An element x ∈ L is called nilpotent if xp = 0 for some positive integerp. An element x is strongly nilpotent if xy is nilpotent for every y ∈ L. TheJacobson radical Rad(L) of L is the set of strongly nilpotent elements of L. Itis not difficult to see that Rad(L) is an ideal of L and L/Rad(L) has no nonzerostrongly nilpotent elements. It can be shown that Rad(L) is a nilpotent ideal :there exists a positive integer p such that x1x2...xp = 0, for all x1, x2, ..., xp ∈Rad(L). An algebra L is semisimple if |L| ≥ 2 and Rad(L) = (0).

A characterization of semisimple algebra is included in the next theorem.

Theorem 2.2 (Wedderburn’s Theorem). If L is a finite-dimensionalsemisimple associative algebra over the field k, then L is expressible as a directsum L = L1 ⊕ L2 ⊕ ... ⊕ Ls, where the Li are exactly the minimal nonzeroideals of L. Moreover, Li is isomorphic to matrix algebra Mni(ki)where ki isa possibly noncommutative extension field of k(1 ≤ i ≤ s).

In the next section we present algorithms in order to compute the Jacobsonradical. The interesting case here is when k (and consequently L) is finite. Weexplain some basic methods of finding the structural ingredients of algebras ina computationally efficient way. These methods are applied in the last section,leading to the computation of the (solvable) radical and the nilradical of Liealgebras.

Recall some basic facts about Lie algebras. Detailed exposition can be foundin Jacobson [10] and Humphreys [9]. A linear space G over the field k is aLie algebra, if G is equipped with a k- bilinear binary operation [, ] such that[x, x] = 0 for every x ∈ G and [[x, y] , z] + [[y, z] , x] + [[z, x] , y] = 0 for everyx, y, z ∈ G (the Jacobi identity). Just like in the associative case, we havethe familiar notions of subalgebra, ideal, factor algebra and homomorphismfor Lie algebras. The derived series of G is the collection G(j) of ideals in Gdefined as G(0) = G and G(i+1) =

[G(i), G(i)

]for i> 0. A Lie algebra is called

solvable if the derived series reaches (0) in finitely many steps: G(n) = 0 forsome natural number n. Here we consider finite dimensional Lie algebra only.In this case G has an unique maximal solvable ideal, denoted by R(G), theradical of G. The descending central series of G is the sequence Gj of ideals ofG, where G0 = G and Gi+1 =

[G,Gi

]for i ≥ 0. A Lie algebra G is nilpotent

if Gn = (0) for some natural number n. If dimk G < ∞, then G has an uniquemaximal nilpotent ideal N(G) the nilradical of G.Example Let L be an associative algebra over k. For two elements a, b ∈ L,the additive commutator is [a, b] = ab − ba. It is easy to check that thisoperation satisfies the identities of a Lie-bracket. As a consequence, if a k-subspace S of L is closed with respect to the operation [, ], then S can beconsidered as a Lie algebra. Particularly important are the Lie subalgebrasof this form which are obtained from G = Mp(k). They are called linear Liealgebras.

Applications to Lie algebras 59

There is a straightforward analogue of the regular representation for a Liealgebra G. For an x ∈ G, let ad(x) : G → G be the linear map that mapsy ∈ G to [x, y] . The map x → ad(x) is a Lie algebra homomorphism fromG to linear Lie algebra gl(G) of all linear transformations of the k-space G.Unfortunately, this map is far from being faithful (if G is simple, then this mapis faithful). We just remark here that, according to a deep theorem of Adoand Iwasuwa [10], every finite-dimensional Lie algebra is actually isomorphicto a linear Lie algebra.

We are interested in exact computations, such that k is either a finite fieldor an algebraic number field. We specify now the input of the algorithmicproblems addressed. In order to obtain sufficient general results, we consideran algebra to be given as a collection of structure constants.

If L is an algebra over the field k and e1, e2, ..., en is a basis of the k - spaceL, then multiplication is completely described if we express the product eiej aslinear combinations of the basis elements eiej = γij1e1 + ... + γijnen.The coefficients γijk ∈ k are called structure constants. When an algebrais given as input, we assume that it is represented as an array of structureconstants. Substructures (such as subalgebras, ideals, subspaces) can then berepresented by bases whose elements are linear combinations of basis elementsof the ambient structure (algebra).

In our cases k can be viewed as an algebra over its prime field P ; thereforek can also be represented with structure constants from P (If k is finite,then P = kp for some prime p, if k is number field, then P = Q). In thesecases k is usually specified by giving the (monic) minimal polynomial f ofa single generating element α over the prime field P . This is a special caseof the representation with structure constants. The coefficients of f give thestructure constants with respect to P - basis 1, α, α1, ..., αn−1 of k where n =dimp k.

Another important way to represent an algebra is in the form of a matrixalgebra. In these cases we are given a collection of matrices which generate thealgebra. The algorithms described in this paper are applicable in this settingas well. From such a matrix representation one can efficiently find a basis ofthe algebra and then calculate structure constants with respect to this basis.

We would like to consider algorithms which have a theoretical guaranteefor their efficiency. From the perspective of computer science these are thepolynomial time algorithms. An algorithm runs in polynomial time if oninputs of length n the computation requires at most nc bit operations. Herec > 0 is a constant independent of n, and n is a positive integer.

60 Camelia Ciobanu

3. COMPUTING THE RADICALSuppose that L is a finite-dimensional algebra over the field k, given as a

collection of structure constants. Our objective is to find a basis of Rad(L), theradical of L, in time polynomial in the input size.

If char k = 0, then the problem is equivalent to solving a system of linearequations over the ground field as follows from the characterization of theradical by Dickson

Theorem 3.1 Let L be a finite dimensional algebra of matrices over a fieldk, and chark = 0. Then Rad(L) = x ∈ L/Tr(yx) = 0 for every y ∈ L.In fact, if e1, e2, ..., en is a linear basis of L over k, then to find Rad(L), itsuffices to solve the linear system Tr(eix) = 0, i = 1, ..., n, where x is an,,unknown” element of L.

We now turn to the case where L (and hence k = kq) is finite. We assumethat p is a prime, q is a power of p, k = kq and that L is a subalgebra ofMn(k). The statement of Dickson’s Theorem is no longer valid in positivecharacteristic. There is, however, a more subtle, and still useful, descriptionof the radical in this case. We explain this in the sequel. Define the naturalnumber l by: pl ≤ n < pl+1. Let M denote the set of matrices L ∪ I ,where I is the identity element of Mn(k). Let a ∈ Mn(k) be a matrix. Itwill be convenient to work with the following variant of the characteristicpolynomial of a: ϕa(X) = det(Xa + I) ∈ k[X]. Consider the expansion of

ϕa(X) as a polynomial in the variable X: ϕa(X) = 1 +n∑

i=1ca,iX

i. The indices

of the form i = pj , j = 0, ..., l play a key role in the following arguments. Forj = 0, ..., l we define the ,,trace functions” Tj by Tj(a) := ca,pj . Obviously,T0(a) = Tr(a) is the trace of the matrix a. We also define a sequence L : R0 ⊇R1 ⊇ ... ⊇ Rl+1 of subsets of L as Rj := a ∈ L|Ti(ma) = 0 for every m ∈M and 0 ≤ i < j (1 ≤ j ≤ l + 1). Alternatively, for every 0 ≤ j ≤ l, we haveRj+1 := a ∈ Rj |Tj(ma) = 0 for every m ∈ M.

At this point we can formulate a characterization of Rad(L) which is themain result of this section, it is useful and reads as follows

Theorem 3.2. Let L ≤ Mn(k) be an algebra of matrices over the finitefield k of characteristic p. Put l = [logp n] and let R0, Rl, ..., Rl+1 be as definedabove. Then:

1 R0, Rl, ..., Rl+1 are ideals of L;

2 Rl+1 = Rad(L);

3 for every j ∈ 0, ..., l the function Tj is pj-semilinear on Rj , i.e.Tj(αa + βb) = αpj

Tj(a) + βpjTj(b) for every α, β ∈ k and a, b ∈ Rj .


Property 3 implies that we can obtain a basis of Rj+1 from a basis Rj bysolving a system of linear equations over k. Indeed, set a0 = I, and leta1, ..., as be a basis of L over k. Suppose that b1, b2, ..., br is a basis ofRj over k, and we are looking for a basis of Rj+1. Semilinearity implies that

an element a ∈ Rj , a =r∑

i=1λibi is in Rj+1 iff

r∑i=1

Tj (aibi) λpj

i = 0, (t = 0, ..., s) .

The inverse of the automorphism λ → λpsof the finite field k = kq can be

computed efficiently, hence the above system can be translated intor∑

i=1

Tj (atbi)1

ps λi = 0 (t = 0, ..., s) .

The latter is a system of linear equations in the variables λ1, λ2, ..., λr. Thus,we start with R0 = L and then in turn proceed to compute Rl, ..., Rl+1.From a basis of Ri we obtain a basis of Ri+1 by solving a system of linearequations over k. The number of equations and the number of variables is atmost n2, hence the system can be solved in time (n + log q)o(l). We obtaina basis of Rad(L) in l + 1 = 0 (log n) such rounds; therefore the overall costof the computation is (n + log q)0(l) bit operations. Below we give a formaldescription of the algorithm.

Radical (L) :=A := I∪ basis of L;B := basis of L;

for j from 1 to[logp n

]+ 1do

if B 6= φ thenC :=

(Tj(ab)

1j

)a ∈ A, b ∈ B

Λ := a basis of KerC :B := λ1b1 + ... + λ2br| (λ1, ..., λr) ∈ Λ

fiodreturn B.

The proof of Theorem 3.2 follows immediately from the next sequence oflemmas. The statement of the first lemma can be considered as a special caseof the theorem, where the underlying module is simple.

Lemma 3.3. Let S be a simple algebra over the finite field k and U be asimple S-module. Then there exists an element a ∈ S with TrU (a) = 1, whereTrU (a) = 1 stands for the ordinary trace of the action of a on U .

Below we show that semilinearity and other useful proprieties hold for thetrace functions Tj on certain ideals.

Lemma 3.4. Let L ≤ Mn(k) be a matrix algebra over the field k of char-acteristic p. Assume that L 6= Rad(L). Let (0) = U0 < U1 < ... < Ur = U be

62 Camelia Ciobanu

a composition series of the L- module U = kn. Let I1, I2, ..., It be the minimalelements of the set of ideals of L properly containing Rad(L). For every indexi ∈ 1, 2, ..., t fix a simple L- module Vi that belongs to the ideal Ii and denotethe multiplicity of Vi in the composition series by mi. Put l =

[logp n

]and

define the ideals R′0, R

′1, ..., R

′l+1 as

Rj = Rad(L) +∑

pp/mi

Ii. (∗)

Then:

i) R′l+1 = Rad(L);

ii) for every j ∈ 0, ..., l the function Tj is pj- semilinear on R′j (in the

sense of Theorem 3.2);

iii) Tj(ab) = Tj(ba) for every j ∈ 0, ..., l, b ∈ L and a ∈ R′j ;

iv) Tj is identically zero on R′j+1 (j = 0, ..., l) ;

v) Tj is not identically zero on ideals Ji so that the multiplicity mi is divis-ible by pj but not by pj+l (j = 0, 1, ...l) .

Lemma 3.4 provides a tool to inductively check that the subsets Rj coincidewith the ideal R′

j defined in that lemma.

Lemma 3.5. Keeping the notation of Lemma 3.4 for each j ∈ 0, ..., l wehave R′

j+1 = a ∈ R′j |Tj(ab) = 0 for every b ∈ I ∪ L.

Remark 3.6. The approach presented here is a simplified and specializedversion of a result from [4 ] where arbitrary fields of characteristic p are al-lowed. In the general case the characterization of the ideals Rj is slightly morecomplicated than formula (∗).

4. COMPUTATIONS IN LIE ALGEBRASJust like associative algebras, Lie algebras can be conveniently described

by structure constants. As we saw, if G is a Lie algebra over a field k ande1, e2, ..., en is a basis of G, then the bracket is described if we have the productseiej as linear combinations of the basis elements: [ei, ej ] = γij1e1 + ...+ γijnen ,γijk

∈ k are called structure constants.Let us now outline algorithms for computing the nilpotent and the solvable

radical of a Lie algebra. These problems can be reduced to associative radicalcomputations. First we consider the nilradical. We need a theorem of Jacobson[10].

Let G be a finite dimensional Lie algebra over an arbitrary field k.


Theorem 4.1 (Jacobson’s Theorem). Let L be the associative (matrix-) algebra generated by the linear transformations ad(x), x ∈ L, i.e.,the imagead(L) of the adjoint representation of L. Then an element x ∈ L is in thenilradical N (L) if and only if ad(x) ∈ Rad(L).

This result offers a reasonable way to computing N(L) if the ground field k isa finite field or an algebraic number field. Indeed, we can compute first a basisof L, and then compute Rad(L) with the algorithms of the previous section.We compute the intersection of the k- subspaces ad(L) and Rad(L) by solvinga system of linear equations. By Jacobson’s Theorem, the inverse image inL of the intersection ad(L) ∩ Rad(L) is N(L). A formal description of ourmethod reads as follows.

Nilradical (L) :=L := associative algebra generated by ad(L)return ad−1Rad(L).

Corollary 4.3. Let G be a finite-dimensional Lie algebra over the field k,where k is either a finite field or on algebraic number field. Suppose that G isgiven as a collection of structure constants. Then the nilradical N(G) can becomputed in time polynomial in the input size.

Next we address the problem of computing the solvable radical R(G). Forfinite k the problem of computing R(G) can be reduced efficiently to theproblem of computing N(G).

First remark that N(G) ≤ R(G) and if N(G) = (0), then R(G) = (0) becausestarting with the next to the last element of the derived series of R(G) they areAbelian, hence nilpotent ideals of G. With these we define the sequence Gi ofLie algebras as follows: let G0 = G; if N(Gi) 6= (0), then let Gi+1 = Gi/N(Gi);if N(Gi) = (0), then Gi+1 is not defined. This sequence of Lie algebras hasno more than dimk G + 1 elements. From Corollary 4.3. it follows that allalgebras Gi can be computed in polynomial time over finite k. Let Gj be thelast algebra of the sequence. We then have Gj

∼= G/R(G). Moreover, we canconstruct a basis for R(G) by keeping track of the preimages of the ideals wefactored out during the computation of the sequence G0, G1, ..., Gj .

It is instructive to view this computation in terms of ideals of G. Fori > 0 let Ji denote the kernel of the composition of the natural maps G0 →G1 → ... → Gi. We then have J1 ⊂ J2 ⊂ ... ⊂ Ji, N(G/Ji) = Ji+1/Ji for0 < i < j and Jj = R(G). From a basis of Ji a basis of Ji+1 is obtained bya single call of the nilradical - algorithm with the algebra G/Ji as input. Asa result, we obtain elements h1, h2, ..., hk ∈ G such that h1 + Ji, ..., hk + Ji

form a basis of Ji+1/Ji. Now the elements hl together with a basis of Ji willconstitute a basis of Ji+1. In such j rounds we obtain a basis of R(G). Belowwe give a formal description of the algorithm:

Solvable Radical(G) :=

64 Camelia Ciobanu

S := (0) :loopS := Nilradical (G/S);φ := natural map G → G/S;S := φ−1(S);until S = (0);return S.

Corollary 4.4. Let G be a finite-dimensional Lie algebra over kq, givenby structure constants. Then (a basis of) the solvable radical R(G) can becomputed in time polynomial in dimkq G and log q.

Variants of the radical algorithms discussed here are implemented by Wil-helm de Graaf in a general library of Lie algebras algorithm called ELIAS(for Eindhoven Lie Algebra System), which is built into the computer algebrasystems GAP4 and MAGMA.

AcknowledgementsThis paper was supported by the CNCSIS grant 1075/2005

References[1] E. H. Bareiss, Sylvester’s identity and multistep integer-preserving Gaussian elimina-

tion, Mathematics of computation, 103 (1968), 565 - 578.

[2] Beck, R. E., B. Kolman, I. N. Stewart, Computing the structure of a Lie algebra, inComputers in nonassociative rings and algebras, Academic, New York, (1977), 167 -188.

[3] Berlenkamp, E. R., Factoring polynominals over large finite fileds, Math. of Computa-tion 24 (1970), 71375.

[4] Cohen, A. M., G. Ivanyos, D. B. Wales, Finding the radical of an algebra of lineartransformations, J. of Pure and Applied Algebra 117&118 (1997) 177 - 193.

[5] Collins G. E., M. Mignotte, F. Winkler, Arithmetic in basic algebraic domains, Com-puter algebra, symbolic and algebraic computation, 2nd ed., Springer, Berlin, (1983),180 - 220.

[6] L.E. Dickson, Algebra and their arithmetic’s, Univeristy of Chicago, 1923.

[7] Edmonds System of distinct representatives and linear algebra, Journal of Research ofthe National Bureau of Standards 718 (1967), 241 - 245.

[8] Ivanyos, G., L. Ronyai, A. Szanto Decomposition of algebras , Applicable Algebra inEngineering, Communication and Computing 5 (1994), 71 - 90.

[9] Humphreys, J. E., Introduction to Lie Algebra and representation theory, GraduateTexts in Mathematics 9, Springer, Berlin, (1980).

[10] Jacobson, N., Lie algebra, John Wiley, (1962).

[11] Knuth, D. E., The art of computer programming, Seminumerical algorithms, 2,Addison-Wesley, New York, (1981).

[12] Lidl, R., Niederreiter, H., Finite fields, Addison-Wesley, New York, (1983).

PERTURBED BIFURCATION IN A DEMAND-OFFER MODEL

ROMAI J., 2, 1(2006),65–70

Elena CodeciCollege ’V. Voda’, Curtea de Argescodeci [email protected]

Abstract For a demand-offer model with cubic demand function the universal unfoldingsand the transient manifolds are determined. The sections in the static bifurca-tion diagram and the universal unfoldings of the manifold S are characterized.The cases are enumerated where these universal unfoldings cannot be found,the respective germs having a greater than three or infinite codimension.

The mathematic model associated with the economical demand-offer modelwith a cubic demand function is the Cauchy problem for the system of ordinarydifferential equations(s.e.d.o.)

x = αx3 + cx2 − y,

y = µx− y − γ,(1)

with α, c, µ, γ- the four parameters and x, y- the state functions. In the

following we use the notation, definitions from [GS] (for perturbed bifurcation)and [GMO] (for general theory of bifurcation and nonlinear dynamics). Themethod of normal form is that presented in [AP] and [Kus].

1. DETERMINATION OF THE EQUILIBRIUMMANIFOLD AND S MANIFOLDS

Proposition 1.1 The bifurcation equation for (1) is

αx3 + cx2 − µx + γ = 0. (2)

It is the geometric locus of equilibria (we considered only x because (0.1)2shows that y is a bijective function of x for µ 6= 0). Sections in the staticbifurcation diagram for the s.e.d.o. (1) in the (x,α,c) space, for µ = γ = 1 arerepresented in fig. 1a; in the (x,c,µ) space, for α = γ = 1, in fig. 1b, in the(x, c, γ) space, for α = µ = −1, in fig.1c.

65

66 Elena Codeci

Fig. 1. Sections in the static bifurcation diagram (2) for: a) µ = γ = 1; b) α = γ = 1; c)α = µ = −1.

Proposition 1.2 For the equilibrium points P1

(9αγ+µc6µα+2c2

, µ(

9αγ+µc6µα+2c2

)− γ

)of

s.e.d.o. (1), the linearized s.e.d.o. has at the origin a saddle-node bifurcation,and the S manifold is

S =(α, c, µ, γ) | 4c3γ − µ2c2 + 18αµγc + 27α2γ2 − 4αµ3 = 0

. (3)

The normal form for the linearized s.e.d.o. around the origin, in the phasespace, for a parameter (α, c, µ, γ) ∈ S fixed, is

(n2

n1

)=

(µ− 1 0

0 0

)(n2

n1

)+

( −2a1µn1n2

−µa1n21

)+ O(n3), (4)

with a1 =3αx0 + c

µ− 1. The graphical representation of S, for a fixed value of the

parameter µ, is drawn in fig.2a.

Fig. 2. a) The graphical representation of S in the (c, α, γ) space for µ = −1; b)The phaseportrait for the linearized system around the origin, for (α, c, µ, γ) = (1, 2,−1, 4

27), when

x0 = − 13.

Expresion (1.2) follows by eliminating x0 between (1.1) and the conditionthat the equilibrium point (x0, y0) has a null eigenvalue.

Fig.2b, reveals the presence of a saddle-node.

Perturbed bifurcation in a demand-offer model 67

2. PERTURBED BIFURCATION IN THE STATICBIFURCATION DIAGRAM

Case α 6= 0.

Proposition 2.1 The static bifurcation diagram is an equation G(x) = 0,where the G function is

G(x, a, b, d) = x3 + ax2 + bx + d. (5)

The germ G, for x selected as a state function, b as a control parameter, a andd as small parameters, is the universal unfolding for the fork at the origin.

Proposition 2.2 For d selected as control parameter, a and b as small para-meters, then G in (5) is a two parameter unfolding of a hysteresis point situatedat the origin.

Let Ex,λ be the set of germs in x and λ variables.

Proposition 2.3 The germ g1(x, a) = x3 + ax2 has an infinite codimension,and, generally, any germ g(x, λ) = x2q(x, λ), with q(x, λ) ∈ Ex,λ has an infinitecodimension.

Case α = 0, c 6= 0. For this case, the unfolding of the static bifurcationdiagram for the s.e.d.o. (1) is

G(x, γ, µ, c) = cx2 − µx + γ, i.e. x = cx2 − µx + γ. (6)

Proposition 2.4 The unfolding (6) factorizes with the universal unfolding ofthe limit point.

2.1. DETERMINATION OF THE UNIVERSALUNFOLDINGS FOR S, CORRESPONDINGTO THE NON-HYPERBOLIC EQUILIBRIAP1

Case γγγ 6= 0.

Proposition 2.5 For the bifurcation equation (3), assuming c as a state func-tion, α as a control parameter and γ 6= 0 as fixed, then the origin (c, α) = (0, 0)is a bifurcation winged-cusp point, if µ = 0. In addition, the function from (3)represents a map strongly equivalent with the normal form of the winged cusp.

68 Elena Codeci

In fig. 3 we give sections in the transient manifolds of the universal unfold-ings H(x, λ, α, β, γ) = x3 + λ2 + α + βλ + γxλ of the winged cusp, with planeγ = const., together with all the persistent bifurcations.

γ < 0 γ = 0 γ > 0

(0) (1) (2) (3)

(4) (5) (6) (7)Fig. 3 Sections in the transient manifolds of the universal unfoldings H(x, λ, α, β, γ) = x3 + λ2 +

α + βλ + γxλ of the winged cusp, with plans γ = const. and the persistent unfoldings.

Proposition 2.6 For the bifurcation equation (3), taking µ = 0, c as statefunction, γ 6= 0 as a control parameter, i.e. for the bifurcation problemG2(c, α, γ) = G1(c, α, γ, 0)/γ = 0, it follows that the origin (c, γ) = (0, 0)is a bifurcation hysteresis point.

Proposition 2.7 For the bifurcation equation (3), with µ = 0, choosing c asa state function, γ as control parameter, and α 6= 0 as fixed, then G1(c, α, γ, 0)is an infinite codimension bifurcation germ.

case γ = 0. The bifurcation problem (3) becomes

G1(c, α, 0, µ) = −4αµ3 − µ2c2 = 0. (7)

Perturbed bifurcation in a demand-offer model 69

Proposition 2.8 In (7), for µ considered as a state function, α 6= 0 as fixed,c as a control parameter, we have that at the origin an infinite codimensionbifurcation occurs.

Proposition 2.9 The bifurcation problem (7), for µ taken as a state function,α 6= 0 as fixed, c as a control parameter, has at the origin a fork bifurcation.

Proposition 2.10 In (7), for c taken as a state function, µ as a controlparameter, α 6= 0 as fixed, then G1(c, α, 0, µ) has an infinite codimension.

Proposition 2.11 For (7) with µ 6= 0 fixed, c chosen as a state function, αas a control parameter, we have at the origin (c, α) = (0, 0) a zero codimensionlimit point.

Case ααα 6= 0 fixed.

Proposition 2.12 The bifurcation problem (3), with µ chosen as a state func-tion, γ as a control parameter, α 6= 0 as fixed, has at the origin (µ, γ) = (0, 0),a cusp bifurcation, for c = 0.

Indeed for c = 0, from (3) we obtain the unfolding

G1(µ, γ, α, 0) = −4αµ3 + 27α2γ2. (8)

Proposition 2.13 i) If in (8) we take µ as a state function, α as a controlparameter and γ as fixed, we obtain that G1(µ, α, γ, 0) is a germ in (µ, α)with an infinite codimension.

ii) If in (8) we take γ as a state function, α 6= 0 as fixed and µ as the controlparameter , then we obtain that G1(γ, µ, α, 0) is a zero codimension germ,strongly equivalent with the limit point germ, g(x, λ) = x2 + λ.

iii) If in (8) we take γ as a state function, we fixe µ 6= 0, and α is thecontrol parameter, then G1(γ, α, µ, 0) is a germ in Eγ,α with an infinitecodimension.

iv) Taking in (8), α as a state function, γ 6= 0 as fixed and µ as a con-trol parameter, we obtain that G1(α, µ, γ, 0) = 0 is equivalent with thebifurcation problem g(x, λ) = x2− λ2 = 0, i.e. it is a simple bifurcation.

References[GMO] Georgescu, A., Moroianu, M., Oprea, I., Bifurcation theory - principles and applica-

tions, Ed. Univ. Pitesti, 1999 (Romanian).

[GS] Golubitsky, M.A., Schaeffer, D.G., Singularities and groups in bifurcation theory,Springer, New York, 1985.

70 Elena Codeci

[AP] Arrowsmith, D.K., Place, C.M., An introduction to dynamical systems, CambridgeUniversity Press, 1990.

[Kuz] Kusnetsov, Yu. A., Elements of applied bifurcation theory, Springer, New York, 1994.

INVERSE PROBLEMS OF TRANSITIVECLOSURE

ROMAI J., 2, 1(2006),71–76

Adrian DeaconuUniv. ”Transilvania” of Brasov, Faculty of Mathematics and [email protected]

Abstract A graph G’ starting from a self-transitive closure graph G with as least aspossible arcs so that the transitive closure of G’ is G is found. Two algorithmsare presented. The number of optimal solutions is computed.

Keywords: graphs, permutations

1. INVERSE PROBLEMS OF TRANSITIVECLOSURE

Let G = (V, E) be an oriented graph.Definition 1. The oriented graph G* = (V, E*) is called the transitive

closure of the graph G, where E∗ = (x, y)| there is a directed path from x toy.

Definition 2. The oriented graph G = (V, E) is a self-transitive closure iffits transitive closure does not differ from G, i.e. G* = G.

Let us consider a self-transitive closure graph G = (V, E). The inverseproblem of transitive closure (ITC) is to find G−1 = (V,E−1), where E−1 hasas least arcs as possible from G and the transitive closure of G−1 is G, i.e.(G−1)∗ = G. Let us present other two definitions, which make the differencebetween the quality of the solutions of the inverse transitive closure problem.

Definition 3. G−1 = (V, E−1) is called a solution for the inverse problemof transitive closure, if and only if (G−1)∗ = G and the elimination of any arc(x, y) from E−1 leads to a graph G′ = (V, E′), where E′ = E−1 − (x, y),for which the transitive closure is not G, i.e. ((G′)∗ 6= G.

Definition 4. G−1 = (V, E−1) is called an optimal solution for the inverseproblem of transitive closure, if and only if any graph G′ = (V, E′) so that(G′)* = G has at least the number of arcs of G−1, i.e. |E′| ≥ |E−1|.

Any optimal solution satisfies the inverse problem of transitive closure.

2. ALGORITHMS FOR THE ITC PROBLEMThe first idea that appears for solving the inverse problem is to find criteria

for elimination of arcs from E so that the resulted graph G′ = (V, E′) (E′ =

71

72 Adrian Deaconu

E − eliminated arcs) still has graph G as its transitive closure. Here it issuch a criterion:

Theorem 1. If an arc (x, y) is eliminated from the graph G and in theresulted graph G′ = (V, E′) (E′ = E − (x, y)) there is still a directed pathfrom x to y, then (G′)* = G*.

Proof. Let (x, y) ∈ E be an arc so that in the graph G′ = (V,E′ =E − (x, y)) there is a directed path P ′ from x to y (1). Suppose that(G′)∗ 6= G∗ ⇒ ∃(u, v) ∈ E∗ and (u, v) /∈ (E′)∗ iff there is a directed path Pfrom u to v in G* and there is no directed path from u to v in (G′)*.

There are two situations:1. if the arc (x, y) /∈ P , then all arcs from P are in E and E′, so P is a

directed path in G′ (contradiction);2. if the arc (x, y) ∈ P , then P = (u, ..., x, y, ..., v). Let P ′′ be the directed

path P, where the arc (x, y) is replaced by the path P ′ (see (1)), i.e. P ′′ = (u,..., P ′, ..., v) = (u, ..., x, ..., y, ..., v), where all arcs are in E′. So, there is adirected path P ′′ in G′ from u to v (contradiction).

In both cases we obtained contradiction, hence the assumption (G′)∗ 6= G∗is false. So, (G′)* = G* and the theorem is proved.

Starting with Theorem 1 and remarking that the arcs (x, x), x ∈ V can bealso eliminated from G, we can easily write an algorithm for finding a solutionof the inverse problem of transitive closure.

Fig. 1. Algorithm 1 (elimination of arcs).

Theorem 2. The algorithm 1 finds a solution G−1 = (V, E−1) of theinverse problem of transitive closure in a complexity of O(m2), where m isthe number of directed arcs in G = (V, E), i.e. m = |E|.

Proof. Obviously G−1 is a solution, because every directed arc was testedfor elimination, such that as a result of application of the algorithm 1 there isno other arc in E−1 which can be eliminated any more. The test if there is adirected path from a node x to a node y can be done in a complexity of O(m),using a search algorithm (BFS or DFS). There are m tests, so the complexityof the algorithm 1 is O(m2).

The initial graph G is a self-transitive closure graph and due to that, usuallyit has many arcs, it is a so-called dense graph. So, algorithm 1 is slow, oftenit runs in a complexity of O(n4), because in a dense graph m is closed to

Inverse problems of transitive closure 73

n2. There is another problem with the algorithm 1. It finds a solution ofthe inverse problem, which is not necessarily optimal. Here is an example toillustrate this case.

Let us apply algorithm 1 to the graph in the fig. 2.

Fig. 2. An example. Fig. 3. Solution found by algorithm 1.

First the algorithm eliminates the arcs (1, 1), (2, 2), (3, 3), (4, 4), (5, 5).Suppose that in the second step the arcs (1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2,4), (4, 1), (4, 2), (4, 3) are eliminated. Finally, it is obtained the graph G−1

from the fig. 3. At the end of the algorithm, the graph G−1 has 7 directedarcs. This is a solution of the inverse problem, but it is not optimal, becausean optimal solution has only 5 directed (fig. 4).

Fig. 4. Optimal solution.

Let us present a faster algorithm and prove that the found solution is op-timal. The idea of this algorithm starts with the remark that the stronglyconnected components of any solution G−1 of the inverse problem are thesame with the strongly connected components of the initial graph G.

Definition 5. K ⊆ V is a strongly connected component of the graph G =(V, E) if for each nodes u, v ∈ K there is a directed path in G from u to v.

Definition 6. The graph Gc = (V c, Ec) is called the condensed graph ofG = (V, E) if it has as nodes all the strongly connected components of G andthe directed arcs of Gc are the connections between the strongly connectedcomponents of G, i.e. V c = K|K is a strongly connected component of Gand Ec = (K1, K2)|K1,K2 ∈ V c and ∃u ∈ K1, v ∈ K2 : (u, v) ∈ E.

Using as few directed arcs from E as possible for each strongly connectedcomponent so that it remains strongly connected and using the directed arcsof the condensed graph, then an optimal solution for the inverse problem ofthe transitive closure is obtained.

74 Adrian Deaconu

Theorem 3. In a self-transitive closure graph G there is a directed pathfrom x to y (x, y ∈ V ), iff there is a directed path from x to y in a solution ofthe inverse problem denoted by G−1.

Proof. ” <= ” If a directed path P is from G−1, as E−1 ⊆ E, it exists inG. ” => ” Let P = (x = u1, u2, ..., ur = y) be a directed path in G. For anydirected arc (ui, ui+1)(i ∈ 1, 2, ?, r − 1) from P, there is a directed path Pi inG−1, from ui to ui+1. We replace every arc (ui, ui+1) with the path Pi in P.It is obtained the path P ′ = (Pu1, Pu2, ?, Pur−1) from x to y.

Theorem 4. K ⊆ V is a strongly connected component of a self-transitiveclosure graph G = (V, E), iff K is a strongly connected component of anysolution G−1 of the ITC problem.

Proof. Directly from Theorem 3.Theorem 5. If K ⊆ V is a strongly connected component of a self-

transitive closure graph G = (V, E), then the graph GK = (K, EK) is complete,where EK = (x, y) ∈ E|x, y ∈ K, i.e. EK = (x, y)|x, y ∈ K.

Proof. Let x and y be two arbitrary nodes from K. Since K is a stronglyconnected component, then there is a directed path from the node x ∈ K tonode y ∈ K and as G = G∗ it follows that (x, y) ∈ EK , whence the theorem.

Theorem 6. Let Gc be the condensed graph of the self-transitive closuregraph G = G*. Any directed arc (Ki,Kj)(i 6= j) of the condensed graph mustbe in the set of arcs E−1 of any solution G−1 = (V, E−1) of the ITC problem,i.e. ∃x ∈ Ki and ∃y ∈ Kj so that (x, y) ∈ E−1.

Proof. Let (G−1)c = V ′, (E−1)c be the condensed graph of G−1. Usingthe Theorem 4, it is obviously that ((G−1)c)∗ = (Gc)∗ (2). Let (Ki,Kj) bean arbitrary chosen directed arc of the condensed graph Gc. Then (Ki, Kj) ∈(Ec)∗ (3).

Suppose that for any x ∈ Ki and for any y ∈ Kj : (x, y) /∈ E−1. It followsthat (Ki,Kj) /∈ (E−1)c, therefore (Ki,Kj) /∈ ((E−1)c)∗ (4). From (2), (3) and(4), a contradiction follows.

Fig. 5. Algorithm 2 (using the condensed graph).

Inverse problems of transitive closure 75

Theorem 7. The algorithm 2 finds an optimal solution G−1 = (V,E−1) ofthe inverse transitive closure problem.

Proof. In the step 3 of the algorithm, in the set E−1 we introduce directedarcs that form directed elementary cycles of each strongly connected compo-nent (which has the minimum number of arcs of all directed cycles). In thestep 4, in the set E−1 we introduce directed arcs that connects the stronglyconnected components of the condensed graph. The algorithm finds an op-timal solution for the inverse problem of transitive closure ((G−1)∗ = G andthere is no other solution with a less number of arcs) - see Theorems 4, 6.

Theorem 8. The complexity of algorithm 2 is O(minm+n, n·p), where mis the number of directed arcs, n is the number of nodes of the self-transitiveclosure graph G = (V, E), i.e. m = |E|, n = |V | and p is the number ofstrongly connected components of G.

Proof. The complexity of the step 1 of the algorithm is O(1). The complex-ity of the step 2 is O(m+n) using the depth first search algorithm (DFS). Thecomplexity of the step 3 is O(n), because s1+s2+ ...+sp = n. The complexityof the step 4 is O(n) (there are at most n-1 arcs found in step 2 that connectthe strongly connected components, because the condensed graph is a forest oftrees). So, the complexity of the whole algorithm is O(m + n). Using the factthat in the self-transitive closure graph G any strongly connected componentis complete (Theorem 5), the condensed graph can be also found in step 2 ina complexity of O(n · p) with the algorithm presented in fig. 6.

Being a self-transitive closure graph, the graph G has many arcs (it is”dense”). In most of the cases m is larger than n · p. So, it is better touse the algorithm from fig. 6 to find the condensed graph.

3. THE NUMBER OF OPTIMAL SOLUTIONSHere we compute the number of optimal solutions for the ITC problem.Theorem 9. The number of optimal solutions of the inverse transitive clo-

sure problem is∏p

i=1(si−1)!·∏1≤i<j≤p Ni,j , where Ni,j = si ·sj, if ∃(Ki, Kj) ∈Ec or ∃(Kj ,Ki) ∈ Ec and Ni,j = 1, otherwise.

Proof. All optimal solutions are generated from the condensed graphGc = (V, Ec). In order to obtain different optimal solutions, the nodes fromevery strongly connected component must be permuted (noncircular). Thenoncircular permutations of k order are obtained by holding one position andpermuting the other k-1. So, there are (k-1)! noncircular permutations of korder (1).

If the directed arcs which connect the strongly connected components linksdifferent nodes from components, different optimal solutions are obtained.Here Ni,j is the number of possible different links between the strongly con-nected components Ki and Kj , if there is an arc (Ki, Kj) or (Kj ,Ki) that

76 Adrian Deaconu

connects the components in the condensed graph (2). So, using (1) and (2),the theorem is proved.

Fig. 6. Algorithm for finding the strongly connected components of a self-transitive closuregraph.

Theorem 10. If the graph G is (weakly) connected (there is a path betweenevery two nodes of G), then the number of optimal solutions of the inverseproblem of transitive closure is

∏pi=1 si! · sp−2

iProof. If the graph G is connected then, between every two components Ki

and Kj there is an arc (Ki,Kj) or (Kj ,Ki) that connects them. This impliesthat

∏

1≤i<j≤p

Ni,j =∏

1≤i<j≤p

si · sj =p∏

i=1

sp−1i .

It follows (by Theorem 9) that the number of optimal solutions of the ITCproblem is

p∏

i=1

(si − 1)! ·∏

1≤i<j≤p

Ni,j =p∏

i=1

(si − 1)! ·p∏

i=1

sp−1i =

p∏

i=1

si! · sp−2i .

FURTHER STUDY OF A MICROCONVECTIONMODEL USING THE DIRECT METHOD

ROMAI J., 2, 1(2006),77–86

Ioana Dragomirescu, Adelina GeorgescuUniv. ”Politehnica” of Timisoara, Univ. of Pitestiioana [email protected], [email protected]

Abstract The analytical study, partially treated in [1] by using the direct method, of thelinear model of natural convection under microgravity conditions for a binaryliquid layer in the presence of the Soret effect is completed. Due to the largenumber of parameters involved, a lot of particular cases are encountered. Theclosed form expressions of the bifurcation points of the characteristic manifoldsare found. The false secular points, occuring in direct formal applications ofnumerical methods, are detected. In the parameter space, they correspond tobifurcation set of the characteristic manifold.

Keywords: thermal convection, microgravity

2000 MSC: 76E06

1. EIGENVALUE PROBLEM.CHARACTERISTIC EQUATION

Many processes performed under microgravity conditions favour a betterunderstanding of the chemical reactions and kinetic and transport processesinvolved. In this respect, the researchers from the Max-Planck Insitute ofExtraterestrial Physics proved that complex plasmas under microgravity con-ditions can self-organize spontaneously to a cristalline-like state (plasma crys-tal). On Earth, such crystals cannot be obtained since the gravity squeezes thecrystals together. The self-organization is modelled by bifurcation emerged asa result of the loss of stability. The new patterns (crystals) correspond to thebifurcating solutions of the equations governing those processes. Combustion,convection, material science, are other domains in which the gravity effectsare studied.

In this paper we are concerned with a class of microconvection models withstrong Soret effect in an infinite horizontal binary liquid layer, bounded bytwo impermeable rigid walls on which the normal heat flux is specified. Thissystem is placed in a uniform gravity field with the acceleration g.

In [1] the linear model of natural convection under microgravity conditionsfor a binary liquid layer in the presence of Soret effect was treated analyticallyusing the direct method based on the characteristic equation. There it wasproved that the characteristic equation associated with the two-point problem

77

78 Ioana Dragomirescu, Adelina Georgescu

for the governing equations has no solution of third algebraic multiplicityorder except for a few particular cases. In these cases the secular equationwas obtained. By simultaneously solving the characteristic and the secularequation, and keeping some parameters constant, the neutral manifolds wereanalytically determined.

When the characteristic equation has multiple roots the straightforward ap-plication of numerical method can lead to false secular points. The false secularmanifolds were analytically detected in [1] for third order multiple character-istic values. Herein we complete the study with the analytical investigation ofthe case of double roots.

The two-point problem for the nondimensional equations linearized aboutthe mechanical equilibrium governing the thermal convection [2] is

(D2 − a2)2Ψ′ + iaG′(T ′ + C ′) = 0,

−ε(iaΨ′ + SDT ′ + LeDC ′) = (D2 − a2)T ′,

εσ(iaΨ′ + SDT ′ + LeDC ′) = Le(D2 − a2)(C ′ − σT ′),

(1)

DΨ′ = ia(ST ′ + LeC ′), Ψ′ = DT ′ = DC ′ = 0, at z = 0 and 1. (2)

where D =d

dz. Here G′ =

G

1 + ε(T0 + C0)2[2] and is assumed to be constant.

0.5 ≤ G′ ≤ 1. The values of the physically admissible parameters are: theLewis number, Le, 0 < Le ≤ 0.5, the Boussinesq parameter, ε, 0.005 ≤ ε ≤0.05 and the separation ratio, σ, 1 ≤ σ ≤ 1.5. Only small wave numbers10−2 ≤ a ≤ 10−1 are considered. These values are chosen so that the problemhas physical meaning.

In [1] we rewrote the two-point problem (1)-(2) in terms of T ′ only in theform

Le(D2−a2)3T ′+εLe(1−σ)D(D2−a2)2T ′+a2εG′[Le+σ(Le−1)]T ′ = 0, (3)

DT ′ = (D2 − a2)T ′ = D3T ′ = 0 at z = 0 and 1. (4)

Denote a1 = ε(1 − σ), a2 = a2εG′[1 + σ(1 − 1/Le)] and take into accountthat the Lewis number is a nonnull physical parameter. Then equation (3)becomes

(D2 − a2)3T ′ + a1D(D2 − a2)2T ′ + a2T′ = 0. (5)

With the notation µ = λ/a, 6b = a1/a, d = (a2 − a6)/a6, the characteristicequation associated with the two-point problem (5)-(4) achieves the simplifiedform [1]

P ≡ µ6 + 6bµ5 − 3µ4 − 12bµ3 + 3µ2 + 6bµ + d = 0, (6)

where the cases b = 0 and/or d = −1 are not considered in the sequel becausethey have been treated in [1].

Further study of a microconvection model using the direct method 79

2. FALSE SECULAR MANIFOLDSThe characteristic equation (6) reads in the equivalent form

(µ2 − 1)3 + 6bµ(µ2 − 1)2 + d + 1 = 0. (7)

If (7) has a double root, then it is a root for its first derivative too, i.e.

(µ2 − 1)[µ(µ2 − 1) + 5b(µ2 − 1) + 4b] = 0. (8)

The case µ = ±1, i.e. d = −1, equivalent to a2 = 0, was treated in [1], so weassume that a2 6= 0. Therefore instead of (8) we study the equation

P1 ≡ µ3 + 5bµ2 − µ− b = 0. (9)

Further we apply a variant of the Euclid algorithm to (7) and (9). Thus,

from (9) we find µ2 − 1 = − 4b

µ + 5b, µ 6= −5b which replaced in (7) yields

P2 ≡ µ3u + µ2(96b3 + 15bu) + µ(480b4 + 75bu) + (125b3u− 64b3) = 0, (10)

The assumption µ 6= −5b holds because if µ = −5b, then (9) implies b = 0and (6) yields d = 0. This case was studied in [1] and it was shown that itcorresponds to false secular points.

The common roots of (7) and (8) are the common roots of (9) and (10). Atfirst, we investigate the possibility that all roots of the equations (9) and (10)are the same, which leads to

1d + 1

=5b

96b3 + 15b(d + 1)=

−1480b4 + 75b2(d + 1)

=−1

61b3 + 125b3d. (11)

The first two equalities from (11) read d+1 = −48b2

5=−480b4

75b2 + 1, i.e. 25b2+

1 = 0. Since b is a real physical parameter the last situation is not possible,so the two equations cannot be equivalent. Thus, we must look for two or onecommon root.

In order to simplify the implied cumbersome computations by reducing thedegree of equations (9) and (10) we perform the linear combination (10)−(d+1)(9) leading to Q1 ≡µ2(96b3+10bd+10b)+µ(480b4+75b2d+75b2+d+1)+(61b3+125db3+db+b) = 0.

(12)and the linear combination µ(12)− (9)(96b3 + 10bd + 10b) implying

Q2 ≡ µ2(25b2+1)(d+1)+µ(11db+11b+157b3+125db3)+(10db2+10b2+96b4) = 0.(13)


Equations (12) and (13) can have two common solutions, only one commonsolution or none. Each of these cases must be treated separately.

Consider first the case of double roots of (12), (13); they correspond to twopairs of equal solutions of (6).Proposition 1. In the five-dimensional (a, ε, σ,G′, Le)− parameter space thetwo hypersurfaces

H1 : aε(1− σ) =

√−3 + 8

√6

375

H2 :εG′[1 + σ(1− 1/Le)]

a4=

16(29 + 6√

6)625

(14)

defined byb∗ = ±

√−3 + 8

√6

375and d∗ =

−161 + 96√

6625

(15)

are false secular manifolds.Proof. For these values of b and d in Appendix it is shown that the

characteristic equation has two double roots µ1 = µ3 and µ2 = µ4 because(12) and (13) have two common roots. The corresponding expressions of µ1,6are given also in Appendix by (21). Since we know the algebraic multiplicity foreach root of the characteristic equation (in fact, we know the roots themselves),we can write the general solution T ′ of the equation. Replacing it into theboundary conditions we obtain the secular equation

λ1 λ2 1 1 λ5 λ6

λ1eλ1 λ2e

λ2 (λ1 + 1)eλ1 (λ2 + 1)eλ2 λ5eλ5 λ6e

λ6

r1 r2 2λ1 2λ2 r5 r6

r1eλ1 r2e

λ2 (2λ1 + r1)eλ1 (2λ2 + r2)eλ2 r5eλ5 r6e

λ6

λ31 λ3

2 3λ21 3λ2

2 λ35 λ3

6

λ31e

λ1 λ32e

λ2 (3λ21 + λ3

1)eλ1 (3λ22 + λ3

2)eλ2 λ3

5eλ5 λ3

6eλ6

= 0 (16)

where λi = µia, ri = λ2i − a2, i = 1, 6. This equation (16) contains a single

variable a and has only the physically nonrealistic root a = 0. On the otherhand, for a = 0 we have the case λ1,6 = 0 for which (16) is not entitled to beconsidered as a secular equation.

Fig. 1. The characteristic equation for b = b∗ < 0 and d = d∗. (a) b = −r−3 + 8

√6

375(b)

d = −−161 + 96√

6

625.


Proposition 2. The characteristic equation has double roots when the realparameters b and d satisfy the relation m = 0, where the expression of m interms of b and d is given by (24).

Proof. In Appendix, the relation m = 0 connecting the parameters b and dis a third degree equation in d, the coefficients of which depend on b. Thereforeits solutions d1, d2 and d3 can be expressed in a simple form by means of thecubic Cardano’s formulas.

For the physical admissible values of the parameters a, Le, σ, ε, G′, thenumerical evaluations show that d < −1. This is way, in the following we areinterested in this case. However, for the sake of completeness, theoretically wetreat also the case d > −1.

Remark that the equation (26) is invariated by a symmetry with respectto the Od-axis, i.e. (µ, b, d) = (−µ,−b, d). This is why, the study can be re-stricted to the half-space µ > 0. In addition, the form (7) of the characteristicequation shows that if |µ| < 1, then d < −1 (fig. 2).

Let Hs be the set of points of the space (b, d, µ) corresponding to d1, d2

and d3. They are situated on three curves di = di(b), i = 1, 2, 3.

Proposition 3. The points on Hs are not secular.

Proof. As can be seen in the Appendix, the discriminant ∆ of the equa-tion m = 0 can be equal to zero, only for b = 0, when d∗1 = 0 , d∗∗2,3 = −1, orfor b = b∗, when d1 = d3 = d∗. For all other values of b and d we have d1 ≥ 0,d3 ≥ −1, d2 ≤ −1. The points (0, 0), (0,−1), (b∗, d∗) are bifurcation pointsfor the set of the roots of m = 0 (fig. 2).

Fig. 2. The curves di(b)

.

For (b, d) = (0,−1) the characteristic equation has three double roots [1],for (b∗, d∗) and (b,−1), (b 6= 0, b∗) it has two double roots while for all otherbifurcation points, i.e. for (b, d) = (b, di(b)), i = 1, 2, 3 (b 6= 0, b∗), only onedouble root corresponds.

Fig. 3 shows that the sheets of the characteristic roots coalesce along thecurves di = di(b) in the (b, d)-plane.

If Le assumes the physically admissible values, i.e. 0 < Le ≤ 0.5, and takinginto account that a, ε, G′ are positive parameters and the expression 1− σ is


Fig. 3. The characteristic surface viewed from four angles.

negative, then it follows that a2 ≡ a2εG′[1 + σ(1 − 1/Le)] ≤ a2εG′(1 − σ) <0, implying d < −1. In this case, only the root d2 of equation m = 0 exists.

For the points (b, d2(b)),with b 6= 0, the general form of the solution of (5)has the form

T ′(z) = (A + Bz)eλ1z +6∑

i=3

Aieλiz,

and, taking into account the boundary conditions, the secular equation reads

λ1 1 λ3 λ4 λ5 λ6

λ1eλ1 (λ1 + 1)eλ1 λ3e

λ3 λ4eλ4 λ5e

λ5 λ6eλ6

r1 2r1 r3 r4 r5 r6

r1eλ1 (r1 + 2λ1)eλ1 r3e

λ3 r4eλ4 r5e

λ5 r6eλ6

λ31 3λ2

1 λ33 λ3

4 λ35 λ3

6

λ31e

λ1 (λ31 + 3λ2

1)eλ1 λ3

3eλ3 λ3

4eλ4 λ3

5eλ5 λ3

6eλ6

= 0. (17)

Solving simultaneously the characteristic and the secular equation, we getthat, for d = d2(b) and for the physically admissible values of the parametersa, Le, σ, ε, our numerical computations indicate that the points on Hs arenot secular.

3. APPENDIXTwo common roots of (12),(13). In this case the ratio of the corre-

sponding coefficients of (12) and (13) must be equal, i.e.

96b3 + 10bu(25b2 + 1)u

=480b4 + (75b2 + 1)u

(11b + 125b3)u + 32b3=

(125b2 + 1)u− 64b2

96b3 + 10bu, (18)

where u = d + 1. Assume, that, the coefficients of (12) and (13)are nonnull.Then system in u, (19), reads

C1 ≡ (−625b4 + 10b2 − 1)u2 + 896b4u + 3072b6 = 0C2 ≡ (−3125b4 − 50b2 − 1)u2 + (3520b4 + 64b2)u + 9216b6 = 0,

(19)


where the coefficients of u2 are nonnull for every b ∈ R. Assume that equations(19) have two common roots. Then

−625b4 + 10b2 − 1−3125b4 − 50b2 − 1

=896b4

3520b4 + 64b2=

3072b6

9216b6=

13,

implying the system in b, 625b4 + 40b2 − 1 = 0, 13b4 + b2 = 0, which has nosolution. Therefore in (19) the two equations cannot be equivalent.

A possible common root of the equations (19) must satisfy the second degreeequation, obtained, for instance, by performing the operations 3(19)1− (19)2),the roots of which are u = 0 and

u =32b2(13b2 + 1)

625b5 + 40b2 − 1. (20)

The root u = 0, i.e. d = −1, is disregarded because we do not study this case.The second root must satisfy one arbitrary equation from (19). Imposing thiscondition, it follows

(1 + 25b2)(625b4 − 10b2 + 1)(375b4 + 6b2 − 1)(125b4 + 22b2 + 1) = 0. (21)

Therefore, we must have 375b4 + 6b2 − 1 = 0 leading to b∗. Then (20) yieldsd∗, whence (15).

Replacing the values (15) of b∗ and d∗ in equation (6) we obtain the followingroots of the characteristic equation

µ1 = µ3 = ±(6 +√

6)√

120√

6− 45150

+√

22

,

µ2 = µ4 = ±(6 +√

6)√

120√

6− 45150

−√

22

,

µ5 = ∓(3 +√

6)√

120√

6− 4575

+2i

25

√45√

6− 95,

µ6 = ∓(3 +√

6)√

120√

6− 4575

− 2i

25

√45√

6− 95,

(21)


the sign − corresponding to b∗ > 0 and + to b∗ < 0. In order to verify therole of the roots µ1, µ2 we apply the Euclid algorithm, i.e.

P (µ + 5b)3 = P1[(µ2 − 1)2(µ + 5b)2 + b(µ2 − 1)(6µ2 + 30bµ− 4)(µ + 5b)−

−4b2(6µ2 + 30bµ− 4)] + P2,

P2 = (d + 1)P1 + Q1,

P1 = ((6b3 + 10bu)−1(µ− 25b2 + 196b3 + 10bu

)Q1 − µC1 − bC2

96b3 + 10bu

which shows that, for b∗2 =−3 + 8

√6

375, we have C1 = C2 = 0, µ1,2 are the

roots of Q1 = 0, of P1, P2 and P .So far, we assumed that none of the coefficients in (12) and (13) are equal

to zero. Now we treat separately the cases in which we have double roots andat least one of these coefficients is null.

a) 96b3 + 10bu = 0. In this case in (19) we must have also (125b2 + 1)u −64b2 = 0 which cannot be satisfied simultaneously with the assumedrelation.

b) 480b4 + u(1 + 75b2) = 0. Similarly, in order for (19) to be satisfied itis necessary to have simultaneously 32b3 + (11b + 125b3)u = 0 implying

b2 =−9 + 2

√39

375, u =

−16(−9 + 2√

39)125

. However, for these expressions,

(19) is not satisfied. As a conclusion, the equations (12), (13) do nothave two common roots in the case b).

c) u =64b2

1 + 125b2. This case, with u > 0 cannot stand because (18) implies

u = −96b2/10 > 0.

We conclude that the equation (7) has two pairs of double roots only for

b2 = b∗2 =8√

6− 3375

and u = u∗ =16(29 + 6

√6)

625.

Only one common root of (12), (13). If this root exists, it must satisfy,in particular, the first degree equation (10b(d+1)+96b3)(13)− (25b2 +1)(d+1)(12), the root of which is µ =

−b[d2(1 + 50b2 + 3125b4) + d(2 + 36b2 + 2730b4) + (1− 14b2 − 395b4 − 9216b6)]d2(625b4 − 10b2 + 1) + d(354b4 − 20b2 + 2) + (−3072b6 − 271b4 − 10b2 + 1)

,

(22)


where we assume that 10b(d + 1) + 96b3 6= 0 and the denominator is differentfrom zero. Imposing to (23) to satisfy (12) and (13) we obtain

−256b5(48b2 + 5d + 5)2 ·m = 0−128b4(d + 1)(25b2 + 1)(48b2 + 5d + 5) ·m = 0,

where

m ≡ d3 + d2(2− 3b2 − 375b4 − 3125b6) + d(1− 6b2 − 366b4 − 3050b6)+

+(−3b2 + 9b4 + 1099b6 + 9216b8). (23)

Owing to the fact that b 6= 0 and d 6= −1, it remains to discuss the situationm = 0. Therefore, consider the relation m = 0 as a third degree equation in d,namely d3 +αd2 +βd+γ = 0. Reduce it to the canonical form y3 +py+q = 0,by putting y = d + α/3, p = β − α2/3, q = 2α3/27 − αβ/3 + γ, whereα = 2 − 3b2 − 375b4 − 3125b6, β = 1 − 6b2 − 366b4 − 3050b6, γ = −3b2 +

9b4 + 1099b6 + 9216b8. Let D = −102427

b6(375b4 + 6b2− 1)2(125b4 + 22b2 + 1)3

be the associated discriminant. As expected, the case D = 0 is equivalent tothe case of two roots of (12), (13), namely 375b4 + 6b2 − 1 = 0, leading to(15). Since D ≤ 0 we must investigate in addition, the case D < 0. In this

case, all roots of the equation m = 0 are real and they read y1 = 2 3√

r cosφ

3,

y2 = 2 3√

r cos(φ

3+ 1200), y3 = 2 3

√r cos(

φ

3+ 2400), where r =

√−p3/27,

cosφ = −q/2r. The corresponding expressions di = di(b) represent threecurves in the parameter space (b, d) (fig. 2).

Remark that for d = −1 and b arbitrary, (6) has two double roots, namelyµ1,2 = 1 and µ3,4 = −1. Consequently, for each point (b, d) on these curvesdifferent from (0, 0), (0,−1), (b∗, d∗), (b,−1) (b 6= 0, b∗) the characteristicequation (6) has one double root only, i.e. λ1 = λ2 = µ ·a, where µ is given by(22). Thus, for any other point (b, d) of the parameter space, the characteristicequation has six mutually distinct roots, situated in the 4-dimensional space(b, d,Reµ, Imµ).

If the denominator of (22) is zero, in order to have a possible root, thenumerator must also be zero. This implies

d2(625b4 − 10b2 + 1) + d(354b4 − 20b2 + 2) + (1− 10b2 − 271b4 − 3072b6) = 0d2(1 + 50b2 + 3125b4) + d(2 + 36b2 + 2730b4) + (1− 14b2−−395b4 − 9216b6) = 0.

(24)From (24)1 we obtain

d =20b2 − 354b4 + 2± 64

√3b6 + 166b8 + 1875b10

2(625b4 − 10b2 + 1)(25)


For d corresponding to the sign plus in front of the radical, the two equations

(25) have a common root for b = ± 175

√−45 + 120

√6, i.e. b = b∗. This case

was already discussed. For the second expression of d in (26), the equation (25)has no real root b. Therefore, the two equations (25) have no common roots.Thus, the vanishing denominator in (23) can lead only to the case (b∗, d∗).

If 10bu + 96b3 = 0, then either b = 0 (which is not our concern), or 10u +96b2 = 0. In the second case, (18) cannot occur, therefore to this case nodouble root of (12) correspond.

4. CONCLUSIONSThe two-point eigenvalue problem governing the microconvection model pre-

sented here depends on five parameters. As a consequence the investigationof the bifurcation of the characteristic manifold is difficult. Partially it wasperformed in [1], while in this paper we deduced analytical expressions for allother bifurcation points of this manifolds. Numerically we found that all ofthem correspond to false secular points.

References[1] Dragomirescu, I., Georgescu, A., Application of the direct method to a microconvection

model, ICTAMI 2005, September, 15-18, Alba-Iulia.

[2] Gaponenko, Yu. A., Zakhvataev, V.E., Microconvection in a binary system, Fluid Dy-namics, 38, 1(2003), 57-68.

[3] Georgescu, A., Characteristic equations for some eigenvalue problems in hydromagneticstability theory, Mathematica, Rev. d’Analyse Num. Theorie de l’Approx., 24(47), 1-2(1982), 31-41.

[4] Georgescu, A., Oprea, I., Pasca, D., Metode de determinare a curbei neutrale ın stabil-itatea Benard, Stud. si Cerc. de Matem. Apl.,52, 4(1993), 267-276.

[5] Georgescu, A., Palese, L., On a method in linear stability problems. Application to anatural convection in a porous medium, Rapp. Int. Dept., Math., Univ. Bari, 9(1996).

[6] Georgescu A., Gavrilescu, M., Palese L., Neutral thermal hydrodynamic and hydromag-netic stability hypersurface for a micropolar fluid layer, Indian J. Pure Appl. Math.,29, 6(1998), 575-582.

[7] Georgescu, A., Palese, L., On a method in linear stability problems. Application tonatural convection in a porous medium, UltraScience, 12,3(2000), 324-336.

[8] Palese, L., Georgescu, A., Pascu, L., Neutral surfaces for Soret-Dufour driven convectiveinstability, Rev. Roum. Sci. Techn., Mec. Appl., 43, 2(1998), 251-260.

[9] Pukhnachov, V.V., Goncharova, O.,N., Mathematical models of microconvection for

isothermally incompressible and weakly compressible liquids, XXI ICTAM, August 15-

21, 2004, Warsaw, Poland.

AN OPTIMIZATION METHOD FOR FREESURFACE STEADY FLOWS

ROMAI J., 2, 1(2006),87–90

Tiberiu Ioana, Titus Petrila”Babes-Bolyai” University, Cluj-Napoca

Abstract The free surface steady flow of an incompressible heavy fluid (inviscid and vis-cous) is studied. The main goal of this work is to develop an efficient approachbased on an optimal shape design formulation that can be used for numericalcalculations.

1. INTRODUCTIONWe consider the two dimensional flow of an incompressible fluid freely mov-

ing along a rigid and impermeable bottom of arbitrary geometry. Apart fromthe influence of gravity, the fluid flow is free of any physical restrictions (thepresence of some obstacles within or above the flow domain has been excluded).

It is assumed that the far field flow has a constant velocity (of unitarymagnitude in the nondimensional form). We also consider, without loosingthe generality, that the uniform stream has a unitary depth.

The flow is referred to a Cartesian coordinates system Oxy with the Ox axisdirected along the unperturbed farfield free line, while Oy is oriented alongthe direction of the outward normal to the unperturbed free surface.

We presume that the flow is steady, the free surface may be written usinga ’height’ function, and the flow domain is bounded.

2. PROBLEM FORMULATIONConsider a two-dimensional domain D of boundary ∂D = S ∪Γ, where S is

an a priori unknown part while Γ is the known part. The boundary Γ containsthe fixed bottom and two artificially boundaries (of inflow and outflow) onwhich appropriate boundary conditions preserving the initial behavior of theunbounded fluid flow are imposed. Assume a steady flow past a rigid andimpermeable wall presenting a smooth bump located within the interval [0, 1].

The Froude number associated with this flow is considered greater thanunity Fr ≥ 1 as well. Within this framework the waves cannot be generatedby the fluid flow [1]. That is why the free surface may be defined by using afunction with zero values at its two edges. Thus, we define the free surfaceby some Lipschitz continuous functions of the type α : [0, 1] → [0, l) , where

87

88 Tiberiu Ioana, Titus Petrila

l ∈ R, 0 < l < 1, α (0) = α (1) = 0 and |α (x1)− α (x2)| ≤ β (x1 − x2) , for anyx1 and x2 belonging to (0, l). In this way, the free boundary becomes S (α) =(x, y) ∈ [0, 1]× [0, l) : y = α (x) . The Lipschitz continuity of α functions isan essential property [2]. Under this condition the flow domain D (α) maybe extended beyond its boundary, that provides a mathematical support forusing an optimization procedure in order to establish the a priori unknown freeboundary and further on all the fluid flow properties. Moreover, the Lipschitzcontinuity condition prevents the occurrence of meaningless physical solutionsproduced by the optimization procedure [6]. Therefore, the family Up

ad of theadmissible functions which may define the boundary S (α), is of the form

Upad =

α ∈ C0,1 [0, 1] : 0 ≤ α (x) ≤ l0 < l, |α (x1)− α (x2)| ≤ β |x1 − x2| ,∨ x1, x2 ∈ [0, 1] , α (0) = α (1) = 0 .

Consequently, in the case of fluid flows with a Froude number above unitywe have defined a set of conditions which have to be fulfilled by the functiondescribing the free boundary such that the flow domain could be extendedbeyond its boundaries (for details of how to prove this assertion, see [2]).

Let n be the number of boundary conditions which has to be imposed on thefixed boundary Γ. Because the free boundary S is a priori unknown, usuallythe number of boundary conditions imposed here should be n + 1.

Denote by a(u, p) = a∗(x, α(x)) a global representation of the first n condi-tions imposed on S while s(u, p) = s∗(x, α(x)) is the (n + 1)th (last one). Letb(u, p) = b∗(x, y) be a compact representation for the boundary conditionsimposed on Γ.

Hence the envisaged problem may be finally formulated in the followingterms: find α(x), u(x, y) and ϕ(x, y) as the solutions of the boundary valueproblem (Re beeing the Reynolds number)

u · ∇u +∇ϕ−Re−1∇ ·(∇u + (∇u)T

)= −Fr−1j,

∇ · u = 0,

s(u, p) = s∗(x, α(x)), a(u, p) = a∗(x, α(x)), b(u, p) = b∗(x, y).

It is this final formulation that will be used in our algorithm.

3. VARIATIONAL FORMULATIONLet us proceed with a representation S0 of the free boundary S, on which we

assume that the conditions a(u, p) = a∗(x, α(x)) are fulfilled. The boundaryS0 would be the right S if the condition s(u, p) = s∗(x, α(x)) is fulfilled aswell.

In order to check whether the condition s(u, p) = s∗(x, α(x)) holds on theboundary S0 and in view of correcting its shape (if necessary), let us introduce

An optimization method for free surface steady flows 89

a positive functional defined on the set of all admissible boundaries. The sur-face we are looking for is that one which minimizes this functional. Denotingthis functional by J , it can be written as

J(S) =∫

S(s(u, p)− s∗(x, α(x)))2 dS.

So, within the optimization framework the problem may be formulated as fol-lows: find the solution S of the problem minS J(S), where u(x, y) si p(x, y) satisfythe boundary value problem

u · ∇u +∇p− Re−1∇ ·(∇u + (∇u)T

)= −Fr−1j,

∇ · u = 0,

a(u, p) = a∗(x, α(x)),b(u, p) = b∗(x, y).

4. THE INVISCID CASELet n and t denote the outward normal unit vector drawn at the free bound-

ary S and the tangent unit vector, respectively.For the case of an inviscid incompressible potential fluid flow (u = ∇φ) the

associated mathematical model reads

∆φ = 0, (x, y) ∈ D, (1)

n · ∇φ = 0, (x, y) ∈ Γ, (2)

n · ∇φ = 0, (x, y) ∈ S, (3)12 −

(12 |∇φ|2 + Fr−2y

)= p1, (x, y) ∈ S,

where Fr stands for the Froude number and p1 is the atmospheric pressure.By introducing an auxiliary function a : Γ → R, the unbounded flow domain

D may be restricted to a bounded domain such that the physical meaning ofthe initial problem be preserved. Keeping the same symbols for the quantities,the governing boundary value problem is (4.1), (4.3) and

p = p1, onS, (4)

n · ∇φ = a, onΓ. (5)

Define the objective functional J(S) =∫S (p− p1)

2 dS. Then the problemis reduced to minS J(S) such that (1), (3), (5) hold.

Using a classical variational technique, the boundary value problem can beassociated with variational problem for the objective functional. So we get anew objective functional, denoted by J∗, of the form

J∗(S) = J(S) +∫

Dψ∆φdD.

90 Tiberiu Ioana, Titus Petrila

Here ψ is an auxiliary variable on the domain D (a Lagrange multiplier). Thenthe optimization problem rereads as: find S the solution of minS J∗(S).

Assume that the free boundary is perturbed along the direction of the nor-mal n with a small quantity ε > 0, such that each point (x, y) of the boundaryS will be mapped onto a point (xε, yε) of Sε, which is related with its originalposition by (x, y) −→ (xε, yε) = (x, y)− εn (this mapping is used, for instance,in [7]).

Within this framework we may use the Taylor series expansion for the po-tential φ

φ ((x, α (x))− εn) = φ (x, α (x))− εn · ∇φ (x, α (x)) + O(ε2

).

Then, on Sε, the velocity potential may written as φε = φ− εn · ∇φ + O(ε2

).

As long as a gradient based approach is employed, in order to solve theabove variational problem, the key element is the determination of the closedform of the gradient of the functional J∗. The main result of this paper,consists in just establishing the gradient expression under the form

gradnJ∗ = −∫

S

(pFr−2n · j + ψnn : ∇∇φ

)dS,

where ψ satisfies the problem

∆ψ = 0, in D,

n · ∇ψ = t · ∇(pt · ∇φ), on S,

n · ∇ψ = 0, on Γ.

References[1] E. H. van Brummelen, A. Segal, Numerical solution of steady free-surface flows by the

adjoint optimal shape design method, International Journal for Numerical Methods inFluids, 41(2003), 3-27.

[2] M. D. Gunzburger, H. Kim, Existence of an optimal solution of a shape control problemfor the stationary Navier-Stokes equations, SIAM Journal on Control and Optimization,36(3)(1998), 895-909.

[3] T. Petrila, On the inverse(design) problem of a free surface gravity flows, Rev. Roum.Math. Pures Appl., 45(2000), 1.

[4] T. Petrila, Mathematical model for the free surface flow under a sluice gate, AppliedMathematics and Computation 2002, 125(2002), 49-58.

[5] T. Petrila, D. Trif, Basics of fluid mechanics and introduction to computational fluiddynamics, Numerical Methods and Algorithms, 3, Springer, Berlin, 2005.

[6] O. Pironneau, Optimal shape design for eliptic systems, Springer, Berlin, 1984.

[7] O. Soto, R. Lohner, On the computation of flow sensitivities from boundary integrals,AIAA-5-8, 2004.

APPROXIMATE INERTIAL MANIFOLDS FORAN ADVECTION-DIFFUSION PROBLEM

ROMAI J., 2, 1(2006),??–108

Anca-Veronica IonUniversity of Pitesti

Abstract In the framework of the infinite-dimensional dynamical systems theory, a familyof approximate inertial manifolds (a.i.m.s) for a problem modelling the Fick-ian diffusion of a substance into a Newtonian fluid is constructed. Estimatesof the distance between these manifolds and the exact solution of the problemare given, proving that, at large times, the solution is kept in some very narrowneighbourhoods of the a.i.m.s.

1. INTRODUCTIONThe concept of approximate inertial manifold (a.i.m.) arose in the frame-

work of the theory of inertial manifolds. First defined in [3], inertial manifoldsare finite-dimensional (at least) Lipschitz invariant manifolds, that attract ex-ponentially all trajectories of an evolution equation.

For many evolution equations the existence of an inertial manifold is notyet proved, since the proof requires the existence of a certain large betweentwo succesive eigenvalues of the linear part (the so-called spectral gap) andthis requirement is not fulfilled [7]. Even if an inertial manifold exists, thisinertial manifold is difficult to construct. In this situation, approximationsof the inertial manifolds were constructed, the a.i.m.s.[3]. An a.i.m. is afinite dimensional (at least) Lipschitz manifold having the property that everytrajectory of the problem enters in a narrow neighbourhood of the a.i.m. ata certain time and remains there for ever. This property inspired some newmethods of approximating the solutions of an evolution equation namely thenonlinear Galerkin and post-processed Galerkin methods. Thus, appart for theproperty of the a.i.m. of keeping the attractor of the problem in some narrowneighbourhood, these methods bring a new and very concrete motivation tostudy the a.i.m.s.

Among several methods of constructing an a.i.m. we choosed the algorithmgiven in [8] for the Navier-Stokes equation. After presenting in Sections 2- 4the problem and the results of existence of the solutions and dissipativity of theproblem, in Sections 5, 6 we splitt the phase space in the direct sum of a finite-dimensional subspace and an infinite-dimensional one and give estimates of thenorm of the projection of the solution on the infinite dimensional subspace.

91

92 Anca-Veronica Ion

We present the algorithm of [8] in Sections 7and 8 and then use it in order toconstruct a family of a.i.m.s for our problem, in Sections 9 and 10. We alsofind the distance between the a.i.m.s and the exact solution.

2. THE PROBLEM, THE FUNCTIONALFRAMEWORK

Consider the generalized problem governing the (Fick-ian) diffusion of asubstance into a Newtonian incompressible fluid. It can be written as thefollowing Cauchy problem for the evolution advection-diffusion equations

∂u∂t

− ν∆u + (u · ∇)u = f (t) , (2.1)

divu = 0, (2.2)∂c

∂t−D∆c + u·∇c = h (t) , (2.3)

u(0) = u0, (2.4)c(0) = c0, (2.5)

where u = u (t,x) is the fluid velocity, x ∈ (0, l)×(0, l) = Ω, u (.,x) : [0,∞) →R2, c (.,x) : [0,∞) → R, ν is the kinematic viscosity, D is the diffusion coeffi-cient and c = c(t,x) is the concentration of the substance that is diffused intothe fluid; f = f (t,x) represent the body forces and h = h (t,x) is a productionfunction for c. These are given functions and they are supposed to dependanalytically on time. We assume that the function h it may describe either aproduction or a consumption of the diffused substance and, as a consequence,it may have either positive or negative values.

We consider periodic boundary conditions. Hence it is assumed that, forevery fixed t, u (t, ·) belongs to the space H1

def=

v ∈ [

L2per (Ω)

]2, divv = 0

,

with the scalar product (u,v) =∫Ω (u1v1 + u2v2) dx, (where u = (u1, u2) , v =

(v1, v2)) and | | the induced norm. For c we assume c (t, ·) ∈ H2def=

L2per (Ω) , where L2

per (Ω) is endowed with the standard scalar product and theinduced norm denoted also by | |. From the context it will be understoodwhether we talk about the norm in H1 or that in H2.

We assume that for every t, f (t) ∈ H1, and h (t) ∈ H2. As far as thedependence on t is concerned the most realistic hypothesis is that of periodicityin time for the functions f(.) and h(.). Anyhow, we suppose that these functionsare bounded: there is a number Mf > 0 such that |f(t)| ≤ Mf , and there isa number Mh > 0 such that |h(t)| ≤ Mh for every t > 0.

We also use the spaces V1def=

u ∈ [

H1per (Ω)

]2 : div u = 0

, with the

scalar product ((u,v)) =∑2

i,j=1

(∂ui∂xj

, ∂vi∂xj

), and V2

def= H1

per (Ω) with the

Approximate inertial manifolds for an advection-diffusion problem 93

scalar product ((c1, c2)) =∑2

j=1

(∂c1∂xj

, ∂c2∂xj

). The induced norms are denoted

by ‖ ‖ .

The operator Adef=−∆ is defined on D(A) =

u ∈ [

H2per (Ω)

]2 : div u = 0

and it is self-adjoint.For the bilinear form B(u,v) = (u · ∇)v the following inequalities [4], [6],

[8]

|B (u,v)| ≤ c1 |u|12 |∆u| 12 ‖v‖ , (∀) u ∈D(A), v ∈V1, (2.6)

|B (u,v)| ≤ c2 ‖u‖ ‖v‖[1 + ln

(|∆u|2‖u‖2

)] 12

, (∀) u ∈D(A), v ∈V1,(2.7)

|B (u,v)| ≤ c3 ‖u‖ ‖v‖[1 + ln

(|∆v|2‖v‖2

)] 12

, (∀) u ∈V1, v ∈D(A).(2.8)

hold [4], [6], [8]. We remind the following properties of the three-linear formb(u,v,w) = (B(u,v),w) (valid for periodic boundary conditions [5]):

b(u,v,w) = −b(u,w,v),b(u,v,v) = 0,

as well as the following inequalities [6], [5]

|b(u,v,w)| ≤ k |u| 12 ‖u‖ 12 ‖v‖ |w| 12 ‖w‖ 1

2 , (∀)u,v,w ∈V1, (2.9)

|b(u,v,w)| ≤ k |u| 12 ‖u‖ 12 ‖v‖ 1

2 |∆v| 12 |w| , (∀)u ∈V1,v ∈ D (A) ,w ∈H1.(2.10)

The operator Adef= −∆ is defined on D(A) = H2

per (Ω) and it is self-adjoint.We define the bilinear form B(u,c) = u∇c. The inequalities (2.6)-(2.8) have

the immediate consequences

|B (u,c)| ≤ c1 |u|12 |∆u| 12 ‖c‖ , (∀) u ∈D(A), c∈V2, (2.11)

|B (u,c)| ≤ c2 ‖u‖ ‖c‖[1 + ln

(|∆u|2‖u‖2

)] 12

, (∀) u ∈D(A), c∈V2,(2.12)

|B (u,c)| ≤ c3 ‖u‖ ‖c‖[1 + ln

(|∆c|2‖c‖2

)] 12

, (∀) u ∈V1, c∈D(A).(2.13)

For the periodic boundary conditions it is usual to consider the averagesof the unknown functions on the periodicity cell, namely ([6] for u (t,x))

um (t) =1l2

∫

Ωu (t,x) dx, cm (t) =

1l2

∫

Ωc (t,x) dx.


These are solutions of the equations

d

dtum (t) = fm (t) , (2.14)

dcm

dt(t) = hm (t) , (2.15)

where fm, hm are the averages of f respectively h over Ω, therefore we canassume that um and cm are given.

The functions u (t,x) = u (t,x) − um (t) , c (t,x) = c (t,x) − cm (t) , aresolutions of the equations

∂u∂t

− ν∆u + (u·∇) u + (um · ∇) u = f − fm, (2.16)

∂c

∂t−D∆c + u·∇c + um·∇c = h− hm. (2.17)

Since um (t) is known, the study of the above equations is very similar tothat of (2.1), (2.3). It is then usual [6] to study equations (2.1), (2.3) with theconditions um (t) = 0, cm (t) = 0. In the following qualitative investigationwe adopt these conditions, but we remind that, when a quanitative study isin view, the equations (2.16), (2.17) should be studied, together with (2.14),(2.15).

Thus, further, in the study of (2.1), (2.3), we use the spaces·

H1 = u ∈H1, um = 0 ,·V1 = u ∈V1, um = 0 ,

·H2 = c∈H2, cm = 0 ,

·V2 = c∈V2, cm = 0.

The operator A is positive-definite on D(A)∩·

H1 and has a compact inverse

[4]. Similarly, the operator A is positive-definite on D(A) ∩·

H2 and has acompact inverse.

3. EXISTENCE OF THE SOLUTIONSThe flow of the incompressible viscous fluid in which the diffusion takes

place is not affected by the substance that is diffused, whence the decouplingof the equations of the model (2.1)-(2.5).

Hence, for the problem (2.1), (2.4) we have the classical existence anduniqueness results for the equations Navier-Stokes in R2, with periodic bound-ary conditions.

Theorem 1 [6]. a) If u0 ∈·

H1, f is analytical in time and for every

t, f (t, ·) ∈·

H1, then the problem (2.1), (2.4) has an unique solution u,

analitycal in time and such that for every t, u (t, ·) ∈·V1. b) If, in addition to

the hypotheses in a), u0 ∈·V1, f (t, ·) ∈

·V1, then u (t, ·) ∈ D(A).


By using the Galerkin-Faedo method we can easily prove the following the-orem.

Theorem 2. a) In the conditions a) of Theorem 1 and if h is analytical

in time and for every t, h(t, ·) ∈·

H2, c0 ∈·

H2, then there is an unique solution

c of the problem (2.3)-(2.5), analytical in time and such that c(t, ·) ∈·

H2.

b) In the conditions b) of Theorem 1 and if c0 ∈·V2, h(t, ·) ∈

·V2 then

c (t, ·) ∈ D(A), ∀t > 0.Since, by Theorems 1, 2 the existence and uniqueness of the solution (u, c)

is global with respect to time, it follows that the problem generates two semi-

dynamical systems: S1 (t)t≥0 , S1 (t) :·

H1 →·

H1, S1 (t)u0 = u (t, .,u0) ,where u (t,x,u0) is the solution of (2.1), (2.2, (2.4), and S (t)t≥0 , S (t) :·

H1×·

H2 →·

H1×·

H2, S (t) (u0, c0) = (u (t, .,u0) , c (t, .,u0, c0)) , where c (t,x,u0, c0)is the solution of (2.3), (2.5), with u solution of (2.1), (2.2), (2.4).

4. DISSIPATIVITY OF THE PROBLEMIt was proved [1], [10], that the semi-dynamical system generated by (2.1),

(2.2), (2.4) is dissipative, in the sense that in·

H1there is an absorbing ball forit. More precisely, there is a ρ0 > 0 such that for every R > 0, there is a

t0(R) > 0 with the property that for every u0 ∈·

H1with |u0| ≤ R, we have

|S1 (t)u0| ≤ ρ0 for t > t0(R) . In addition, there is an absorbing ball in·V1 for

S1 (t), i.e. there is a ρ1 > 0 such that for every R > 0, |u0| ≤ R implies‖S1 (t)u0‖ ≤ ρ1 for t > t1(R) .

The component c of the solution (u, c) is also dissipative both in H2 andV2. More precisely, the following result holds.

Theorem 3. a) There is a η0 > 0 with the property that for every Rc > 0there is a tc0 (Rc) > 0 such that for |c0| ≤ Rc

|c (t, .,u0, c0)| ≤ η0, t ≥ tc0 (Rc) .

b) There is a η1 > 0 with the property that for every R, Rc > 0 there is atc1 (R, Rc) > 0 such that

‖c (t, .,u0, c0)‖ ≤ η1

for |u0| ≤ R, |c0| ≤ Rc, t ≥ tc1 (R, Rc) .The proof of this theorem is contained in the proof of Theorem 2 and we

do not insist on it.


5. SPLITTING OF SPACES, PROJECTION OFEQUATIONS

The eigenvalues of A = −∆ in·

H1 are

λj1,j2 =4π2

l2(j21 + j2

2

), (j1, j2) ∈ Z2\ (0, 0) . (5.18)

For·

H1there is a total system f whose elements are eigenfunctions of A = −∆.

They have the form [8]√

2l

ej|j| sin

(2π jx

l

),

√2

l

ej|j| cos

(2π jx

l

), where j = (j1, j2) , j =

(j2,−j1) , |j| = (j21 + j2

2

) 12 , jx

l := j1x1

l + j2x2

l , (j1, j2) ∈ Z2\ (0, 0) .The eigenvalues of A = −∆ are also (5.18) and there is a total system

for·

H2 formed of eigenfunctions of A = −∆, namely those having the form√2

l sin(2π jx

l

),√

2l cos

(2π jx

l

).

Due to the simetries of the functions sin and cos to every ordered pair ofnatural numbers (j1, j2) , for both operators A and A, four eigenfunctions asabove (with ”+” or ”-” between the two terms j1x1

l and j2x2

l ) correspond.We fix a value m ∈ N and consider the set Γm of eigenvalues λj1,j2 having

|j1| , |j2| ≤ m.Denote λ := λ1 = 4π2

l2, Λ := 4π2

l2(m + 1)2 , δ := λ

Λ = 1(m+1)2

.

Let w1, ...,wm1 ( resp. w1, ..., wm1) be the eigenfunctions of A (resp. A )corresponding to λj1,j2 ∈ Γm.

Let L (u,v, ...) stand for the linear space spanned by u,v, ... .Denote by P the projection operator on L (w1, ...,wm1) , with Q the projec-

tion operator on L (wm1+1, ...) , by P the projection operator on L (w1, ..., wm1) ,and by Q the projection operator on L (wm1+1, ...) .

For the solution u of the Navier-Stokes equation, p = Pu, q = Qu, and,similarly, for the diffusion component c, cp = Pc, cq = Qc.

By projecting the equations (2.1), (2.3) on the above introduced spaces, wehave,

dpdt− ν∆p + PB(p + q) = Pf , (5.19)

dqdt− ν∆q + QB(p + q) = Qf , (5.20)

dcp

dt−D∆cp + PB (u, c) = Ph, (5.21)

dcq

dt−D∆cq + QB (u, c) = Qh. (5.22)


6. ESTIMATES FOR THE SMALL COMPONENT OFTHE SOLUTION

It is already known that for the Navier Sokes equations the componentq (t) of the solution is small in the sense that it has the order of δ for t largeenough. In [8] it is proved that for every initial state u0 with |u0| ≤ R thereis a moment t2 (R) such that, for t ≥ t2 (R) ,

|q (t)| ≤ K0L12 δ, ‖q (t)‖ ≤ K1L

12 δ

12 ,∣∣q′ (t)∣∣ ≤ K2L

12 δ, |∆q (t)| ≤ K3L

12 ,

where, for the chosen projection spaces, we have L = ln(1 + 2m2). Here andin the sequel the prime means time derivative.

Let us prove a similar result for cq. We assume in the following that for afixed R, we have |u0| ≤ R.

Theorem 4. There is a moment tc2 (R) such that for every t ≥ tc2 (R) ,the following inequalities

|cq (t)| ≤ J0L12 δ, ‖cq (t)‖ ≤ J1L

12 δ1/2, (6.23)

∣∣c′q (t)∣∣ ≤ J2L

12 δ, |∆cq (t)| ≤ J3L

12 , (6.24)

hold, where J0, J1, J2, J3 are independent of m.Proof. In the sequel we frequently use the embedding inequalities

‖cq‖ ≥ Λ12 |cq| , |∆cq| ≥ Λ |cq| ,

easy to derive by considering the Fourier series of cq and the fact that Λ is theleast eigenvalue of A on L (wm1+1, ...) .

Taking the scalar product of (5.22) by cq we have(

dcq

dt, cq

)−D (∆cq, cq) + (u∇c, cq) = (Qh, cq) ,

such that, by using (2.12) and (2.13), we obtain

12

d

dt|cq|2 + DΛ |cq|2 ≤ |(Q (p∇cp) , cq)|+ |(Q (q∇cp) , cq)|+ |(Qh, cq)|

≤ CL1/2 ‖p‖ ‖cp‖ |cq|+ CL1/2 ‖q‖ ‖cp‖ |cq|+ |Qh| |cq|

≤ C2L

DΛρ21η

21 +

DΛ4|cq|2 +

C2L

DΛδη2

1 +DΛ4|cq|2 +

+M2

h

DΛ+

DΛ4|cq|2

and (since 1 < L for m > 1) it follows


d

dt|cq|2 + DΛ |cq|2 ≤ δ

(C2L

Dλρ21η

21 +

C2L

Dλδη2

1 +M2

h

Dλ

)

= K0Lδ,

whence, by applying Gronwall Lemma, (6.23)1, where J0 =√

K0Dλ .

Taking the scalar product of (5.22) by −∆cq, we obtain

12

d

dt‖cq‖2 + D |∆cq|2 ≤ |(u∇c,∆cq)|+ |(Qh,∆cq)|

≤ |(p∇cp,∆cq)|+ |(p∇cq,∆cq)|++ |(q∇cp, ∆cq)|+ |(q∇cq, ∆cq)|+ |(Qh,∆cq)|

≤ CL12 (‖p‖ ‖cp‖+ ‖p‖ ‖cq‖+ ‖q‖ ‖cp‖) |∆cq|+

+C |q| 12 |Aq| 12 ‖cq‖ |∆cq|+ |Qh| |∆cq| .The use of the inequality ‖cq‖ ≤ ‖c‖ ≤ η1 further leads to

d

dt‖cq‖2 + DΛ ‖cq‖2 ≤ CL

D

(ρ21 + δ

)η21 +

1D

M2h ,

and

d

dt‖cq (t)‖2 ≤ ‖cq (0)‖2 e−DΛt +

1Λ

(CL

D2

(ρ21 + δ

)η21 +

1D2

M2h

),

where C is a generic notation for a constant not depending on m.Since, usually, m is large, L is large too and we obtain (6.23)2, for t great

enough.Then, like for the Navier-Stokes equation [6], we can prove that the exten-

sion of the semigroup to the complex time variable is analytic in time in aneighbourhood of the time axis, and by using the Cauchy formula, we obtain(6.24)1.

Finally, by using (5.22) and the above estimates, the inequality (6.24)2follows.¤

7. THE FAMILY OF A.I.M.S FOR THE NAVIERSTOKES EQUATIONS

The family of a.i.m.s for the Navier Stokes problem, is defined in [8] as

follows. The first one is the graph M0 of the function Φ0 : P·

H1→Q·

H1, thatsatisfies the relation

−ν∆Φ0 (X) + QB(X) = Qf , where X ∈ P·

H1.


We remark that this relation is similar to the equation −ν∆q+QB(p) = Qf ,obtained from (5.20) by neglecting dq

dt in the presence of ∆q and q in thepresence of p. Therefore, Φ0 (X) has the following form

Φ0 (X) = (−ν∆)−1 (Qf −QB(X)) .

The next a.i.m. defined in [8] is M1, the graph of the function

Φ1 : P·

H1→Q·

H1, given by the solution of the problem

−ν∆Φ1 (X) + QB(X) + QB(X,Φ0 (X)) + QB(Φ0 (X) ,X) = Qf ,

that isΦ1(X) = − (ν∆)−1 [Qf −QB(X)−QB(X,Φ0 (X))−QB(Φ0 (X) ,X)] .

For j ≥ 2, the a.i.m. Mj is defined as the graph of Φj : P·

H1→Q·

H1, withΦj (X) the solution of

−ν∆Φj (X) + QB(X) + QB(X,Φj−1 (X)) + QB(Φj−1 (X) ,X)++QB(Φj−2 (X)) + DΦj−2 (X) Γu,j−2 (X) = Qf ,

where DΦj−2 (X) is the differential of Φj−2 (X), and

Γu,j−2 (X) = ν∆X−PB (X + Φj−2 (X)) + Pf .Hence

Φj (X) = − (ν∆)−1 [Qf −QB (X)−QB (X,Φj−1 (X))−QB (Φj−1 (X) ,X)−QB(Φj−2 (X))−DΦj−2 (X) Γu,j−2 (X)]

where DΦj−2 (X) is the differential of Φj−2 (X), and

Γu,j−2 (X) = ν∆X−PB (X + Φj−2 (X)) + Pf .

This construction is better understood with the definition of the so-called”induced trajectories”.

8. INDUCED TRAJECTORIES FOR THENAVIER-STOKES PROBLEM

In [8] a family of functions, kj,m; j ∈ N , is defined by the equalities

−ν∆k0,m + QB(p) = Qf , (8.25)−ν∆k1,m + QB(p,k0,m) + QB(k0,m,p) = 0,


−ν∆kj,m + QB (p,kj−1,m) + QB (kj−1,m,p)+

+j−3∑s=0

QB(kj−2,m,ks,m) +j−3∑r=0

QB(kr,m,kj−2,m)

+QB(kj−2,m,kj−2,m) + k′j−2,m = 0, j ≥ 2,

where p (t) is, as before, the projection through P of the solution u(t).Then the functions qj are defined [8] through the relations qj = k0,m+...+kj,m,and therefore, satisfy the equations

−ν∆q0 + QB (p) = Qf ,−ν∆q1 + QB (p) + QB (p,q0) + QB (q0,p) = Qf ,

−ν∆qj + q′j−2 + QB (p) + QB (p,qj−1)

+QB (qj−1,p) + QB (qj−2,qj−2) = Qf , for j ≥ 2.

The connection between the definitions of the a.i.m.s and those of the func-tions qj is obvious. Moreover, it is clear that the for every t ≥ 0, (p (t) ,qj (t))lies on Mj for j ≥ 0.

The functions of time uj(t) = p (t) + qj (t) are called induced trajectoriesassociated with the trajectory u(t) = p (t) + q (t) .

The following inequalities

|kjm| ≤ κjδ1+j/2L(j+1)/2, ‖kjm‖ ≤ κjδ

1/2+j/2L(j+1)/2, (8.26)∣∣k′jm∣∣ ≤ κjδ

1+j/2L(j+1)/2, (8.27)

|qj | ≤ κjδL1/2, ‖qj‖ ≤ κjδ

1/2L1/2,∣∣q′j

∣∣ ≤ κjδL1/2, (8.28)

are proved in [8], where κj are independent of m, but depend on j, ν, |f | , λ.

They will help us to estimate the distance between the trajectories of theNavier-Stokes problems and the a.i.m.s.

Indeed, since uj(t) ∈ Mj ,

distH1 (u(t), Mj) ≤ dist (u(t),uj(t)) = |q(t)− qj(t)| .

By estimating this last norm and the similar one for V1, in [8] and [9] it isproved that

distH1 (u (t) , Mj) ≤ κjL1+j2 δ

3+j2 ,

distV1 (u (t) , Mj) ≤ κ′jLj2 δ

2+j2 .


9. ”INDUCED TRAJECTORIES” FOR THEDIFFUSED SUBSTANCE

Following the above reasoning, we define the functions hj,m : R+ → Q·

H2, j ∈N, as solutions of the equations

−D∆h0,m + QB (p, cp) = Qh, (9.29)−D∆h1,m + QB (p, h0,m) + QB (k0,m, cp) = 0, (9.30)

h′j−2,m −D∆hj,m + QB (p, hj−1,m) + QB (kj−1,m, cp) + (9.31)

+j−3∑r=0

QB(kr,m, hj−2, m)+j−3∑s=0

QB(kj−2,m, hs, m)+QB (kj−2,m, hj−2, m) =

0,

for j ≥ 2.In these definitions p = p(t) = Pu(t) and cp = cp(t) = Pc(t), where

(u (t) , c (t)) is the solution of the diffusion problem, and kj,m are given by(8.25).

We prove the followingTheorem 5. There are some constants κj ,ςj independent of m, but de-

pending on j and on ν, |f | , λ1, D, and there is a tc2(R)(≥ tc1(R)) such thatfor m, j, and every t ≥ tc2(R), the estimates

|hjm| ≤ κjLj+12 δ

j2+1, ‖hjm‖ ≤ κjL

j+12 δ

j+12 , (9.32)

∣∣h′jm∣∣ ≤ κjL

j+12 δ

j2+1, |Ahjm| ≤ κjL

j+12 δ

j2 (9.33)

hold, (∀) j ≥ 0.Proof. We look for an estimate for the norm of h0m. From (9.29), by using

(2.12), we succesively obtain

|D∆h0m| = |Q (p∇cp)|+ |Qh| ≤ |p∇cp|+ |Qh|≤ cL

12 ‖p‖ ‖cp‖+ Mh,

whence

|h0m| ≤(cρ1η1

D+ Mh

) L12

Λ=

cρ1η1 + MhD

DλL

12 δ, ‖h0m‖ ≤ cρ1η1 + MhD

Dλ12

δ12 .

With κ0 = cρ1η1+MhDDλ , κ′0 = cρ1η1+MhD

Dλ12

, the relations (9.32) are proved forj = 0 .


Since u and c are analytic in time, so are p and cp. Moreover, these lastfunctions are the restrictions to the real axis of functions of a complex variablethat are analytic in a neighbourhood of the real axis. It follows that h0m hasthe same property. Then, by using the same reasoning as for the Navier-Stokeseuations [6], we prove the first relation of (9.33), for j = 0.

From the defintion of h1,m, with (2.12), (2.13), and by taking into accountthe estimates for h0,m, we succesively deduce

|D∆h1,m| ≤ c ‖p‖ ‖h0,m‖L12 + c ‖k0,m‖ ‖cp‖L

12

≤ cρ1κ′0Lδ12 + cκ′0δ

1/2Lη1 ≤(cρ1κ′0 + cκ′0η1

)Lδ1/2,

|h1,m| ≤ (cρ1κ′0 + cκ′0η1)Dλ

Lδ3/2,

‖h1,m‖ ≤ (cρ1κ′0 + cκ′0η1)

Dλ12

Lδ.

We find as for h0,m, that h1,m is analytic in time and, then, h′1,m is majorizedby an expression of the order of Lδ

32 .

For k ≥ 2 we proceed by induction. We assume that the inequalities (9.32),(9.33) are true and that hj,m is the restriction to the real axis of an analyticfunction of t for every j ≤ k − 1. From the definition of hk,m, we have

|D∆hk,m| ≤ ∣∣h′k−2,m

∣∣ + |Q (p∇hk−1,m)|+ |Q (kk−1,m∇cp)|+k−3∑

r=0

|Q(kr,m∇hk−2, m)|+

+k−3∑

s=0

|Q(kk−2,m∇hs, m)|+ |Q (kk−2,m∇hk−2, m)|

≤ CLk−12 δ

k2 + CL

12 ‖p‖ ‖hk−1,m‖+ CL

12 ‖kk−1,m‖ ‖cp‖+

+CL12

k−3∑

r=0

‖kr,m‖ ‖hk−2, m‖+ CL12

k−3∑

s=0

‖kk−2,m‖ ‖hs, m‖+

+CL12 ‖kk−2,m‖ ‖hk−2, m‖ ,

whence

|D∆hk,m| ≤ CLk−12 δ

k2 + CL

12 ρ1κ′k−1L

k21 δk/2 + CL

12 η1κ

′k−1δ

k2 L

k+12 +

+κ′k−2Lk−12 δ

k−12

k−3∑

r=0

κrδ1+r2 L

r+12 +

+κk−2Lk−12 δ

k−12

k−2∑

s=0

κ′sLs+12

1 δs+12 ≤ C ′′

kLk+12

1 δk2 .


Consequently,

|hk,m| ≤C ′′

k

DΛL

k+12

1 δk2 =

C ′′k

DλL

k+12

1 δk2+1 = κkL

k+12

1 δk2+1,

where κk = C′′kDλ , hence, the conclusion. The other inequalities are proved as

for j = 0 and j = 1..¤We define now cqj : R+ → Q

·H2, j ∈ N by

cq,j = h0,m + h1,m + ... + hj,m.

As a consequence of their definition, these new functions satisfy the relations

−D∆cq,0 + QB (p, cp) = Qh, (9.34)−D∆cq,1 + QB (p, cp) + QB (p, cq,0) + QB (q0, cp) = Qh, (9.35)

c′q,j−2 −D∆cq,j + QB (p, cp)+ (9.36)

+QB (p, cq,j−1) + QB (qj−1, cp) + QB(qj−2, cq,j−2) = Qh,

for j ≥ 2. By summing the relations proved for hk,m we obtain

|cq,j | ≤ ςjL12 δ, ‖cq,j‖ ≤ ς ′jL

12 δ

12 ,

∣∣c′q,j

∣∣ ≤ ς ′′j L12 δ, j ≥ 0.

Now, along the lines of [8], we define the induced trajectories for c

cj (t) = cp (t) + cq,j (t)

and find the distance between these functions and c (t). We define ξj (t) =cj (t)− c (t) = cq,j (t)− cq (t) .

Theorem 6. There are some constants, κj , independent of m but depend-ing on j and ν, |f | , λ1, D and there is a moment t3 such that for every m,j and for t ≥ t3 the inequalities

|ξj | ≤ κjLj+12 δ

j+32 , (9.37)

‖ξj‖ ≤ κ′jLj+12 δ

j2+1, (9.38)

∣∣ξ′j∣∣ ≤ κ′jL

j+12 δ

j+32 , (9.39)


hold for ∀j ≥ 0.Proof. We start with the definition of cq,0 and from the equation for cq,

written in the form

∆cq,0 = − 1D

[Qh−Q (p∇cp)] ,

∆cq = − 1D

[Qh−Q (u∇c)− dcq

dt

].

By substracting the two relations, we obtain

∆ (cq,0 − cq) =1D

[Q (p∇cp)−Q (u∇c)− dcq

dt

](9.40)

= − 1D

[Q (p∇cq) + Q (q∇cp) + Q (q∇cq) +

dcq

dt

],

whence, by using the same inequalities as above, we have

|ξ0| = |cq,0 − cq| ≤ 1DΛ

∣∣∣∣Q (p∇cq) + Q (q∇cp) + Q (q∇cq) +dcq

dt

∣∣∣∣

≤ CL12

DΛδ

12 =

CL12

Dλδ

32 = κ0L

12 δ

32 ,

i.e. the inequality (9.37) for j = 0. Similarly, we obtain (9.38) for j = 0.As we can see from (9.40), ξ0 is analytical in t and can be extended to a

function of a complex variable, analytical in a neighbourhoord of the t axis.Then, by using the Cauchy formula, it follows that |ξ′0| is of the same order as|ξ0| .

Now, for |ξ1| we have

∆ (cq,1 − cq) =1D

[Q (p∇cp) + Q (p∇cq,0) + Q (q0∇cp)−Q (u∇c)− dcq

dt

]

=1D

[Q (p∇ (cq,0 − cq)) + Q ((q0 − q)∇cp)−Q (q∇cq)− dcq

dt

],

and, by taking the norm and using the previous results, we get (9.37) and(9.38) for j = 1.

The inequality for |ξ′1| is obtained as that for |ξ′0| .From this point on, we proceed by induction, assuming that (9.37), (9.38)

and (9.39) hold for every j ≤ k − 1.


We have

∆ (cq,k − cq) =1D

[Q (p∇ (cq,k−1 − cq)) + Q ((qk−1 − q)∇cp)+

+Q((qk−2 − q)∇cq,k−2) + Q(q∇ (cq,k−2 − cq)) + c′q,k−2 − c′q],

hence

|ξk| = |cq,k − cq| ≤ 1D

δ[CL

12 ρ1L

k2 δ

k+12 + CL

12 η1L

k2 δ

k+12

+ CLk−12 δ

k+14 δ

k−14 δ

12 + CL

12 δ

12 L

k−12 δ

k2 + CL

k−12 δ

k+12

]

≤ CLk+12 δ

k+32 ,

‖ξk‖ ≤ CLk+12 δ

k+22 .

By using the Cauchy formula for the extension of ξk (t) to a complex variable(analytical in a neighbourhood of the time axis), we obtain for |ξ′k| the sametype of estimate as for |ξk| .¤

10. APPROXIMATE INERTIAL MANIFOLDSFOR OUR PROBLEM

Here we construct the a.i.m.s for the diffusion problem (2.1)-(2.5).

The first a.i.m. is the manifold M0 defined as the graph of the Ψ0 : P·

H1 ×P

·H2→Q

·H1 ×Q

·H2,

Ψ0 (X, X) =(− (ν∆)−1 (Qf −QB(X)) , − (D∆)−1 [Qh−QB (X, X)]

).

Denote the components of Ψ0 (X, X) by (Ψu0 (X) ,Ψc0 (X, X)) (obviouslyΨu0 (X) = Φ0(X)).

The second a.i.m., denoted by M1, is the graph of Ψ1 : P·

H1×P·

H2→Q·

H1×Q

·H2, Ψ1 (X, X) = (Ψu,1 (X) , Ψc,1 (X, X)), where Ψu,1 (X) = Φ1(X) and

Ψc,1 (X, X) =−1D

∆−1 [Qh−QB (X, X)−QB (X, Ψc0 (X,X))−QB (Ψu,0 (X) , X)] .

For j ≥ 2 we define the function Ψj : P·

H1×P·

H2→Q·

H1×Q·

H2, Ψj (X, X) =(Ψu,j (X) , Ψc,j (X, X)), where Ψu,j (X) = Φj(X) and


Ψc,j (X,X) = − (D∆)−1 [Qh−QB (X, X)−QB (X,Ψc,j−1 (X,X))−−QB (Ψu,j−1 (X) , X)−QB(Ψu,j−2 (X) ,Ψc,j−2 (X,X))−−DΨc,j−2 (X,X) (Γu,j−2 (X) , Γc,j−2 (X,X))]) ,

where DΨc,j−2 (X,X) is the differential of Ψc,j−2 (X,X) and

Γc,j−2 (X,X) = D∆X − PB (X + Ψu,j−2 (X) , X + Ψc,j−2 (X,X)) + Ph.

The graph of this function is the a.i.m. Mj of the diffusion problem.Theorem 7. For t large enough, the distance between the solution of the

diffusion equations and the a.i.m. Mj is bounded by

distH1xH2

((u (t) , c (t)), Mj

)≤ K1L

1+j2 δ

3+j2 . (10.41)

Proof. For j = 0 and j = 1, by definition, Ψu,j (p) = qj (p) , Ψc,j (p,p) =cq,j (p,p) ; the inequality (??) follows directly from the result (??) of [8] andfrom (2.1), j = 0, resp. j = 1 of our Theorem 6. For j ≥ 2, we need a specialproof for (??). We take as initial step in our inductive reasoning the casej = 2. We have

Ψc,2 (p,cp)− cq = Ψc,2 (p,cp)− cq,2 + cq,2 − cq == Ψc,2 (p,cp)− cq,2 + ξ2,

such that the difference between Ψc,2 (p,cp) and cq,2 satisfies the identities

D∆[Ψc,2 (p,cp)− cq,2] = QB (p,Ψc,1 (p,cp)− cq,1) ++QB (Ψu,1 (p)− q1, cp) ++QB(Ψu,0 (p) ,Ψc,0 (p,cp))−QB(q0, cq,0) ++DΨc,0 (p,cp) (Γu,0 (p) , Γc,0 (p,cp))− c′q,0

= DΨc,0 (p,cp) (Γu,0 (p) ,Γc,0 (p,cp))− c′q,0,

whence

|Ψc,2 (p,cp)− cq,2| ≤∣∣∣(D∆)−2 [Q (Γu,0 (p)∇cp) + Q (p∇Γc,0 (p,cp))

−Q(p′∇cp

)−Q(p∇c′p

)]∣∣

≤ D

λ2δ2

∣∣QB(Γu,0 (p)− p′, cp

)+ QB

(p,Γc,0 (p,cp)− c′p

)∣∣

≤ D

λ2δ2CL

12(∥∥Γu,0 (p)− p′

∥∥ ‖cp‖+ ‖p‖∥∥Γc,0 (p,cp)− c′p∥∥)

.


Now,∥∥Γu,0 (p)− p′

∥∥ = ‖PB (p + Φ0 (p))−PB (p + q)‖= ‖PB (p,Φ0 (p)− q) + PB (Φ0 (p)− q,p) +

+PB (Φ0 (p) ,Φ0 (p))−PB (q,q)‖≤ Λ

12 CL

12 (‖p‖ ‖Φ0 (p)− q‖+ ‖Φ0 (p)− q‖ ‖q‖ +

+ |PB (Φ0 (p) ,Φ0 (p))−PB (q,q)|)≤ CLδ

12 ,

and∥∥Γc,0 (p,cp)− c′p

∥∥ =∥∥ PB

(p + Ψu,0 (p) , cp + Ψc,0 (p,cp)

)− PB (u,c)∥∥

≤ ‖PB (Ψu,0 (p)− p, cp) + PB (p,Ψc,0 (p,cp)− cq)‖+ ‖PB (Ψu,0 (p) ,Ψc,0 (p,cp))− PB (cp, cp)‖

≤ Λ12 CL

12 (‖Ψu,0 (p)− p‖ ‖cp‖+ ‖p‖ ‖Ψc,0 (p,cp)− cq‖) +

+Λ12 C

(L

12 ‖Ψu,0 (p)− cp‖ ‖cp‖ +

+ |PB (Ψu,0 (p) ,Ψc,0 (p,cp))− PB (cp, cp)|)≤ CLδ

12 ,

whence|Ψc,2 (p,cp)− cq,2 (p,cp)| ≤ CL

32 δ

52 .

Assume that for every k ≤ j − 1

|Ψc,k (p,cp)− cq,k (p,cp)| ≤ CLk+12 δ

k+32 .

For j > 2, we have

|D∆ (Ψc,j (p,cp)− cq,j (p,cp))| ≤

≤ CL12 ‖p‖ ‖Ψc,j−1 (p,cp)− cq,j−1‖+ CL

12 ‖cp‖ ‖Ψu,j−1 (p)− qj−1‖+

+C |Ψu,j−2 (p)− qj−2|12 |∆(Ψu,j−2 (p)− qj−2)|

12 ‖cq,j−2‖+

+C |Ψu,j−2 (p)| 12 |∆Ψu,j−2 (p)| 12 ‖Ψc,j−2 (p,cp)− cq,j−2‖++

∣∣DΨc,j−2 (p,cp) (Γu,j−2 (p) ,Γc,j−2 (p,cp))− c′q,j−2

∣∣≤ CL

j+12 δ

j+12 +

∣∣DΨc,j−2 (p,cp) (Γu,j−2 (p) ,Γc,j−2 (p,cp))− c′q,j−2

∣∣ .(10.42)

A careful analysis of the order of magnitude and the use of the inductionhypothesis and of the inequalities (2.6)-(2.8), (2.11)-(2.13) leads us to theconclusion that the terms in the right-hand side of (10.42) have the sameorder of magnitude with respect to δ, as δ → 0.


Hence, finally,

|Ψc,j (p,cp)− cq| = |Ψc,j (p,cp)− cq,j + cq,j − cq| ≤≤ |Ψc,j (p,cp)− cq,j |+ |ξj | ≤ CL

j+12 δ

j+32 .

Since

dist((u(t), c(t)) , Mj

)≤ C (|Φj (p)− q|+ |Ψc,j (p,cp)− cq (p,cp)|) ,

the conclusion of the theorem follows.¤AcknowledgementsWork supported by the Romanian Ministry of Education and Research

within the frame of the grant CEEX-05-D11-25/5.11.2005

References[1] P. Constantin, C. Foias, Navier-Stokes equations, Univ. of Chicago Press, Chicago,

1988.

[2] C. Foias, G. R. Sell, R. Temam, Varietes inertielles des equations differentielles, C. R.Acad. Sci., Ser.I, 301(1985), 139-141, .

[3] C. Foias, O. Manley, R. Temam, Modelling of the interactions of the small and largeeddies in two dimensional turbulent flows, Math. Modelling and Num. Anal., 22(1988),93-114.

[4] A. Georgescu, Hydrodynamic stability theory, Kluwer, Dordrecht, 1985.

[5] J. Robinson, Infinite-dimensional dynamical systems - an introduction in dissipativeparabolic PDE’s and the theory of global attractors, Cambridge Univ. Press, Cambridge,2001.

[6] R. Temam, Navier-Stokes equations and nonlinear functional analysis, CBMS-NSFReg. Conf. Ser. in Appl. Math., SIAM, 1995.

[7] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics,Springer, New Yotk, 1997.

[8] R. Temam, Induced trajectories and approximate inertial manifolds, Math. Mod. Num.Anal., 23, (1989), 541-561.

[9] R. Temam, Attractors for the Navier-Stokes equations, localization and approximation,J. Fac. Sci. Univ. Tokyo, Soc. IA, Math., 36(1989), 629-647.

[10] R. Temam, Infinite-dimensional dynamical systems in Mechanics and Physics, Springer

Appl. Math. Sci., 68, Springer, New-York, 1997.

FINITE STATE MACHINE TESTING FROMAN OR STATE REFINEMENT DESIGN

ROMAI J., 2, 1(2006),109–117

Florentin Ipate, Tudor BalanescuDepartment of Mathematics and Computer Science, University of [email protected], tudor [email protected]

Abstract The paper formalizes the OR refinement of finite state machines and devel-ops an efficient method to integrate the test sets of the component machinesin order to test the flattened machine. The approach supports component-software development since it constructs test sets for an assembly of systemsfrom readily available test sets of the prefabricated parts.

Keywords: finite state machine, OR state refinement, test generation, verification, W−method

2000MSC: 68N32

1. INTRODUCTIONThe test set generation problem for protocols modeled by deterministic

finite state machines has attracted a great deal of interest and has generated asubstantial literature. Given two deterministic finite state machines S and I,the former representing the specification and the latter the implementation, atest set is a set of input sequences that, if S and I are not equivalent, containsat least one sequence that produces different results when applied to the twomachines. The test set is generated from the specification S and, in principle,little information is available about the implementation I. For instance, inthe so called W- method [7] the only information about the implementationis an upper bound on the number of its states. Originally, the W -methodwas developed in the context of completely specified deterministic finite statemachines and was recently revised for the case in which both the specificationand the implementation may be partially specified. The W -method has alsobeen adapted to finite state machines based specification languages, such asstream X-machines [11], [10] and Harel statecharts [3], [4].

On the other hand, in practice a finite state machine specification is usuallyconstructed through a process of refinement; the equivalent flattened finitestate machine of such a refinement can be constructed and the W -method canbe applied to test the flattened machine. This approach might not alwaysbe practical, for large scale systems the flattened machine can be extremelycomplex and this will result in a test set of unmanageable size.

109

110 Florentin Ipate, Tudor Balanescu

The paper formalizes the OR refinement of Finite State Machines (in section4) and develops an efficient testing strategy for this kind of refinement.

2. PRELIMINARIESDefinition 2.1 A Deterministic Finite State Machine (DFSM for short) is aquintuple M = (Q,Li, Lo, h, M0), where Q is a finite set of states, Li is a finiteset of input symbols, Lo is a finite set of output symbols, h : Q×Li → Q×Lois the behavior (partial) function, M0 is the initial state.

A DFSM is often described by a state-transition diagram as in Example 2.1.Let q, r ∈ Q, a ∈ Li, o ∈ Lo. We write q → a/o → r to denote a transition

from q to r with label a/o if h(q, a) = (r, o). As a response to an arbitraryinput sequence x = a1 · · · · ·an, n ≥ 0, ai ∈ Li for all i, 1 ≤ i ≤ n, a state q ∈ Qof the DFSM M may produce an output sequence t = o1 · · · · · on, oi ∈ Lofor all i, 1 ≤ i ≤ n, if there are the following n transitions qi → ai/oi → qi+1,1 ≤ i ≤ n and q1 = q. It is said that the input a1 ·· · ··an leads the machine fromthe state q to the state r or that the state r is reachable from q. We denotethis by q ⇒ a1/o1 · · · · · an/on ⇒ qn+1 or simply by q ⇒ a1/o1 · · · · · an/on ⇒if the reached state qn+1 may be omitted. Moreover, if the produced outputstring o1 · · · · · on is not significant, it may be omitted, writing either q ⇒a1 · · · · · an ⇒ qn+1 or q ⇒ a1 · · · · · an ⇒ .

For an input sequence a1·· · ··an, we say that two states q, r ∈ Q give identicalresponse to a1 · · · · · an if: q ⇒ a1/o1 · · · · · an/on ⇒ iff r ⇒ a1/o1 · · · · · an/on ⇒for all o1, . . . , on ∈ Lo. Note that the definition includes the case where nosuch o1, . . . , on exist (i.e. for partially specified DFSM).

Let A ⊆ Li∗ be an arbitrary set of input sequences. Two states q, r ∈ Qare said to be A− equivalent if q and r give identical responses for each inputsequence x ∈ A. We denote this by q ∼=A r. Otherwise, they are said to be A−distinguishable (denoted by q 6∼=A r). If A = Li∗ (i.e. q and r give identicalresponses for any input sequence) then q and r are said to be equivalent.

Let M and N be two DFSMs with the same sets of input (Li) and output(Lo) symbols and A ⊆ Li∗. Then M and N are said to be A-equivalent if theirinitial states M0 and N0 are A− equivalent. Otherwise, M and N are said tobe A-distinguishable. If A = Li∗ then M and N are said to be equivalent.

A finite set R ⊆ Li∗ is said to be a state cover of a DFSM M if the emptysequence ε is contained in R, and, for every state q ∈ Q other than M0, Rcontains an input sequence x ∈ R that may lead the machine from the initialstate M0 to q (i.e. M0 ⇒ x ⇒ q). A finite set W ⊆ Li∗ is said to be acharacterization set of a DFSM M if any two distinct states q, r ∈ Q areW -distinguishable.

Example 2.1 Consider a simple tape recorder capable of playing and record-ing a tape. For simplicity, we consider the tape to be infinite. The DFSM

Finite state machine testing from an OR state refinement design 111

rec / Rec play / Play

stop / Idle

stop/Idle play / Play

OFF IDLE

PLAYING

off / Off

on / Idle

RECORDING off / Off

off / Off

stop /

Fig. 1. DFSM model of a tape recorder.

presented in fig. 1 models the control of this tape recorder. The inputs to thisDFSM are events: Li = play, stop, rec, on, off. The outputs are the opera-tions performed by the tape recorder: Lo = Play, Idle, Rec,Off. The initialstate OFF is pointed at by a transition from a blob.

Then R = ε, on, on · rec, on · play is a state cover of this DFSM andW = on, rec, play is a characterization set.

A DFSM M is said to be minimal if any other equivalent DFSM has at leastthe same number of states as M ; a DFSM A is minimal if and only if thereexists a state cover and a characterization set of A [8].

Let M = (Q,Li, Lo, h,M0) and M ′ = (Q′, Li, Lo, h′, M ′0) be two DFSMs.

Then a bijective function f : Q −→ Q′ is called an isomorphism from M toM ′ if for every states q1, q2 ∈ Q, input symbol a ∈ Li and output symbolo ∈ Lo, q1 → a/o → q2 if and only if f(q1) → a/o → f(q2). Any twoequivalent minimal DFSMs are isomorphic [8], so for any DFSM M, thereexists an unique (up to an isomorphism) minimal DFSM M ′ equivalent to M .M ′ is called the minimal DFSM of M.

3. THE W -METHODLet S be a DFSM specification and I a DFSM model of the implementa-

tion. The W -method generates a test set (a finite set of sequences) that candetermine any error in I with respect to S, provided that the number of statesof I is bounded by an integer m that may be larger than the number n ofstates of S. The set is generated from the specification S and no informationis available about I except the above mentioned upper bound.


The W -method was first developed in the context of completely specifiedDFSMs [7]. In [1], the method was extended to cope with (possibly) partiallyspecified DFSMs.

Definition 3.1 Let S = (QS , Li, Lo, hS , S0) and I = (QI , Li, Lo, hI , I0) betwo DFSMs. A finite set Y ⊆ Li∗ is called a test set for S and I if: S and Iare Y -equivalent =⇒ S and I are equivalent.

The W -method involves the selection of two sets of input sequences:

W ⊆ Li∗, a characterization set of S

R ⊆ Li∗, a state cover set of S.

The method makes the following assumptions about the specification S andthe implementation I :

S is deterministic and minimal

I is deterministic and the number of states of I is bounded by an integerm.

For the case where S and I are assumed to be completely specified the W -method provides a test set

Ycs = R · Li[m− n + 1] ·W,

with Li[j] = ε ∪ Li ∪ · · · ∪ Lij for j ≥ 0 [7].However, Y is not always a valid test set when partially specified DFSMs

are considered, as shown by the following example. Consider the partiallyspecified DFSM specification S = (0, 1, a, b, a, b, h, 0) with h(0, a) =(1, a) and h undefined elsewhere. Then R = ε, a is a state cover of S andW = a is a characterization set of S. Let I = (0, 1, a, b, a, b, h′, 0) bean implementation of S with h′(0, a) = (1, a), h′(0, b) = (1, b) and h′ undefinedelsewhere. Since S and I have the same number of states (n = m = 2), theW -method gives Ycs = R·Li[1]·W = a, a∪a, b∪a, a∪a∪a, a∪b∪a. However,the two DFSMs are Ycs- equivalent but they are not equivalent. Indeed, thesingle word which distinguishes them is b, which is not contained in Ycs.

This situation occurs because when S and I are completely specified when-ever S and I are s-equivalent for an input sequence s, they are also t-equivalent for any prefix t of s. This is no longer true when at least one of Sand I is partially specified. Therefore, one way of extending the W -method tothe general case where the machines may be partially specified is to take allprefixes of the sequences in Ycs. It has been proved, however, that a smallersubset of prefixes is sufficient [1], this is

Y = R · Li[m− n + 1] · (W ∪ ε).


stop / rec / Rec

continue / Rec

play / Play

stop / Idle

stop/Idle

play / Play

OFF

IDLE

off / Off

on / Idle

RECORDING

RECORD PAUSE_REC

pause / Pause

continue / Play

PLAYING

PLAY PAUSE_PLAY pause / Pause

off / Off

off / Off

Fig. 2. Refinement of OR states.

4. OR STATE REFINEMENTOne way of refining an existing DFSM specification is to replace one or

more states with other DFSMs. The semantics of this transformation is thatgiven by the expansion of OR states in state charts [3] and is illustrated bythe following example.Example 4.1 The original model in Example 2.1 can be refined by allowingthe tape recorder to pause while playing or recording. For instance, the behaviorof the RECORDING state is a two-state machine describing whether thetape recorder is actively recording or waiting for the user to press the continuecommand. The refinement of the two states, PLAY ING and RECORDING(called in what follows OR states), is graphically illustrated in fig. 2. Theflattened DFSM for the refined model is presented in fig. 3. Any transitionarriving to the OR state in the original specification will now be directed tothe initial state of the DFSM that replaces that state. Any transition leavingthe OR state will now leave all the states of the replacement DFSM.

Definition 4.1 A DFSM M ′ = (Q′, Li′, Lo′, h′,M ′0) is called an OR state

refinement of a DFSM M = (Q,Li, Lo, h, M0) if Q ⊆ Q′, Li ⊆ Li′, Lo ⊆ Lo′,M0 = M ′

0 and there is a partition E = Eqq∈Q of Q′ with q ∈ Eq for allq ∈ Q such that the following hold:

h(q, a) = h′(q′, a) for all q ∈ Q, a ∈ Li, q′ ∈ Eq

π1(h′(q′, a)) ⊆ Eq for all q ∈ Q, a ∈ Li′ − Li, q′ ∈ Eq, where π1 :Q′ × Lo′ −→ Q′ denotes the projection function


play / Play

stop /

stop / Idle

stop / Idle

pause /

rec / Rec

continue / Rec

play / Play

stop / Idle

stop/Idle

play / Play

OFF IDLE

off / Off

on / Idle

RECORD PAUSE_REC pause /

continue / Play

PLAY PAUSE_PLAY

off / Off

off / Off

off / Off

off / Off

Fig. 3. Flattened DFSM for OR state refinement.

Li0 = Li′ − Li is called the refining input alphabet.For q ∈ Q, the DFSM Mq = (Eq, Li0, Lo′, hq, q), where hq is the restriction

of h′ to Eq × Li0, is called the refining machine for q.

For simplicity, in Definition 4.1, the refined state and the initial state of thecorresponding refining machine are considered to be identical. For clarity, theyare given different names in our examples, where only the states PLAY INGand RECORDING are refined. So the partition E is defined by EPLAY ING =PLAY, PAUSE PLAY , ERECORDING = RECORD, PAUSE REC andEq = q for the other states. The refining input alphabet is Li0 = continue, pause.

5. OR STATE REFINEMENT TESTINGConsider the problem of generating test sets from a specification of the form

of an OR state refinement of a DFSM S = (QS , Li, Lo, hS , S0). The W methodcan be directly applied to the flattened specification S′ = (Q′

S , Li′, Lo′, h′S , S′0)and, furthermore, a state cover RS′ and a characterization set WS′ , can beconstructed from a state cover RS and a characterization set WS of the originalDFSM S and state covers Rq, q ∈ QS and characterization sets Wq, q ∈ QS ofthe refining machines using the following formulae

RS′ = RS ⊗ R,

where R = Rq | q ∈ QS is a set that contains a state cover RMq for eachstate q ∈ Qs and for A ⊆ Li∗ and B = Bq | q ∈ QS ⊆ 2Li∗ , A⊗B = a · b |a ∈ A, b ∈ Bq, S0 ⇒ a ⇒ q. That is RS′ is obtained by concatenating each


sequence in RS that forces S into q from the initial state S0 with all sequencesin Rq

WS′ = W ∪ (⋃

q∈QS

Wq).

The proofs are straightforward.On the other hand, since the specification has been developed through a

refinement process, one would also expect the implementation to be build in-crementally: initially a developer may only have a skeleton of the system thenstep by step fill it with details. For instance the DFSM in fig. 1 can be builtfirst. Afterwards, when the functionality contained in the RECORDING (orPLAY ING) refining machine is added it can be considered not to have aneffect outside this state, (i.e. any transition leaving an OR state in I will leaveall the states of the refining machine in I ′ and have the same destination as theoriginal transition), so implementation can also be modelled by an OR staterefinement I ′ = (Q′

I , Li′, Lo′, h′I , I′0) of some DFSM I = (QI , Li, Lo, hI , I0).

We can now take advantage of this situation and show how the test sets forS and the refining machines can be reused in the construction of the new testset.

Let S′, the specification, be an OR state refinement of S and I ′, the imple-mentation, be an OR state refinement of I, as above. Our aim is to constructa test set Y ′ for S′ and I ′. Let n be the number of states of S and m theupper bound for the number of states of I. For q ∈ QS let Sq be the refiningmachine for q and for p ∈ QI let Ip be the refining machine for p. Let nq bethe number of states of Sq, mp the upper bound for the number of states of Ip

and k = maxp∈QImp. We assume that S is minimal and Sq is minimal for all

q ∈ Qs. Without loss of generality all states of I are assumed to be reachablefrom the initial state I0 (otherwise the non-reachable states may be removed).Similarly, for p ∈ QI , all states of Ip are assumed to be reachable from theinitial state p.

Then the test set Y ′ we are after is

Y ′ = Y ∪ U,

where Y = R · Li[m − n + 1] · (W ∪ ε) is a test set for S and I and U isconstructed using the following sets:

Yq = Rq · Li0[k − nq + 1] · (Wq ∪ ε) for all q ∈ QI , where Rq is a statecover and Wq a characterization set of Sq; then Yq is a test set for Sq

and Ip for all p ∈ Qp

T = R · Li[m− n]

ThenU = T ⊗ Y,


where Y = Yq | q ∈ QS is a set that contains a set Yq for each state q ∈ Qs.For our example consider m = 5 and k = 2. Then T = R ·Li[1] = ε, on, on ·

rec, on · play · ε, play, stop, rec, on, off, RRECORDING = RPLAY ING =ε, pause, WRECORDING = WPLAY ING = pause so YRECORDING = YPLAY ING =ε, pause · Li0[1] · pause; ROFF = RIDLE = WOFF = WIDLE = ε, soYOFF = YIDLE = Li0[2]. Thus U = ε, on · off, on · rec · off, on · play ·off · YOFF ∪on, on · stop, on · rec · stop, on · play · stop · YIDLE ∪on · rec ·YRECORDING ∪ on · play · YPLAY ING.

Lemma 5.1 For S and I as above, if S and I are equivalent, then T is astate cover of I.

Proof. We prove by induction on 0 ≤ j ≤ m−n that Tj = R ·Li[j] reachesat least n + j states of I. For j = 0 the statement follows since R is a statecover of S, which is the minimal DFSM of I. Assume the statement true for j.Then either Tj reaches all the states of I or Tj+1 will reach at least one morestate than Tj . Hence Tj+1 reaches at least n + j + 1 states of I. 2

Theorem 5.1 For S, S′, I, I ′ and Y ′ as above, Y ′ is a test set for S′ and I ′.Proof. Assume that S′ and I ′ are Y ′-equivalent. Then S′ and I ′ are Y -equivalent, hence S and I are Y -equivalent. Since Y is a test set for S and I,S and I are equivalent. Furthermore, from Lemma 5.1 it follows that T is astate cover of I.

Let q ∈ QS and p ∈ QI be two equivalent states of S and I, respectively.Then there exists t ∈ T such that S0 ⇒ t ⇒ q and I0 ⇒ t ⇒ p. Since S′ andI ′ are U -equivalent it follows that Sq and Ip are Yq-equivalent. Hence Sq andIp are equivalent.

Therefore, we have proved that S and I are equivalent and for any pairof equivalent states q ∈ QS , p ∈ QI , Sq and Ip are equivalent. Then fromDefinition 4.1 it follows that S′ and I ′ are equivalent. 2

It can be shown that the implementation refinement approach considerablyreduces the size of the test set generated.

6. CONCLUSIONSThe paper formalizes the OR refinement for finite state machines and it

shows how if the implementation is also constructed through a process ofrefinement, then the test process can be distributed into smaller chunks (i.e.the original machine and the refining machines), thus diminishing the effortdevoted to testing and the size of the final test set.

The partial OR (p-OR) state refinement of finite state machines will alsobe considered in a future paper. The generalization of these test generationmethods for refined DFSMs to other finite state machine based specificationlanguages, such as Harel statecharts, will also be investigated.


References[1] Balanescu, T., Gheorghe, M., Ipate, F., Holcombe, M., Formal black box testing for

partially specified deterministic finite state machines, Foundations of Computing andDecision Systems, 28 (2003).

[2] Balanescu, T., Ipate, F., The Wp method for partially specified deterministic finitestate machines, Annals of Bucharest University, Informatica, 52, 1 (2004), 47-61.

[3] Bogdanov, K., Automatic testing of Harel’s statecharts, PhD thesis, The University ofSheffield, 2000.

[4] Bogdanov, K., Holcombe, M., Statechart testing method for aircraft control systems, J.Softw. Test. Verific. Reliability, 11 (2001), 39-54.

[5] Boiten, E., Derrick, J., Unifying concurrent and relational refinement, Electronic Notesin Theoretical Computer Science, 70 (2002), 35 pp.

[6] Brookes, S.D., Roscoe, A.W., An improved failures model for communicating processes,Pittsburgh Symposium on Concurrency, Lecture Notes in Computer Science, 197(1985), 281-305.

[7] Chow, T.S., Testing software design modeled by finite-state machines, IEEE Trans.Softw. Engrg, 4 (1978), 178-187.

[8] Eilenberg, S., Automata, languages and machines, Academic, New York, 1974.

[9] Fujiwara, S., von Bochmann, G., Khendek, F., Amalou, M. and Ghedamsi, A., Testselection based on finite state models, IEEE Trans. Softw. Engrg, 17 (1991), 591-603.

[10] Holcombe, M. and Ipate, F., Correct systems: Building a business process solution,Springer, Berlin, 1998.

[11] Ipate, F. and Holcombe, M., An integration testing method that is proved to find allfaults, Intern. J. Comput. Math, 69 (1997), 159-178.

[12] Ipate, F. and Holcombe, M., An integrated refinement and testing method for streamX-machines, Appl. Algebra Engrg., Comm. Comput, 13 (2002), 67-91.

[13] Lee, D. and Yannakakis, M., Principles and methods of testing finite state machines -A survey, Proceed. IEEE, 84 (1996), 1090-1123.

[14] Luo, G., v. Bochmann, G. and Petrenko, A., Test selection based on communicatingnondeterministic finite-state machines using a generalised Wp-method, IEEE Trans.Softw. Engrg, 20 (1994), 149-161.

[15] Milner, R. Communication and Concurrency, Prentice-Hall, 1989.

[16] Petrenko, A., Fault model-driven test derivation from finite state models: Annotatedbibliography, Modelling and Verification of Parallel Processes, 4th Summer School,MOVEP 2000, Springer Verlag, Lecture Notes in Comput. Sci. 2067 (2001), 196-205.

[17] Urbasek, M., Net Transformations for Petri Net Technology, Bulletin of the EATCS,80 (2003), 77-94.

[18] v. Bochmann, G. ,Petrenko, A., Protocol testing: Review of methods and relevance forsoftware testing, ACM International Symposium on Software Testing and Analysis,Seattle USA, 1994, 109-123.


SHARPENING OF CRITICAL SPECTRALPARAMETER AT DYNAMIC BIFURCATIONBY PSEUDOPERTURBATION METHOD

ROMAI J., 2, 1(2006),119–126

Boris V. Loginov*, Olga V. Makeeva**, Elena V. Foliadova***Ulyanovsk State Technical University, **Ulyanovsk State Pedagogical University, Russia

Abstract Taking into account our previous works [2]-[5] at the suggested by the M.K.Gavurin [1] pseudoperturbation method the problem of sharpening the ap-proximately given critical value of the spectral parameters and relevant Jordanchains in Poincare-Andronov-Hopf bifurcation is solved.

1. THE STATEMENT OF THE PROBLEMLet E1, E2 be two real Banach spaces and consider the differential equa-

tionA

dx

dt= Bx−R(x, ε). (1)

Here A,B : E1 → E2 are, for simplicity, linear (bounded) operators, R(x, ε)is definite and continuous in a neighborhood of the point (0, 0) ∈ E1+R1

together with its Frechet derivative Rx(x, ε), and R(0, 0) = 0, Rx(0, 0) = 0.We consider the sufficiently general case when the A-spectrum σA(B) of theoperator B intersects the imaginary axis at the points ±iα of multiplicity n

[6, 7, 11, 13]. Let the elements u(1)1k , u

(1)2k ∈ E1 (elements v

(1)1k , v

(1)2k ∈ E∗

2) besuch that

B(α)Φ(1)k ≡

(B αA−αA B

)(u

(1)1k

u(1)2k

)= 0,

(B∗(α)Ψ(1)

k ≡(

B∗ −αA∗αA∗ B∗

) (v

(1)1k

v(1)2k

)= 0

), k = 1, . . . , n,

(2)

i.e. the zero-subspace N(B(α)) and N(B∗(α)) of the operators B(α) andB∗(α) have the forms

N(B(α)) = span

Φ(1)

1k =

(u

(1)1k

u(1)2k

), Φ(1)

2k =

(−u

(1)2k

u(1)1k

), k = 1, . . . , n

,

N(B∗(α)) = span

Ψ(1)

1k =

(v

(1)2k

−v(1)1k

), Ψ(1)

2k =

(v

(1)1k

v(1)2k

), k = 1, . . . , n

.

119

120 Boris V. Loginov*, Olga V. Makeeva**, Elena V. Foliadova**

Carrying out the complexification of the equation (1), consider it in thespaces Ek = Ek + iEk, k = 1, 2, and suppose that the operator R(x, ε) admitsa sufficiently smooth extension on these spaces. Then the elements u

(1)k =

u(1)1k + iu

(1)2k , u

(1)k and v

(1)k = v

(1)1k + iv

(1)2k , v

(1)k are the eigenelements of the

following eigenvalue problem [11]

Bu(1)k = iαAu

(1)k , Bu

(1)k = −iαAu

(1)k , B∗v(1)

k = −iαA∗v(1)k , B∗v(1)

k = iαA∗v(1)k .

(3)The Poincare substitution t = τ

α+µ , x(t) = y(τ) in the equation (1) allowsus to extend the classical Lyapounov-Schmidt method for the investigationof the Poincare-Andronov-Hopf bifurcation problem [11, 13], by introducingthe spaces Y (Z) of 2π-periodic continuously differentiable functions of τ withvalues in E1 (2π-periodic continuous functions of τ with values in E2) andduality

〈〈y, f〉〉 =12π

∫ 2π

0〈y(τ), f(τ)〉dτ y ∈ Y, f ∈ Y ∗(y ∈ Z, f ∈ Z∗). (4)

Then the operators (By)(τ) ≡ −αAdydτ + By(τ) and (B∗z)(τ) ≡ αA∗ dz

dτ +B∗z(τ) have 2π-dimensional zero subspaces

N(B) = span

ϕ(1)k = ϕ

(1)k (τ) = u

(1)k eiτ , ϕ

(1)k = u

(1)k e−iτ , k = 1, . . . , n

,

N(B∗) = span

ψ(1)k = ψ

(1)k (τ) = v

(1)k eiτ , ψ

(1)k = v

(1)k e−iτ , k = 1, . . . , n

. (5)

Let now the zero-subspaces N(B) and N(B∗) have A-Jordan (A∗-Jordan)chains ϕ

(j)k = u

(j)k eiτ , ψ

(j)k = v

(j)k eiτ , j = 1, . . . , pk of the length pj . According

to the lemma on the biorthogonality of the Jordan chains (for linear operator-functions of spectral parameter) [8, 12], they can be chosen to satisfy theequalities

〈〈ϕ(j)k , γ(l)

s 〉〉 = δksδjl, j(l) = 1, . . . , pk(ps), k, s = 1, . . . , n, γ(l)s = A∗ψ(ps+1−l)

s ,

z(j)k = Aϕ

(pk+1−j)k . (6)

For the operator-function (2) the relations 6 read (see (3)–(5))

〈Au(pk+1−j)1k , v

(l)1s 〉+ 〈Au

(pk+1−j)2k , v

(l)2s 〉 = δksδjl, 〈Au

(pk+1−j)2k , v

(l)1s 〉 = 〈Au

(pk+1−j)1k , v

(l)2s , 〉

〈Au(j)1k , v

(ps+1−l)1s 〉+ 〈Au

(j)2k , v

(ps+1−l)2s 〉 = δksδjl, 〈Au

(j)2k , v

(ps+1−l)1s 〉 = 〈Au

(j)1k , v

(ps+1−l)2s , 〉

(7)

Sharpening of critical spectral parameter by pseudoperturbation method 121

which mean the biorthogonality property for A-Jordan chains of the operator-

function B(α− ε) = B(α)− εA, A =(

0 A−A 0

), A∗ =

(0 −AA 0

), i.e.

〈AΦ(pk+1−j)νk , Φ(l)

µs〉 = δµνδksδjl, 〈Φ(j)νk , A∗Φ(ps+1−l)

µs 〉 = δµνδksδjl (8)

The aim of this article is the sharpening of sufficiently good approx-imations α0, u

(s)j0 , v

(s)j0 , |α − α0| ≤ ε, ‖u(s)

j − u(s)j0 ‖ ≤ ε, ‖v(s)

j − v(s)j0 ‖ ≤ ε,

s = 1, . . . , pj , j = 1, . . . , n to the critical value of the bifurcation parameter αand generalized Jordan chains

(B − iαA)u(k)j = Au

(k−1)j , (B + iαA)u(k)

j = −Au(k−1)j ,

(B∗ + iαA∗)v(k)j = −A∗v(k−1)

j , (B∗ − iαA∗)v(k)j = A∗v(k−1)

j ,(9)

where 〈Au(k)j , v

(pσ+1−l)σ 〉 = δjσδkl, by means of the pseudoperturbation method

[2, 3, 5].The above-mentioned discussion allows us to solve this problem both on

the base of operator-function B(α) with relations (8) and operator-functionB ± iαA with relations (9). The cases of nonlinear dependence on α can beinvestigated by means of the linearization [5] of the spectral parameter.

The obtained results are supported by RFBR grant 06-01-01692.

2. GENERAL SCHEME OFPSEUDOPERTURBATION OPERATORCONSTRUCTION

A. Operator-function B − iαA.Similarly to [3, 5] the following assertion can be provedLemma 1A. Passing if necessary to linear combinations, the systems γ(l)

k0n,pkk,l=1,

γ(l)k0 = A∗v(pk+1−l)

k0 , z(j)i0 n,pi

i,j=1, z(j)i0 = Au

(pk+1−l)k0 satisfying the biorthogonality

relations 〈u(j)i0 , γ

(l)k0 〉 = δikδjl, 〈z(j)

i0 , v(l)k0〉 = δikδjl, can be determined.

Computing the expressions σ(1)k0 = (B − iα0A)u(1)

k , σ(j)k0 = (B − iα0A)u(j)

k −Au

(j−1)k0 , τ

(1)k0 = (B∗ + iα0A

∗)v(1)k0 , τ

(j)k0 = (B∗ + iα0A

∗)v(j)k0 + A∗v(j−1)

k0 , 1 < l ≤pk, define two pseudoperturbation operators

D0x =n∑

k=1

pk∑

j=1

〈x, γ(j)k0 〉[σ(j)

k0 −n∑

r=1

pr∑

s=1

〈σ(j)k0 , v

(s)r0 〉z(s)

r0 ] +n∑

k=1

pk∑

j=1

〈x, τ(j)k0 〉z(j)

k0 ,

D0x =n∑

k=1

pk∑

j=1

〈x, γ(j)k0 〉σ(j)

k0 +n∑

k=1

pk∑

j=1

[〈x, τ(j)k0 −

n∑

r=1

pr∑

s=1

〈u(s)r0 , τ

(j)k0 〉γ(s)

r0 〉]z(j)k0 . (10)


Theorem 1. The pseudoperturbation operators (10) have the followingproperties

D0u(j)k0 = σ

(j)k0 , D∗

0v(j)k0 = τ

(j)k0 , j = 1, . . . , pk, k = 1, . . . , n (11)

Corollary 1B. The pseudoperturbation operators have the required proper-ties

(B − iα0A−D0)u(1)k0 = 0, (B − iα0A−D0)u

(j)k0 = Au

(j−1)k0 ,

(B∗ + iα0A∗ −D∗

0)v(1)k0 = 0, (B∗ + iα0A

∗ −D∗0)v

(j)k0 = −A∗v(j−1)

k0 , j > 1. (12)

The proofs are performed by a direct computation [3, 5]. Thus the givensufficiently good approximation to pure imaginary eigenvalue and generalizedJordan chains become exact for the perturbed operator.

B. Operator-function B(α).Lemma 1B. Passing, if necessary, to linear combinations, the systems

γ(l)µk0n,pk

k,l=1, γ(l)µk0 =

(0 −AA 0

)Ψ(ps+1−l)

µs0 , z(j)µi0n,pi

i,j=1, z(j)µi0 =

=(

0 A−A 0

)Φ(pi+1−l)

µi0 , µ = 1, 2, satisfy the boirthogonality relations (8).

Computing the expressions σ(1)µk0 = B(α0)Φ

(1)µk0, σ

(j)µk0 = B(α0)Φ

(j)µk0−AΦ(j−1)

µk0,

τ(1)µk0 = B∗(α0)Ψ

(1)µk0, τ

(j)µk0 = B∗(α0)Ψ

(j)µk0 − A∗Ψ(j−1)

µk0 , 1 < l ≤ pk, µ = 1, 2introduce four pseudoperturbation operators Dµ0, µ = 1, 2 of the form (10).

Theorem 1B and its Corollary 1B are proved by direct computation.

3. NEWTON-KANTOROVICH ITERATIONALPROCEDURE FOR THE DETERMINATIONOF CRITICAL VALUE OF PARAMETRER αAND GENERALIZED JORDAN CHAINS

Since up to the notation the description of the Newton-Kantorovichiterative process for the operator-function B(α), repeats the relevant part ofthe articles [3, 5], here we present it only for the operator-function B − iαA.Using the first equality of Corollary 1A rewrite the equation (B− iαA)uk = 0in the form of the system

(B− iα0A−D0)u− i(α−α0)Au + D0u +n∑

k=1

〈u, γ(1)k0 〉z

(1)k0 =

n∑

k=1

ξ(1)k z

(1)k0 , (13)

ξ(1)s = 〈u, γ

(1)s0 〉. (14)

According to the E. Schmidt lemma and general perturbation theory fora sufficient accuracy of the initial approximations there exists the bounded


operator Γ0 = B0−1

= [(B − iα0A−D0) +∑n

k=1〈·, γ(1)k0 〉z(1)

k0 ]−1. Following thescheme of the articles [3, 5], set

ljk(it− iα0) = 〈((it− iα0)A−D0)[I − Γ0(it− iα0)A + Γ0D0]−1u(1)j0 , v

(1)k0 〉 =

= 〈I − [I − Γ0((it− iα0)A−D0)]−1u(1)j0 , γ

(1)k0 〉.

The function f(t) = det[ljk(it− iα0)] has t=α as K=n∑

s=1ps− multiple root,

since − 1s!

dsljk(it−iα0)dts = ljk,s(it− iα0) =

〈[I− (it− iα0)Γ0A+Γ0D0]−1(Γ0A[I− (it− iα0)Γ0A+Γ0D0]−1)su(1)j0 , γ

(1)k0 〉 = 0

for s < K, i.e. dsf(α)dαs = 0 and dKf(α)

dαK = det[dpj ljk(α−α0)

dαpj ] 6= 0 by virtue of the

relation − 1pj !

dpj ljk(0)

dαpj = 〈u(1)j0 , γ

(1)k0 〉+ O(‖D0‖) = δjk + O(‖D0‖).

In order to determine of α it is possible to apply to the equation f (K−1)(t) =0 the modified or basic Newton-Kantorovich method, taking as the initial ap-proximation t = α0. In fact [9], since |f (K)(α1) − f (K)(α2)| ≤ L|α1 − α2| forα1, α2 ∈ S(α0, ρ) – the ball of small radius ρ and centered at α0, L is some con-

stant, then the equation f (K−1)(t) = 0 in the ball S(α0, r), r = 1−√

(1−4h)

2h η ≤ρ, h = |f (K)(α0)|−2L|f (K−1)(α0)| ≤ 1

4 , η = |f (K)(α0)|−1|f (K−1)(α0)|, has theunique solution t = α.

The construction of the iteration αν of the modified Newton-Kantorovichmethod αν = αν−1 − [f (K)(α0)]−1f (K−1)(αν−1), ν = 1, 2, ... is reduced, atevers step, to the solution of Kn linear equations. In fact, the calculationof f(iαν−1 − iα0) is reduced to the calculation of ljk(αν−1 − α0), i.e. to thesolving of the equation x

(ν−1)j1 −Γ0(iαν−1− iα0)A + Γ0D0x

(ν−1)j1 = u

(1)j0 , which

is equivalent to the equation

(B − iαν−1A)x(ν−1)j1 +

n∑

r=1

〈x(ν−1)j1 , γ

(1)r0 〉z(1)

r0 = z(1)j0 . (15)

In order to compute the sequential derivatives ljk,s(iαν−1 − iα0), j, k =1, . . . , n, s = 1, . . . , pj − 1 we must solve the equations

(B − iαν−1A)x(ν−1)j2 +

n∑

r=1

〈x(ν−1)j2 , γ

(1)r0 〉z(1)

r0 = Ax(ν−1)j1 ,

(B − iαν−1A)x(ν−1)j3 +

n∑

r=1

〈x(ν−1)j3 , γ

(1)r0 〉z(1)

r0 = Ax(ν−1)j2 ,


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(B − iαν−1A)x(ν−1)jpj

+n∑

r=1

〈x(ν−1)jpj

, γ(1)r0 〉z(1)

r0 = Ax(ν−1)jpj−1.

The basic Newton-Kantorovich method αν = αν−1−[f (K)(αν−1)]−1f (K−1)(αν−1)requires n(K + 1) linear equations.

Remark 1. According to results in [10] these equations are stable withrespect to computation error of the operator and right-hand sides.

After the computation of the exact value α, the elements of the gener-alized Jordan chains can be found directly from the equations

(B − iαA)xjs +n∑

r=1

〈xjs, γ(1)r0 〉z(1)

r0 = Axjs−1 =

z(1)j0 , s = 1

Axjs−1, s = 2, pj

(B∗ + iαA∗)yjs +n∑

r=1

〈z(1)r0 , yjs〉γ(1)

r0 =

γ

(1)j0 , s = 1

−A∗yjs−1, s = 2, pj

Remark 2. The cases of nonlinear dependence on α, which arise in Poincare-Andronov-Hopf bifurcation for differential equations of the order m > 1, canbe investigated on the base of the linearization [4].

4. THE CASE OF DIFFERENTIAL EQUATIONOF HIGHER ORDER

Consider the Poincare-Andronov-Hopf bifurcation for the s-th order dif-ferential equation

Asdsx

dts+ As−1

ds−1x

dts−1+ . . . + A1

dx

dt= Bx−R(x,

dx

dt, . . . ,

ds−1x

dts−1, ε) (16)

with bounded linear operators Ak, B : E1 → E2, k = 1, s. The nonlinearoperator R(x, x′, . . . , x(s−1), ε) is definite and continuous in a neighborhood ofthe point (0, . . . , 0, 0) ∈ u

sE1 uR1 together with its Frechet derivatives Rx(k)

and R(0, . . . , 0, 0) = 0, Rx(k)(0, . . . , 0, 0) = 0. The linearized equation (17)can be reduced to the equation AdX

dt = BX in the space of vector-functionsX = (x1, x2, . . . , x− s) ∈ u

sE1

A =

C 0 0 . . . 0 00 C 0 . . . 0 0...

......

. . ....

...0 0 0 . . . C 00 0 0 . . . 0 As

, B =

0 C 0 . . . 0 00 0 C . . . 0 0...

......

. . ....

...0 0 0 . . . 0 CB −A1 −A2 . . . −As−2 −As−1

.


where C ∈ L(E1, E2) is an arbitrary invertible operator and xk = dk−1x1

dtk−1 ,k = 2, s. Therefore the nonlinear operator R(x, x′, . . . , xs−1, ε) should beconsidered as representable in the form R(X, ε). According to Section 1, it isnatural to consider in u

sE1 → u

sE2 the eigenvalue problem

B(α)(

u1

u2

)≡

(B αA

−αA B

)(u1

u2

)= 0, uj = (uj1, uj2, . . . , ujs), j = 1, 2.

As before, we consider the sufficiently general case when the A-spectrumσA(B) of the operator B intersects the imaginary axis at the points ±iα of themultiplicity n, i.e. there exist elements u(1)

jk = (u(1)j1k, u

(1)j2k, . . . , u

(1)jsk) ∈ u

sE1,

j = 1, 2 (elements v(1)jk = (v(1)

j1k, v(1)j2k, . . . , v

(1)jsk) ∈ u

sE∗

2) such that

B(α)Φ(1)k =

(B αA

−αA B

) (u(1)

1k

u(1)2k

)= 0,

(B∗(α)Ψ(1)k =

(B∗ αA∗

−αA∗ B∗

) (v(1)

1k

v(1)2k

)= 0)

k = 1, n and the zero-subspaces N(B(α)) and N(B∗(α)) of the operators B(α)and B∗(α) have the forms (3).

The construction of the pseudoperturbation operator by means of thecorresponding Newton-Kantorovich iterative process for the operator-functionB(α), up to the notation, repets the relevant parts of the articles [3, 5] and,therefore, is omitted.

References

[1] M. K. Gavurin, On pseudoperturbation method for eigenvalues determination, J. Com-put. Math. Math. Phys. 1, 5 (1961), 751-770.

[2] B. V. Loginov, D. G. Rakhimov, N. A. Sidorov, Development of M.K. Gavurin’spseudoperturbation method, Fields Inst. Communs, 25(2000), 367-381.

[3] B. Karasozen, B. V. Loginov, V. A. Trenogin, Pseudoperturbation method for sharp-ening of approximately given generalized Jordan chains, Bul. St. Ser. Mat.-Info., Univ.din Pitesti, Romania, 9 (2003), 183-190.

[4] I. V. Konopleva, B. V. Loginov, O. V. Makeeva, Linearization on spectral parameterin branching theory, Proc. Middle-Volga Math. Soc.,7, 1 (2005), 105-113.

[5] B. V. Loginov, O. V. Makeeva, A. V. Tsyganov, Sharpening of approximately given Jor-dan chains of linear operator-function of spectral parameter on the base of perturbationtheory, Interuniv. Proc. ”Functional Analysis”, 38 (2003), 53-62.


[6] B. V. Loginov, Yu. B. Rousak, Generalized Jordan structure in the problem of thestability of bifurcating solutions, Nonlinear Analysis, TMA, 17, 3 (1991), 219-232.

[7] B. V. Loginov, Determination of the branching equation by its group symmetry –Andronov-Hopf bifurcation, Nonlinear Analysis, TMA, 28, 12 (1997), 2033-2047.

[8] B. V. Loginov, Yu. B. Rousak, Generalized Jordan structure in branching theory, Rightand Inverse Problems for Partial Differential Equations, Salakhitdinov M.(ed), Fan,AN Uzbek SSR, Tashkent, (1978), 133-148.

[9] L. A. Ljusternik, V. I. Sobolev, Elements of functional analysis, Nauka, Moscow, 1964.

[10] N. A. Sidorov, General questions of regularization in branching theory problems, IrkutskUnic. Proc.,1982.

[11] V. A. Trenogin, Periodical solutions and passage type solutions in abstract reaction-diffusion equations, Questions on Qualitative Theory of Differential Equations, Nauka,Novosibirsk, Siberian Branch Acad. Sci. USSR, (1988), 134-140.

[12] M. M. Vainberg, V. A. Trenogin (1969), Branching theory of solutions of nonlinearequations, Nauka, Moscow; Engl. transl. Volters-Noordorf, Leyden, 1974.

[13] V. I. Yudovich, Investigation of autooscillations of continua arising at the loosing of

stability of stationary regime, Prikl. Math. Mech 36, 3 (1972), 450-459; Engl. transl. J.

Appl. Math. Mech 36 (1972).

EXISTENCE RESULTS FOR A CLASS OFNONLINEAR VOLTERRA INTEGRO-DIFFERENTIAL PROBLEM

ROMAI J., 2, 1(2006),127–134

Rodica Luca–TudoracheDepartment of Mathematics, ”Gh.Asachi” Technical [email protected]

Abstract The existence of the strong and weak solutions of a Volterra integrodifferentialsystem with some nonlinear boundary conditions and initial data is proved.

Keywords: Volterra integrodifferential equations, boundary condition, maximal monotone

operator, compact operator, weak and strong solutions.

2000 MSC: 47H05, 45K05

1. INTRODUCTIONWe shall investigate the nonlinear Volterra integrodifferential system

(S)

∂u

∂t+ a2(x)

∂2v

∂x2+ a1(x)

∂v

∂x+ a0(x)v + α(x, u) +

∫ t

0b(t− s)A(u(s, x)) ds = f(t, x)

∂v

∂t− ∂2

∂x2(a2(x)u) +

∂

∂x(a1(x)u)− a0(x)u + β(x, v) +

∫ t

0c(t− s)B(v(s, x)) ds = g(t, x),

t > 0, 0 < x < 1,

with the boundary condition

(BC)

(col

(a1(0)u(t, 0)− ∂

∂x(a2u)(t, 0),−a1(1)u(t, 1) +

∂

∂x(a2u)(t, 1),

a2(0)u(t, 0),−a2(1)u(t, 1))

, S(w′(t)))∈

∈ −G

col

(v(t, 0), v(t, 1),

∂v

∂x(t, 0),

∂v

∂x(t, 1)

)

w(t)

, t > 0

and the initial data

(IC)

u(0, x) = u0(x), v(0, x) = v0(x), 0 < x < 1w(0) = w0.

127

128 Rodica Luca–Tudorache

Here S is a positive diagonal m-matrix and G is an operator in the space IRm+4

which satisfies some appropriate assumptions. This problem has applicationsin the theory of integrated circuits and in the elastic beam theory (see [2,5,7]for further references). For A = B ≡ 0 the above problem has been studiedin [3] concerning the existence, uniqueness and regularity properties of thesolutions, and in [4,2] for the asymptotic behavior and the existence of periodicsolutions. In this paper we deduce some existence results for the weak andstrong solutions for the problem (S)+(BC)+(IC), by applying several theoremsfrom [8]. For the basic concepts and results in the theory of nonlinear evolutionequations of monotone type and of Volterra integrodifferential equations werefer the reader to [1,6,8].

Introduce the assumptions used in the sequel.(H1) The functions ak ∈ W k,∞(0, 1), k = 0, 2 and a2(x) 6= 0, ∀x ∈ [0, 1].(H2) The functions x → α(x, p) and x → β(x, p) are in L2(0, 1) for any fixed

p ∈ IR. Moreover, the functions p → α(x, p) and p → β(x, p) are continuousand nondecreasing from IR into IR, for a.a. x ∈ (0, 1).

(H3) The functions b, c : [0, a] → IR are continuous (a > 0).(H4) The functions A, B : IR → IR are continuous and there exist the

constants d, d > 0 and e, e ∈ IR such that|A(u)| ≤ d|u|+ e, |B(u)| ≤ d|u|+ e, ∀u ∈ IR.

(H5) G : D(G) ⊂ IRm+4 → IRm+4 is a maximal monotone mapping (possi-

bly multivalued), D(G) 6= ∅. Moreover, G =(

G11 G12

G21 G22

)with

G11 : D(G11) ⊂ IR4 → IR4, G12 : D(G12) ⊂ IRm → IR4,G21 : D(G21) ⊂ IR4 → IRm, G22 : D(G22) ⊂ IRm → IRm.

(H6) S = diag(s1, . . . , sm) with sj > 0, j = 1,m.The assumption (H5) is a technical one; it is automatically satisfied if G is

a matrix.

2. PRELIMINARY RESULTSLet us write the problem (S)+(BC)+(IC) as a Volterra integrodifferential

problem in a certain Hilbert space. For this, let us consider the spaces X =(L2(0, 1))2, IRm and Y = X × IRm with the corresponding scalar products

〈f, g〉X =∫ 1

0f1(x)g1(x) dx+

∫ 1

0f2(x)g2(x) dx, f = col (f1, f2), g = col (g1, g2) ∈

X,

〈x, y〉s =m∑

i=1

sixiyi, x, y ∈ IRm,

Existence results for a class of nonlinear Volterraintegrodifferential problem 129

〈(

fx

),

(gy

)〉Y = 〈f, g〉X + 〈x, y〉s,

(fx

),

(gy

)∈ Y .

Now we define the operator A : D(A) ⊂ Y → Y ,D(A) =

y = col (u, v, w); u, v ∈ H2(0, 1), w ∈ IRm, col(δ0v, w) ∈ D(G),

δ1u ∈ −G11(δ0v)−G12(w)

,where w = col (w1, . . . , wm), δ0v = col (v(0), v(1), v′(0), v′(1)), δ1u = col (a1(0)u(0)−(a2u)′(0),−a1(1)u(1) + (a2u)′(1), a2(0)u(0),−a2(1)u(1)),

A

uvw

=

a2v′′ + a1v

′ + a0v−(a2u)′′ + (a1u)′ − a0u

S−1G21(δ0v) + S−1G22(w)

,

uvw

∈ D(A).

We also define the operator B : D(B) ⊂ Y → Y , B(col (u, v, w)) =col (α(·, u), β(·, v),0), col (u, v, w) ∈ D(B), with D(B) = y = col (u, v, w) ∈ Y, B(y) ∈ Y .

If the assumptions (H1), (H2), (H5) and (H6) hold, then D(A) 6= ∅, D(A) =X ×D(G12) ∩D(G22) and D(A) ⊂ D(B) [3].

Lemma 2.1 If (H1), (H2), (H5) and (H6) hold, then the operator A + B ismaximal monotone.

For the proof of Lemma 1 see [3].

Lemma 2.2 If (H1), (H2), (H5) and (H6) hold, then for every λ > 0 theoperator (I + λ(A + B))−1 is compact in Y .

Proof. Without loss of generality we suppose that G is single-valued. Let beλ > 0 and M = col (f i, gi, hi); i ∈ I a bounded set in Y . We prove that theset (I + λ(A + B))−1(M) is bounded in (H2(0, 1))2 × IRm. For this purpose,we firstly remark that the set col (pi, qi, ri); i ∈ I is bounded in Y , where

pi

qi

ri

= (I+λ(A+B))−1

f i

gi

hi

⇔

f i

gi

hi

=

pi

qi

ri

+λ(A+B)

pi

qi

ri

, i ∈

I,(the operator (I + λ(A + B))−1 is nonexpansive in Y ).

The last relation is equivalent to the system

1λ

pi + a2(qi)′′ + a1(qi)′ + a0qi + α(x, pi(x)) =

1λ

f i

1λ

qi − (a2pi)′′ + (a2p

i)′ − a0pi + β(x, qi(x)) =

1λ

gi

1λ

rij + [S−1G21(δ0q

i) + S−1G22(ri)]j =1λ

hij , j = 1,m

δ1pi = −G11(δ0q

i)−G12(ri).

(1)

We use similar arguments as those used in [3, Theorem 1] and by (1) wededuce that the sets −(a2p

i)′′+(a1pi)′−a0p

i; i ∈ I and a2(qi)′′+a1(qi)′+


a0qi; i ∈ I are bounded in L1(0, 1). Then, by [7, Lemma B] it follows that

the sets(pi)(j); i ∈ I and (qi)(j); i ∈ I, j = 0, 1 are bounded in C([0, 1])

and the sets(pi)′′; i ∈ I and (qi)′′; i ∈ I are bounded in L1(0, 1).

By (H2) we deduce that the sets α(·, pi); i ∈ I, β(·, qi); i ∈ I arebounded in L2(0, 1) and then (1)1,2 give us that the sets (pi)′′; i ∈ I and(qi)′′; i ∈ I are bounded in L2(0, 1). So, the set col (pi, qi, ri); i ∈ I isbounded in (H2(0, 1))2 × IRm and therefore the operator (I + λ(A + B))−1 iscompact in Y . Q.E.D.

We define now the operator K : [0, a] → L(Y ),

K(t) =

b(t) 0 00 c(t) 00 0 0

I, ∀ t ∈ [0, a],

(obviously, for every t ∈ [0, a], K(t) ∈ L(Y )) and the operator C : Y → Y ,

C

uvw

=

A(u)B(v)

0

,

where A, B : L2(0, 1) → ÃL2(0, 1) are defined by A(u)(x) = A(u(x)), B(v)(x) =B(v(x)), for a.a. x ∈ (0, 1).

Using the above operators, problem (S)+(BC)+(IC) can be equivalentlyexpressed as a nonlinear Volterra integrodifferential problem in the space Y

(P)

dy

dt(t) + (A + B)(y(t)) +

∫ t

0K(t− s)C(y(s)) ds 3 F (t, ·), t > 0,

y(0) = y0,

where y(t) = col (u(t), v(t), w(t)), F (t, ·) = col (f(t, ·), g(t, ·), 0), y0 = col (u0, v0, w0).

Definition 2.1 A function y = col(u, v, w) : [0, b] → Y (b > 0) is called a weak(strong) solution on [0, b] to the problem (P) with y0 = col(u0, v0, w0) ∈ D(A)if y(t) ∈ D(A), for a.a. t ∈ (0, b), the function h : [0, b] → Y defined by

h(t) = −∫ t

0K(t− s)C(y(s))ds + F (t, ·), ∀ t ∈ [0, b]

belongs to L1(0, b; Y ) and y is a weak (respectively strong) solution in theknown sense [1] to the problem

dy

dt(t) + (A + B)(y(t)) 3 h(t), 0 ≤ t ≤ b

y(0) = y0.

For the definition in a general case (for mild solutions in a Banach space) see[8].


We say that y = col(u, v, w) is a weak (strong) solution of the problem(S)+(BC)+ (IC) if y is a weak (respectively strong) solution in the abovesense to the problem (P).

3. EXISTENCE OF WEAK AND STRONGSOLUTIONS TO PROBLEM (S)+(BC)+(IC)

First we present an existence result for the weak solutions of our problem.

Theorem 3.1 Suppose that the assumptions (H1)-(H6) hold. If col(u0, v0, w0) ∈D(A), f, g ∈ L1

loc(IR+; L2(0, 1)), then there exists T > 0, T ∈ (0, a] such thatthe problem (S)+(BC)+(IC) has at least one weak solution y = col(u, v, w) ∈C([0, T ];Y ).

Proof. Under the assumption (H3) the operator K is continuous. Indeed, lett0 ∈ [0, a] and tj → t0, as j →∞, in IR. Then‖K(tj)−K(t0)‖L(Y ) = sup

‖y‖Y ≤1‖(K(tj)−K(t0))(y)‖Y ≤

≤ sup‖y‖Y ≤1

[‖[b(tj)− b(t0)]u‖L2(0,1) + ‖[c(tj)− c(t0)]v‖L2(0,1)

]=

= sup‖y‖Y ≤1

[| b(tj)− b(t0) | ·‖u‖L2(0,1)+ | c(tj)− c(t0) | ·‖v‖L2(0,1)

] ≤≤| b(tj)− b(t0) | + | c(tj)− c(t0) |→ 0, as t →∞

(b, c are continuous in t0). This yieldsK(tj) → K(t0), as j →∞, in L(Y ).

If the assumption (H4) holds, then the operator C is well-defined, that is,for every y ∈ Y it follows that C(y) ∈ Y . Indeed, let y = col(u, v, w) be anarbitrary element from Y , fixed for the moment. Then the functions A(u) andB(v) are measurable and

‖C(y)‖Y =√‖A(u)‖2

L2(0,1)+ ‖B(v)‖2

L2(0,1)=

√∫ 1

0|A(u(x))|2 dx +

∫ 1

0|B(v(x))|2 dx ≤

≤√

2∫ 1

0(d2|u(x)|2 + e2) dx + 2

∫ 1

0(d|v(x)|2 + e2) dx < +∞,

(u, v ∈ L2(0, 1)). Therefore C(y) ∈ Y . Moreover, the operator C is continuous,that is if yj → y, as j →∞ in Y then C(yj) → C(y), as j →∞.

We shall prove that if uj → u as j →∞ in L2(0, 1), then A(uj) → A(u) asj →∞, in L2(0, 1). The proof of this conclusion is based on Vitali’s theorem:first, we show that A(uj) → A(u), as j → ∞, in measure and then, using(H4) we show that the sequence (A(uj) − A(u))2 is uniformly integrable.Then, it follows that (A(uj) − A(u))2 → 0, as j → ∞ in L1(0, 1), henceA(uj)− A(u) → 0, as j →∞, in L2(0, 1).


By Lemma 1 and Lemma 2 the operator A + B is maximal monotone andfor every λ > 0 the operator (I +λ(A+B))−1 is compact in the space Y . Now,using [8, Theorem 5.1.1] we deduce that for col(u0, v0, w0) ∈ D(A), f, g ∈L1

loc(IR+; L2(0, 1)), there exists T > 0, T ∈ (0, a] such that the problem(P) (and also the problem (S)+(BC)+(IC)) has at least one weak solutiony = col(u, v, w) ∈ C([0, T ];Y ). Q.E.D.

By Lemma 1, Lemma 2 and [8, Corollary 5.1.1] we obtain

Corollary 3.1 Suppose that the assumptions (H1), (H2), (H4)-(H6) hold. Ifb, c ∈ C1([0, a]), y0 = col(u0, v0, w0) ∈ D(A), f, g ∈ W 1,1

loc (IR+; L2(0, 1)),then there exists T > 0, T ∈ (0, a] such that the problem (S)+(BC)+(IC)has at least one strong solution y = col(u, v, w) ∈ W 1,∞(0, T ; Y ), that isy(t, ·) ∈ D(A), ∀ t ∈ [0, T ), y satisfies the equations of the system (S), theboundary condition (BC) and the initial data (IC) for a.a. x ∈ (0, 1) and forall t ∈ [0, T ), where ∂u/∂t, ∂v/∂t and ∂w/∂t are replaced by ∂+u/∂t, ∂+v/∂t,respectively d+w/dt. Moreover u, v ∈ L∞(0, T ; H2(0, 1)).

Now we introduce the assumptions(H3)’ The functions b, c : [0,∞) → IR are continuous.(H7) There exists at least one solution y = col(u, v, w) ∈ D(A) to the

equationA(y) + B(y) 3 0. (2)

Theorem 3.2 Assume that the assumptions (H1)-(H2), (H3)’, (H4)-(H7)hold. If col(u0, v0, w0) ∈ D(A), f, g ∈ L1

loc(IR+; L2(0, 1)), then the prob-lem (S)+(BC)+(IC) has at least one weak solution y = col(u, v, w) defined onthe positive half-axis.

Proof. By (H7) we have that F = (A+B)−1(0) 6= ∅ and using (H4) we obtain

‖C(y)‖Y ≤√

d0

(‖u‖2

L2(0,1)+ ‖v‖2

L2(0,1)

)+ e0 ≤ d0‖y‖Y + e0, ∀ y ∈ Y ,

where d0 = 2 maxd2, d2, e0 = 2(e2 + e2), d0 =√

d0, e0 =√

e0.In what follows, we shall prove that every weak solution of the problem

(P), col(u, v, w) : [0, l) → D(A) (l > 0) is bounded on [0, l). For, let γ =col(p, q, r) ∈ F . We multiply the equation (P)1 by y(t)− γ in the space Y

12

d

dt‖y(t)−γ‖2

Y + <

∫ t

0K(t−s)C(y(s)) ds, y(t)−γ >Y≤< F (t, ·), y(t)−γ >Y

⇒ 12

d

dt‖y(t) − γ‖2

Y ≤(∫ t

0‖K(t− s)‖L(Y )‖Cy(s)‖Y ds + ‖F (t, ·)‖Y

)‖y(t) −

γ‖Y

⇒ 12

d

dt‖y(t)− γ‖2

Y ≤[∫ t

0(|b(t− s)|+ |c(t− s)|)

(d0‖y(s)‖Y + e0

)ds+


+‖F (t, ·)‖Y

]‖y(t)− γ‖Y , 0 < t < l

⇒ d

dt‖y(t)−γ‖Y ≤

∫ t

0

(|b(t− s)|+ |c(t− s)|)(d0‖y(s)− γ‖Y + d0‖γ‖Y + e0

)ds+

+∥∥∥∥(

f(t, ·)g(t, ·)

)∥∥∥∥X

, 0 < t < l.

We integrate the above inequality over [0, t]

‖y(t) − γ‖Y ≤ ‖y0 − γ‖Y +∫ t

0

∫ s

0[|b(s − θ)| + |c(s − θ)|](d0‖y(θ) − γ‖Y +

d0‖γ‖Y +

+e0) dθ ds +∫ t

0

∥∥∥∥(

f(s, ·)g(s, ·)

)∥∥∥∥X

ds, 0 ≤ t < l.

From the obtained inequality, using the assumptions of the theorem we deducethat there exist the constants C1, C2 > 0 such that

‖y(t)− γ‖Y ≤ C1 + C2

∫ t

0‖y(s)− γ‖Y ds,

(C1, C2 are dependent on b, c|[0,l], γ and l).Now, by Bellman’s inequality, it follows that y is bounded on [0, l). Us-

ing Lemma 1, Lemma 2 and [8, Theorem 5.1.3] we deduce that the problem(S)+(BC)+(IC) has at least one weak solution defined on the positive half-axis(that is a weak solution from Theorem 1 can be extended on [0,∞)). Q.E.D.

Remark. If, in addition to the assumptions (H1)-(H2), (H5)-(H6), wesuppose that

a) there exist m1, m2 > 0 such that(α(x, p1)− α(x, p2))(p1 − p2) ≥ m1(p1 − p2)2,(β(x, p1)− β(x, p2))(p1 − p2) ≥ m2(p1 − p2)2,

for a.a. x ∈ (0, 1) and for all p1, p2 ∈ IR,b) there exists M > 0 such that for all x, y ∈ D(G), x = col(xa, xb),

y = col(ya, yb) ∈ IR4 × IRm and w1 ∈ G(x), w2 ∈ G(y) we have< w1 − w2, x− y >IRm+4≥ M‖xb − yb‖2

IRm ,then the equation (2) has a unique solution y = col(u, v, w) ∈ D(A) and (H7)is fulfilled. Indeed, under the above assumptions the operator A+B is stronglymonotone, so it is coercive. Therefore R(A+B) = Y and we obtain the aboveconclusion.

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SURVEY ON COMPUTER REALIZATION OFBRANCHING EQUATION CONSTRUCTIONON ALLOWED GROUP SYMMETRY ANDSUBGROUP INVARIANT SOLUTIONS

ROMAI J., 2, 1(2006),135–148

Oleg V. MakeevUlyanovsk State Technical University, Russia

Abstract In order to study bifurcation problems, the nonlinear equations of station-ary and dynamical branching are written in operator form in Banach spaces.The Lyapounov-Schmidt asymptotical method reduces their solving to the con-struction and investigation of finite-dimensional nonlinear algebraic systems.Multiple degeneracy of linearized operator frequently stipulated by the groupof symmetries of the nonlinear problem entails technical difficulties at the con-struction of the branching equation (BEq). Herein the computer programs aresuggested for the construction of the general form of the Lyapounov-Schmidtbranching equation both for stationary and Andronov-Hopf bifurcation on theallowed group of symmetries with its subsequent investigation. Special at-tention is paid to planar and spatial crystallographic groups. For a discretegroup of symmetries the program for the determination of subgroup structureand dual to it structure of branching systems is presented with applications tononlinearly perturbed Helmholtz equation.

1. INTRODUCTIONMany applied problems of critical phenomena can be investigated as bifurca-

tional symmetry breaking problems. At multivariate branching the nonlinearproblem often has one or several families of solutions depending on free para-meters. The existence of such solutions as a rule is related to the presence ofthe original problem symmetry. In bifurcational symmetry breaking problemsthe original nonlinear equation is invariant with respect to the Euclidean spaceRs (s ≥ 1) motion group and its solution invariant with respect to this groupis the rest state or uniform motion. At the loss of stability there arise solutionswith cellular structure, i.e. periodical solutions with crystallographic group ofsymmetries (semi-direct product G = G1 o G1 of s–parametrical continuousshift group G1(α1, . . . , αs) and the group G1 of the elementary cell of period-icity), which are mutually transformed by the action of the group G1. Theseproblems can be solved by the construction and investigation of the relevantbranching equation (BEq).

135

136 Oleg V. Makeev

The general scheme for the construction of sufficiently smooth (analytical)BEq of a general form on the allowed group of symmetries both for stationaryand dynamic situations was developed in a series of B. V. Loginov’s papers [2,9] on the base of group analysis methods for differential equations [8]. However,at the realization of this scheme to concrete applications serious difficultieshave arisen. Usually, because of the high order of degeneracy, undergone bythe linear operator at the bifurcation point, the investigator can not accountfor all possible factors and it is impossible to perform this scheme in manual.Whence the need for a computer realization of the indicated scheme. Such arealization is considered here. Computer programs for BEq construction andinvestigation are realized for BEq with simple cubic lattice symmetry. Theapplications to statistical crystal theory are known [2, 3, 9].

In various applications the problem of subgroup invariant solutions arises.Computer program is proposed by which all subgroups of discrete symmetrieswith known composition law can be found. Here it is applied to the concreteproblem of the nonlinearly perturbed Helmholtz equation for square latticesymmetry.

The base consists of the group symmetry inheritance theorem for branchingequation and the methods of group analysis of differential equations. Partially,the description of the proposed programs of the construction of general BEqis presented in [5]. In the following the terminology, designation of indicatedworks and results of program application are used without any additionalexplanation. For the sake of simplicity of presentation here only dynamic(Andronov-Hopf) bifurcation is considered.

Let E1 and E2 be two Banach spaces. The problem about periodical solu-tions of differential equation with a small parameter ε reads

Adx

dt= Bx−R(x, ε), R(0, ε) ≡ 0, Rx(0, 0) = 0. (1)

Here A and B are densely defined closed linear Fredholm operators A : DA ⊂E1 → E2, B : DB ⊂ E1 → E2. Moreover DA ⊃ DB and A is subordinated toB (i.e. ‖Ax‖E2 ≤ ‖Bx‖E2 +‖x‖E1 on DB) or DB ⊃ DA and B is subordinatedto A (i.e. ‖Bx‖E2 ≤ ‖Ax‖E2 + ‖x‖E1 on DA), R (x, ε) is nonlinear operatorsufficiently smooth in a neighborhood of (0, 0) ⊂ E1+R1.

The theorem about symmetry inheritance by relevant Lyapounov-SchmidtBEq 0 = f(ξ, ξ, µ, ε) = fj(ξ, ξ, µ, ε)n

1 : Ξn → Ξn at Andronov-Hopf bifurca-tion is expressed by the equality

f(Agξ, Agξ, µ, ε) = Bgf(ξ, ξ, µ, ε). (2)

Here Ag and Bg are n-dimensional representations of the group G, which isallowed by the original nonlinear equation, in the zero-subspace N and defectsubspace N∗ of the operator linearized at the bifurcation point .

Computer realization of the branching equation construction 137

The equality (2) means that the manifold F : f − f(ξ, ξ) = 0 is invari-ant to the transformation group ξ = Agξ, f = Bgf . Consequently, for theconstruction of the general BEq on the allowed group of symmetries the Lie-Ovsyannikov invariants and invariant manifolds techniques [8] may be applied.

If the left-hand side of the branching system is assumed to be continuous onξ-variables, then for the construction of the general BEq it is sufficient to haveonly a complete system of functional independent invariants. In the analyticcase, at the BEq expansion on homogeneous forms not all invariants can beexpressed via powers of basic ones. But the use of additional invariants leads tothe repetition of BEq summands. Therefore, when using additional invariants,the expansion of the BEq by invariant monomials should be factorized subjectto constraints between the used invariants. This factorization with respect tothe expression inside the brackets will be designated by the symbol [· · · ]out.

Thus, according to the group-analysis scheme of the BEq, for the construc-tion of its general form one must: a) find the complete system of functionallyindependent invariants, allowed by BEq continuous group, b) introduce theadditional invariants, c) establish the constraints between the used invariantsand d) make the factorization of the BEq expansion by these constraints, i.e.construct the symbol [· · · ]out. All tasks of this logic-combinatorial problemare realized by means of computer programs for examples of construction ofBEqs, which admit the simple cubic lattice symmetry.

The obtained results are entered into RFBR- and INTAS-06 applications.

2. LYAPOUNOV-SCHMIDT BEQCONSTRUCTION WITH SIMPLE CUBICLATTICE SYMMETRY

Here the construction of BEq for symmetry breaking problems of Andronov-Hopf bifurcation is presented when spatial cell of periodicity is an octahedron.Let dim N = 12 for the choice of its basis of the form ϕj = ei[〈lj ,q〉+t], ϕj , 6

1

q = (x1, x2, x3) with inverse lattice vectors l2k−1 = 2πek, l2k = −l2k−1, k =1, 2, 3. The BEq is invariant with respect to the reflection-rotation octahedrongroup, which is generated by

C(1)4∼= (1)(2)(3546), C

(2)4∼= (1625)(3)(4),

C(3)4∼= (1324)(5)(6), J ∼= (12)(34)(56).

The BEq continuous symmetry group has the form

Ag(α) = diagei(2πα1+α0), e−i(2πα1+α0), ei(−2πα1+α0), e−i(−2πα1+α0), ei(2πα2+α0),

e−i(2πα2+α0), ei(−2πα2+α0), e−i(−2πα2+α0), ei(2πα3+α0), e−i(2πα3+α0),

ei(−2πα3+α0), e−i(−2πα3+α0).

138 Oleg V. Makeev

Fig. 1. Octahedron lattice.

By differentiating on α0, . . . , α3 and reducing on common multiplier we receivethe Lie algebra basis that corresponds to rotations in the coordinate planes(ξ2k−1, ξ2k), k = 1, 2, 3

X0 = ( ξ1, −ξ1, ξ2, −ξ2, ξ3, −ξ3, ξ4, −ξ4, ξ5, −ξ5, ξ6, −ξ6 ),X1 = ( ξ1, −ξ1, −ξ2, ξ2, 0, 0, 0, 0, 0, 0, 0, 0 ),X2 = ( 0, 0, 0, 0, ξ3, −ξ3, −ξ4, ξ4, 0, 0, 0, 0 ),X3 = ( 0, 0, 0, 0, 0, 0, 0, 0, ξ5, −ξ5, −ξ6, ξ6 ).

Then the basic invariants are defined by the next system of first order partialdifferential equations

Xiν

∂I

∂ξi+ F j

ν

∂I

∂fj= 0, ν = 0, 1, 2, 3. (3)

Hereinafter in every monomial expression the symbol ξ will be omitted, i.e.for example, the notation ξ1ξ2ξ3ξ4 = 1 2 3 4 is used. The computer programselects six invariants of second order ξkξk, k = 1, . . . , 6 and next six invariantsof fourth order

1 2 3 4; 1 2 5 6; 1 2 3 4;1 2 5 6; 3 4 5 6; 3 4 5 6.


Three standard constraints between fourth order invariants

1 2 3 4× 1 2 5 6 = 1 1× 2 2× 3 4 5 6,1 2 3 4× 3 4 5 6 = 3 3× 4 4× 1 2 5 6,1 2 5 6× 3 4 5 6 = 5 5× 6 6× 1 2 3 4

are separating the system of three fourth order invariants that are used forthe BEq general construction

1 2 3 4, 1 2 5 6, 3 4 5 6.

This system is subordinated to one nonstandard constraint

1 2 3 4× 1 2 5 6× 3 4 5 6 = 1 1× 2 2× 3 3× 4 4× 5 5× 6 6.

Thus the BEq corresponding to adopted symmetry takes the form f1(ξ, ξ, µ, ε)

≡∑pα;qβ

apαqβ(ξ1ξ1)p1 . . . (ξ6ξ6)p6 [ξ1(ξ1ξ2ξ3ξ4)q1(ξ1ξ2ξ5ξ6)q2(ξ3ξ4ξ5ξ6)q3 ]out = 0.

The symbol [· · · ]out occurs in the following way

f1(ξ, ξ, µ, ε) ≡∑

pα;qβ ;k1;k2

(ξ1ξ1)p1 . . . (ξ6ξ6)p6 [ap;0ξ1+ap;k1ξ1(ξ3ξ4ξ5ξ6)k1+

+bp;k1ξ1(ξ3ξ4ξ5ξ6)k1 + cp;k1 ξk1−11 (ξ2ξ5ξ6)k1 + dp;k1ξ

k1+11 (ξ2ξ5ξ6)k1+

+ep;k1ξk1+11 (ξ2ξ3ξ4)k1+fp;k1 ξ

k1−11 (ξ2ξ3ξ4)k1+gp;k1,k2 ξ

k1−11 ξk1

2 (ξ3ξ4)k1+k2(ξ5ξ6)k2+

+hp;k1,k2ξk1+11 ξk1

2 (ξ3ξ4)k2(ξ5ξ6)k1+k2 + mp;k1,k2ξk2−11 ξk2

2 (ξ3ξ4)k1(ξ5ξ6)k1+k2+

+np;k1,k2 ξk1+k2−11 ξk1+k2

2 (ξ3ξ4)k1(ξ5ξ6)k2+rp;k1,k2ξk1+k2+11 ξk1+k2

2 (ξ3ξ4)k2(ξ5ξ6)k1+

+up;k1,k2ξk2+11 ξk2

2 (ξ3ξ4)k1+k2(ξ5ξ6)k1 ] = 0, f2j(ξ, ξ, µ, ε) ≡ Jf2j−1(ξ, ξ, µ, ε) = 0,

j = 1, 2, 3, f2j−1(ξ, ξ, µ, ε) ≡ pj−1f1(ξ, ξ, µ, ε) = 0, j = 2, 3 (4)

where pj consecutively takes the values C(3)4 , C

(2)3

4 . The first equationof the written branching system is invariant with respect to the octahedronsubgroup generated by substitutions e, C

(1)4 , C

(1)2

4 , C(1)3

4 , J C(2)2

4 , J C(3)2

4 ,J u13 = J C

(1)4 C

(2)2

4 , J u58 = J C(1)4 C

(3)2

4 that preserve the number1 that gives the symmetry relation between its coefficients.

140 Oleg V. Makeev

3. BEQ INVESTIGATIONAt the Andronov-Hopf bifurcation investigation (i.e. in dynamical branch-

ing) difficulties arise too. They are generated by the fact that the branchingsystem includes, except for n complex variables, the unknown small additionalcontribution µ to the oscillation frequency that defines the limit cycle onset.All these variables should be calculated from the branching system. In theV. I. Yudovich articles (for example see [7]) the asymptotic approach for thedetermination of periodical branching solution was suggested. It is based onsubspaces invariant techniques relatively to the BEq left-hand side. After thesolutions determination in a certain subspace as a consequence of group sym-metry we have the solutions as trajectories of this subspace. Hence there arisethe need of computer program for the construction of the subspace system.For the description of this program we refer to [5].

In the example of BEq (2.1) with simple cubic lattice symmetry in caseof the octahedron cell the computer program gives 1795 invariant subspaces.However their complete system consists of the following 46 subspacesΞ1 = (ξ1, 0, 0, 0, 0, 0), Ξ2 = (ξ1, ξ1, 0, 0, 0, 0), Ξ3 = (ξ1, 0, ξ1, 0, 0, 0), Ξ4 =(ξ1, ξ2, 0, 0, 0, 0), Ξ5 = (ξ1, 0, ξ2, 0, 0, 0), Ξ6 = (ξ1, ξ1, ξ2, 0, 0, 0), Ξ7 = (ξ1, 0, ξ1, 0, ξ2, 0),Ξ8 = (ξ1, 0, ξ1, ξ1, 0, 0), Ξ9 = (ξ1, 0, ξ1, 0, ξ1, 0), Ξ10 = (ξ1, ξ2, ξ3, 0, 0, 0), Ξ11 =(ξ1, 0, ξ2, ξ3, 0, 0), Ξ12 = (ξ1, 0, ξ2, 0, ξ3, 0), Ξ13 = (ξ1, ξ1, ξ1, ξ1, 0, 0), Ξ14 =(ξ1, ξ1, ξ1, 0, ξ1, 0), Ξ15 = (ξ1, ξ1, ξ2, ξ3, 0, 0), Ξ16 = (ξ1, ξ1, ξ2, 0, ξ3, 0), Ξ17 =(ξ1, 0, ξ1, 0, ξ2, ξ3), Ξ18 = (ξ1, ξ2, ξ1, ξ1, 0, 0), Ξ19 = (ξ1, ξ2, ξ1, 0, 0, ξ1), Ξ20 =(ξ1, 0, ξ1, ξ1, ξ2, 0), Ξ21 = (ξ1, ξ1, ξ2, ξ2, 0, 0), Ξ22 = (ξ1, ξ1, ξ2, 0, ξ2, 0), Ξ23 =(ξ1, 0, ξ1, 0, ξ2, ξ2), Ξ24 = (ξ1, ξ2, ξ3, ξ4, 0, 0), Ξ25 = (ξ1, ξ2, ξ3, 0, ξ4, 0), Ξ26 =(ξ1, 0, ξ2, ξ3, ξ4, 0), Ξ27 = (ξ1, 0, ξ2, 0, ξ3, ξ4), Ξ28 = (ξ1, 0, ξ1, ξ1, ξ2, ξ2), Ξ29 =(ξ1, 0, ξ1, ξ1, ξ1, ξ1), Ξ30 = (ξ1, ξ2, ξ1, ξ1, ξ3, 0), Ξ31 = (ξ1, 0, ξ1, ξ1, ξ2, ξ3), Ξ32 =(ξ1, ξ1, ξ2, ξ2, ξ3, 0), Ξ33 = (ξ1, ξ1, ξ2, ξ3, ξ4, 0), Ξ34 = (ξ1, ξ1, ξ2, 0, ξ3, ξ4), Ξ35 =(ξ1, ξ2, ξ3, ξ4, ξ5, 0), Ξ36 = (ξ1, ξ2, ξ3, 0, ξ4, ξ5), Ξ37 = (ξ1, 0, ξ2, ξ3, ξ4, ξ5), Ξ38 =(ξ1, ξ1, ξ2, ξ2, ξ3, ξ3), Ξ39 = (ξ1, ξ1, ξ2, ξ2, ξ3, ξ4), Ξ40 = (ξ1, ξ1, ξ1, ξ1, ξ2, ξ2), Ξ41 =(ξ1, ξ2, ξ1, ξ1, ξ1, ξ1), Ξ42 = (ξ1, ξ1, ξ1, ξ1, ξ1, ξ1), Ξ43 = (ξ1, ξ2, ξ1, ξ1, ξ2, ξ2), Ξ44 =(ξ1, ξ3, ξ1, ξ1, ξ2, ξ2), Ξ45 = (ξ1, ξ1, ξ2, ξ3, ξ4, ξ5), Ξ46 = (ξ1, ξ2, ξ3, ξ4, ξ5, ξ6).

The main part of the BEq (2.1) has the formiµξ1+aξ1ε+bξ1|ξ1|2+cξ1|ξ2|2+dξ1(|ξ3|2+|ξ4|2+|ξ5|2+|ξ6|2)+eξ2(ξ5ξ6+ξ3ξ4)+· · · = 0;iµξ2+aξ2ε+bξ2|ξ2|2+cξ2|ξ1|2+dξ2(|ξ3|2+|ξ4|2+|ξ5|2+|ξ6|2)+eξ1(ξ5ξ6+ξ3ξ4)+· · · = 0;iµξ3+aξ3ε+bξ3|ξ3|2+cξ3|ξ4|2+dξ3(|ξ1|2+|ξ2|2+|ξ5|2+|ξ6|2)+eξ4(ξ5ξ6+ξ1ξ2)+· · · = 0;iµξ4+aξ4ε+bξ4|ξ4|2+cξ4|ξ3|2+dξ4(|ξ2|2+|ξ1|2+|ξ5|2+|ξ6|2)+eξ2(ξ5ξ6+ξ1ξ2)+· · · = 0;iµξ5+aξ5ε+bξ5|ξ5|2+cξ5|ξ6|2+dξ5(|ξ4|2+|ξ3|2+|ξ1|2+|ξ2|2)+eξ6(ξ1ξ2+ξ3ξ4)+· · · = 0;iµξ6+aξ6ε+bξ6|ξ6|2+cξ6|ξ5|2+dξ6(|ξ3|2+|ξ4|2+|ξ1|2+|ξ2|2)+eξ5(ξ1ξ2+ξ3ξ4)+· · · = 0.

As an example of invariant subspaces techniques consider this branchingsystem in the subspace Ξ21 = (ξ1, ξ1, ξ2, ξ2, 0, 0)

iµξ1 + aξ1ε + (b + c)ξ1|ξ1|2 + 2dξ1|ξ2|2 + eξ1ξ22 = 0,


iµξ2 + aξ2ε + (b + c)ξ2|ξ2|2 + 2dξ2|ξ1|2 + eξ21 ξ2 = 0. (5)

Assume that ξ1 6= 0, ξ2 6= 0, and let us search for the solutions in the formξ1 = r1ε

1/2 exp(iθ1), ξ2 = r2ε1/2 exp(iθ2), µ = νε. Dividing the first equation

by ε3/2r1 exp(iθ1), the second one by ε3/2r2 exp(iθ2), we get the system

(b + c)r21 + (2d + e exp(iα))r2

2+ = −iν − a,

(2d + e exp(−iα))r21 + (b + c)r2

2 = −iν − a,

α = 2(θ2 − θ1),

whencer21 =

∆1

∆=

∆1∆|∆|2 , r2

2 =∆2

∆=

∆2∆|∆|2 ,

where∆ = (b + c)2 − (2d− b− c + e exp(iα))(2d− b− c + e exp(−iα)),∆1 = (iν + a)(2d− b− c + e exp(iα)),∆2 = (iν + a)(2d− b− c + e exp(−iα)).

Since r21 and r2

2 are real, it follows the next system of two equations forthe determination of ν and α: Im (∆1∆) = Im (∆2∆) = 0, Im(∆1∆) =Im ((b + c)2 − 4d2 − e2 − 4de cosα)·(iν+a)(2d−b−c+e(cosα+i sinα)) = 0,

Im(∆2∆) = Im ((b + c)2 − 4d2 − e2 − 4de cosα)·(iν+a)(2d−b−c+e(cosα−−i sinα)) = 0.

This system reads, equivalently, in the form

pν + (u + sν) cos α + (v + rν) sin α + q = 0,

pν + (u + sν) cos α− (v + rν) sin α + q = 0,

wherep = Re(2d− b− c), u = Im(a e), s = Re e,q = Im(a(2d− b− c)), v = Re(a e), r = −Im e.

After the determination of ν from every equation of the system, it follows thesystem for the determination of α, i.e.

ν = −q + u cosα + v sinα

p + s cosα + r sinα= −q + u cosα− v sinα

p + s cosα− r sinα,

Hereinafter by using the scheme [1] in Maple 6.0 one gets

α = 0, π, ±Arg

(√2 qrvp− v2p2 + u2r2 − q2r2 + v2s2 − 2urvs

ur − vs− i

qr − vp

ur − vs

).

142 Oleg V. Makeev

Finally, |ξ1| = ∆1∆ , |ξ2| = ∆2

∆ .The construction of the asymptotic periodical branching solution in other

subspaces is made as indicated above and it is not given here in view of itsawkwardness.

4. SOLUTIONS INVARIANT TO SUBGROUPSOF DISCRETE SYMMETRIES

The general theory for finding the subgroup invariant solutions of ordinarydifferential equations was suggested in [10]. Then this theory was developedfor bifurcation problems and it was realized in concrete problems for the con-struction of solutions invariant to normal divisors of discrete symmetry [2].Here this theory is realized in the form of computer program for the construc-tion of the subgroup structure for the discrete symmetry of nonlinear equationwith subsequent formation of the relevant branching systems of solutions in-variant to arbitrary subgroups. As a simple illustration of this abstract theorywe consider here the equations

∆ u + λ2 sinh u = 0, (6)

∆ u + λ2 sin u = 0 (7)

and search for periodic solutions with square lattice of periodicity.Applications of these equations to low temperature plasma theory are given

in [6].At the choice of the base N(B) = ϕj4

1 = 12aeiλ0(x+y), 1

2ae−iλ0(x+y),12aeiλ0(y−x), 1

2ae−iλ0(y−x) the four dimensional BEq inheriting the square sym-metry group D4 has the form [9]

f1(ξ, ε) ≡ a0(ε)ξ1 +∑

|p|≥2

ap1p2(ε)ξ1(ξ1ξ2)p1(ξ3ξ4)p2 = 0,

f2(ξ, ε) ≡ r2 f1(ξ, ε) = f1(r2 ξ, ε) = 0,f3(ξ, ε) ≡ r f1(ξ, ε) = f1(r ξ, ε) = 0,

f4(ξ, ε) ≡ r3 f1(ξ, ε) = f1(r3 ξ, ε) = 0; r = (1324), s = (13)(24). (8)

The leading terms of the branching system (4.0) are the following

ξ1ε + Aξ21ξ2 + Bξ1ξ3ξ4 + . . . = 0,

ξ2ε + Aξ22ξ1 + Bξ2ξ3ξ4 + . . . = 0,

ξ3ε + Aξ23ξ4 + Bξ1ξ2ξ3 + . . . = 0,

ξ4ε + Aξ24ξ3 + Bξ1ξ2ξ4 + . . . = 0,

(9)

where A = ±λ20/4, B = ±λ2

0/2, λ20 = π2/a2, a is the lattice width (the upper

sign is related to equation (6) and the lower sign to (7))


The origin is the discrete square-group D4 = e, r, s, r2, r3, sr, sr2, sr3,(rks = sr4−k, k = 1, 2, 3) generated by the permutations of basic elementindices and the structure L(D4) of all its subgroups. If H0 = D4 ⊃ H1 ⊃H2 ⊃ . . . ⊃ Hæ = Hkæ

1 is some chain of subgroups of the length æ then thereexists the basis Ræ in N , in which the representation Ag for every subgroupHi is splitting into irreducible ones. The set of all BEqs for H–invariantsolutions forms the dual to inclusion structure L′ to L(D4): BEq of solutionswhich are invariant with respect to the more narrow subgroup contains theBEq of solutions which are invariant with respect to a more wide subgroup.For two chains A = Hkæ

1 and g−1Ag = g−1Hkgæ1 of similar subgroups

the connection between the Hk–invariant element subspaces and BEqs of Hk–invariant solutions respectively is realized by the element g.

The basis application of this abstract theoretical result application is thecomputer program for the subgroup structure of discrete symmetry construc-tion. By means of this program, for every subgroups chain from the completebranching system the subsystem determining solutions invariant with respectto every of its representatives is selected.

Consider its algorithm. The program for the subgroups construction offinite discrete symmetry uses the Kelly table. The proposed program beginsthe subgroups structure construction with the bottom upwards from trivialsubgroup e. Furthermore the ability for subgroups verification on normaldivisor property is realized. Let describe the algorithm steps.1. Let some subgroup Hk = e, gr1 , . . . , grk

be found. Add to this subsetsome element g that doesn not enter the set Ik (in the first step Ik = Hk).2. Form the set that consists of all kinds of products of elements in Hk+1 =Hk ∪ g. As the result of the formation of such products one gets some setHk+s = e, gr1 , . . . , grk

, g, grk+1, . . . , grk+s

. The order of this set is the initialgroup order or its divisor.3. In case of equality the set Hk+s coincides with the investigated group. Letus come back to the point 1 and add to the set Ik the element g.4. Otherwise we have find the subgroup. Add to the set Ik elements fromthe subgroup Hk+s. For the found subgroup Hk+s repeat all actions from thepoint 1.

Thus the suggested algorithm contains the recursion, leading to the deter-mination of the subgroup structure.

144 Oleg V. Makeev

In the structure L(D4) the suggested program selects the following subgroupchains

A1 : D4 = A1,0 ⊃ A1,1 = e, rs, sr, r2 ⊃ A1,2 = e, rs,s−1A1s = sAs : D4 ⊃ sA1,1s = e, rs, sr, r2 ⊃ sA1,2s = e, sr,

conjugate with A1 meaning the reflection sA2 : D4 = A2,0 ⊃ A2,1 = A1,1 = e, rs, sr, r2 ⊃ A2,2 = e, r2;A3 : D4 = A3,0 ⊃ A3,1e, s, r2, r2s ⊃ A3,2 = e, r2s,

r−1A3r = r3A3r : D4 ⊃ r3A3,1r = e, s, r2, r2s ⊃ r3A1,2r = e, s, conjugate with A3

meaning the counterclockwise rotation rA4 : D4 = A4,0 ⊃ A4,1 = A3,1 = e, s, r, r2s ⊃ A4,2 = e, r2;A5 : D4 = A5,0 ⊃ A5,1e, r, r2, r3 ⊃ A5,2 = A2,2e, r2.

The subgroups e, rs, sr, r2, e, s, r2, r2s, e, r, r2, r3, e, r2 are normaldivisors (on the figure they are shown by semiboldface lines).

Fig. 2. Structure of L(D4).

Let H be a subgroup of D4. The subspace of N(B) consisting of elementswhich are invariant to H is determined by means of the projection operator[4]

P (H) =1|H|

∑

g∈H

T (g), (10)

where T (g) is the representation of D4 by the matrix group generated by

T (s) =

0 0 1 00 0 0 11 0 0 00 1 0 0

and T (r) =

0 0 0 10 0 1 01 0 0 00 1 0 0

.

For every chain of subgroups the projection operators and hence the subgroup-invariant elements in N(B) are determined. On the base of the formulas for


the basis of N(B) changing and relevant transformations of BEq [2] we aregoing to construct the branching systems for the solution of the equationswhich are invariant with respect to subgroups.

L e t u s c o n s i d e r t h e c h a i n A1. The projection operatorP (A1,1) transforms the N(B) basis into the subspace of A1,1-invariant elementsspanϕ×1 = ϕ1+ϕ2√

2, ϕ×2 = ϕ3+ϕ4√

2. Adding to the base of this subspace the

elements ϕ×3 = −iϕ1+iϕ2√2

, ϕ×4 = −iϕ3+iϕ4√2

the BEq in the new base

a0η1 +∑

|p|≥1

ap1p2

2p1+p2η1(η2

1 + η23)

p1(η22 + η2

4)p2 = 0,

a0η2 +∑

|p|≥1

ap1p2

2p1+p2η2(η2

2 + η24)

p1(η21 + η2

3)p2 = 0,

a0η3 +∑

|p|≥1

ap1p2

2p1+p2η3(η2

1 + η23)

p1(η22 + η2

4)p2 = 0,

a0η4 +∑

|p|≥1

ap1p2

2p1+p2η4(η2

2 + η24)

p1(η21 + η2

3)p2 = 0 (11)

is obtained. Then the BEq for A1,1-invariant solutions follows from (4.0) bysetting η3 = 0, η4 = 0 in accordance with the form of the projector P (A1,1)

a0η1 +∑

|p|≥1

ap1p2

2p1+p2η2p1+11 η2p2

2 = 0,

a0η3 +∑

|p|≥1

ap1p2

2p1+p2η2p21 η2p1+1

2 = 0 (12)

.The leading part of (4.0) is

η1ε +A

2η31 +

B

2η1η

22 = 0

η2ε +A

2η32 +

B

2η21η2 = 0 (13)

with the solutions1) η2 = 0, η1 = ±

√−2ε

A , sign ε = −sign A; u = η1ϕ×1 = ± 1

a

√− ε

A cosλ0(x +y) + O(|ε|),2) η1 = 0, η2 = ±

√−2ε

A , sign ε = −sign A; u = η2ϕ×2 = ± 1

a

√− ε

A cosλ0(y −x) + O(|ε|),

146 Oleg V. Makeev

Fig. 3. Relief of the solution 3.

3) η21 = η2

2 = − 2εA+B , sign ε = −sign (A+B); u = ± 2

a

√− ε

A+B cosλ0x cosλ0y+O(|ε|).

For the chain sA1s of conjugate subgroups the transformation s transformsthe branching equations for solutions invariant with respect to subgroups ofthe first chain into the BEqs for solutions which are invariant relatively tosubgroups of the second chain.

The subspace of N(B) elements invariant to the subgroup A1,2 spanϕ×1 =ϕ1+ϕ2√

2, ϕ×3 = ϕ3, ϕ

×4 = ϕ4 is transformed by s in the subspace spanϕ×1 =

ϕ1, ϕ×2 = ϕ2, ϕ

×3 = ϕ3+ϕ4√

2, of N(B) elements invariant with respect to the

subgroup sA1,2s. The BEq for solutions invariant to A1,2

a0η1 +∑

|p|≥1

ap1p2

2p1η2p1+11 ηp2

3 ηp24 = 0

a0η3 +∑

|p|≥1

ap1p2

2p2η2p21 ηp1+1

3 ηp14 = 0

a0η4 +∑

|p|≥1

ap1p2

2p2η2p21 ηp1

3 ηp1+14 = 0 (14)

is transformed by s in the BEq for solutions invariant to sA1,2s

a0η1 +∑

|p|≥1

ap1p2

2p2ηp1+11 ηp1

2 η2p23 = 0,

a0η2 +∑

|p|≥1

ap1p2

2p2ηp11 ηp1+1

2 η2p23 = 0,

a0η3 +∑

|p|≥1

ap1p2

2p1ηp21 ηp2

2 η2p1+13 = 0. (15)


Moreover the branching systems form the structure dual to the structureof subgroups of D4: BEq of solutions which are invariant to more narrowsubgroup contains the BEq of solutions which are invariant to a more widesubgroup. In fact, the BEq of A1,1-invariant solutions (sA1,1s-invariant solu-tions) is resulting from the BEq of A1,2-invariant solutions (sA1,2s-invariantones) setting in (4.0) η3 = η4 = η2√

2(correspondingly in (4.0) η1 = η2 = η4√

2or

η1 = η2 = η3√2).

The relevant leading part of (4.0) has the form

η1

(ε +

A

2η21 + Bη3η4

)= 0,

η3

(ε + Aη3η4 +

B

2η21

)= 0,

η4

(ε + Aη3η4 +

B

2η21

)= 0 (16)

.Consequently, the A11–invariant solutions of the equations (6), (7) are repre-

sented by the formula η∗1ϕ×1 +η∗3ϕ

×3 +η∗4ϕ

×4 , where the vector (η∗1, η

∗3, η

∗4) passes

the solution set of the previous system. The sA11s–invariant solutions are,s(η∗1ϕ

×1 +η∗3ϕ

×3 +η∗4ϕ

×4 ) = η∗1 sϕ×1 +η∗3 sϕ×3 +η∗4 sϕ×4 = [η∗1ϕ1+η∗3ϕ2+η∗4

(ϕ3+ϕ4)√2

]respectively. Among the set of the solutions of the system (4.0) should be men-tioned1) η3 = η4 = 0, η1 = ±

√−2ε

A , sign ε = −sign A ,

2) η3 = η4 = ±√− ε(A+B−1)

A(A+B) , η1 = ±√− ε

A+B , sign ε = −sign (A + B),

sign (A + B − 1) = sign A,

3) η1 = ±√− ε

A+B , η3η4 = − εA+B , sign ε = −sign (A + B).

References[1] I. S. Iohvidov, Hankel and Toeplitz matrix and forms, Nauka, Moscow, 1974.

[2] B. V. Loginov, Branching theory of solutions of nonlinear equations under group in-variance conditions, Fan, Tashkent, 1985.

[3] B. V. Loginov, H. R. Rakhmatova, N. N. Yuldashev: On the construction of branchingequation by its group of symmetries (crystallographic groups), in Mixed type equationsand free boundary value problems, Fan, Tashkent, Acad. Sci. Uzbek SSR, 1987, 183–195.

[4] G. Ya. Lyubarsky Group theory and its applications in physics, GITTL, Moscow, 1958

[5] O. Makeev, B. Loginov, Computer realization of the branching equation constructionon allowed group symmetry, Wang Y., Hutter K. (eds.), Trends in Applications ofMathematics to Mechanics, Proc. of the International Symposium, STAMM, Seeheim,Germany, August 22-28, Shaker. Berichte aus der Mathematik Aachen, 2005, 277-288.

148 Oleg V. Makeev

[6] D. Montgomery, G. Joyce, Statistical mechanics of ”negative temperature” states inplasma, Physics of Fluids, 17,6(1974), 1139–1145.

[7] I. V. Morshneva, V. I. Youdovich: On cycles bifurcation from equlibria for systems withinversion and rotation symmetries, Siberian Math. J. 26,1(1985), 124–133.

[8] L. V. Ovsyannikov, Group analysis of differential equations, Nauka, Novosibirsk, 1978.Engl. transl. Academic, New-York, 1982.

[9] N. Sidorov, B. Loginov, A. Sinitsyn, M. Falaleev, Lyapunov-Schmidt methods in non-linear analysis and applications, MIA 550, Kluwer, Dordrecht, 2002.

[10] S. A. Vladimirov, Ordinary differential equations with discrete group of symmetriesDiferentsialnye Uravnenya, 12, 7(1975), 180-183.

ON A SPARSE MATRIX MEMORIZING METHOD

ROMAI J., 2, 1(2006),151–151

Vlad Monescu, Silviu Dumitrescu, Bogdana Pop”Transilvania” University of Brasov, Dept. of Computer Sci.

Abstract A method to memorize a sparse matrix is discussed. In order to access theelements, the matrix′s binary converted blocks are taken into consideration.The method is compared with some classic ones and computational results arepresented.

Keywords: sparse matrix, binary conversion, block matrix.

1. INTRODUCTIONThe idea to take into account the large number of zeros of a matrix and

their location was initiated in the second half of nineteenth century by elec-trical engineers. A n×m matrix is a sparse matrix if the number of nonzeroentries is much smaller than n ×m. There are two problems: how to retainin minimum memory space a sparse matrix and how to access its elements.Sparse matrices arise in optimization problems, solutions to partial differentialequations, structural and circuit analysis and computational fluid dynamics.Sparse matrices can be huge; dimensions on the order of 100,000 are not un-common. Only by exploiting sparsity can we hope to be able to manipulatesuch a matrix on a computer.

2. THE ALGORITHMWe propose a method to memorizing a sparse matrix. To this end, let A

be a sparse matrix, A ∈ Mn,m(R). For the following construction choose thenumbers p, q ∈ N∗. The rows of matrix are divided in groups of p rows andthe columns are divided in groups of q columns. If n mod p 6= 0 or m modq 6= 0 then supplementary rows, respectively columns are added containingnull values. The matrix we obtain by adding rows or columns is equivalentwith the initial matrix. Each of ([n−1

p ] + 1)× ([m−1q ] + 1) blocks of elements is

p× q binary converted. All positions containing nonzero values are considered[1]. The block which has in the upper - left corner the element ai,j is

ar,s ar,s+1 · · · ar,s+q−1

ar+1,s ar+1,s+1 · · · ar+1,s+q−1

· · · · · ·ar+p−1,s ar+p−1,s+1 · · · ar+p−1,s+q−1

.

149

150 Vlad Monescu, Silviu Dumitrescu, Bogdana Pop

We transform this block in a sequence of bits bpqbpq−1...b1b0, where

bpq−i(q−1)−j =

1, ar+i,s+j 6= 00, ar+i,s+j = 0

Thus we obtain a new matrix T which has nonnegative integer elements as aresult of conversion in p×q bits in a matricial disposal. If we denote by N thenumber of nonzero elements in A, then N + ([n−1

p ] + 1)× ([m−1q ] + 1) memory

locations are needed to memorize the sparse matrix A using this method.Example.

For the matrix A =

11 0 0 14 5 0 0 80 0 0 5 0 0 0 10 0 0 6 0 0 0 04 0 0 7 0 1 0 00 0 0 0 0 0 0 40 0 0 0 10 0 0 017 0 0 0 0 0 58 0

, if p = 4 and q = 4

we have the following configuration

A′=

1 0 0 1 · 1 0 0 10 0 0 1 · 0 0 0 10 0 0 1 · 0 0 0 01 0 0 1 · 0 1 0 0· · · · · · · · ·0 0 0 0 · 0 0 0 10 0 0 0 · 1 0 0 01 0 0 0 · 0 0 1 00 0 0 0 · 0 0 0 0

.

It has been added a row with 0. Each group of 16 bits represents a nonneg-ative integer number in binary conversion. Thus we obtain

T =(

37145 37124128 6176

).

The vector of nonzero elements (ordered by their appeareances in their blocks)is

wT =(

11 14 5 6 4 7 5 8 1 1 17 4 10 58).

Once we have T , and w it is necessary to access the elements of matrix A.Let it be v, b, k ∈ N. We denote

Nb(v) - number of nonzero bits of v in the b-bits binary representation,

Pb,k(v) - position of the k-th nonzero bit of v in the b-bits binary repre-sentation,

On a sparse matrix memorizing method 151

Bb,k(v) - value of the k-th bit of v in the b-bits binary representation.

If we have to access the element ai,j and

Bb,pq−(i mod p)q−j mod q(t[ i−1p

]+1,[ j−1q

]+1) = 0

then ai,j = 0, else the position s of the element ai,j in w is given by

s =

[ i−1p

]∑

k=1

[m−1q

]+1∑

l=1

Npq(tk,l)+

[ j−1q

]∑

l=1

Npq(t[ i−1p

]+1,l)−N(i mod p)q+j mod q(t[ i−1p

]+1,[ j−1p

]+1)

This method is very easy to be implemented in a programming languageusing bitwise operations. Also, the memory space required to retain the matrixusing this algorithm is

N + ([n− 1

p] + 1)× ([

m− 1q

] + 1).

This storage method is obviously better than classical methods if the num-bers p and q are big enough.

Denoting

r, the number which represents the memorising index of the given matrixA, (r = N

nm , N 6= nm) and

r′, the memorising index of the matrix A through this method,

then the inequalitypq >

nm

(r − 1)Nis a condition to obtain a better memorising index.

3. Conclusions

A new method to memorizing a sparse matrix was developed in this paper.Blocks of matrices were binary converted and were taken into considerationwhen the elements were accessed. The solving method was compared withother algorithms and a condition of efficiency was formulated. Computationalresults were presented in order to authenticate the proposed algorithm.

References[1] Y. Saad, Iterative methods for linear systems, PWS Publishing, Boston, 1996.

[2] N. Andrei, C. Rasturnoiu, Matrice rare si aplicatiile lor, Bucuresti, 1983.

[3] S. Pissanetzky, Sparse matrix technology, Academic, London, 1984.


REGENERATIVE PROCESS WITH A RAREEVENT

ROMAI J., 2, 1(2006),153–156

Bogdan-Gh. Munteanu”Transilvania” Univ. of Brasov, Dept. of Mathematical Analysis and [email protected]

Abstract The distribution of time to failure in a system with reparable components,under the assumption that the repair time of failed components has an arbitrarycumulative distribution function is investigated.

Keywords: regenerative process, law limit, asymptotic theorems

2000 MSC: 62E17, 60F15, 60G50, 60K05, 60K10.

1. INTRODUCTIONFor a repair time with an arbitrary cumulative distribution function, the

techniques of birth and death processes is needed. We will assume that thesystem with renewable components enters from time to time into the brandnew state and that the time instants of these entrances generate a regenerativeprocess. Let the entrances take place at epochs T0 = 0, Ti, i = 1, 2, .... Thenthe random intervals ξi = Ti − Ti−1, i = 1, 2, ... are independent identicallydistributed random variables. Then the random intervals Tk =

∑ki=1 ξi, k ≥

1, are called the regeneration points, and the sequence ξi, i ≥ 1 is called aregenerative process (fig.1). We assume that ξi are positive random variables.

Fig. 1. The regenerative process ξi; • designates regeneration points.

153

154 Bogdan-Gh. Munteanu

2. DISTRIBUTION OF A REGENERATIVEPROCESS DURATION

Consider a simple model based on a regenerative process leading to theexponential distribution. Assume that some event A can happen with prob-ability p during a regeneration period. The probability p remains constantand is not affected by the previous history of the process. Consider a ran-dom variable Y = ξ1 + ξ2 + ... + ξN , where N has a geometric distributionP (N = k) = (1 − p)k−1p , k ≥ 1. Therefore, Y represents the duration ofa regenerative process which is stopped at the end of that period on whichthe event A had appeared for the first time. We stress that random variableN is independent of random variables ξj j ≥ 1. The next proposition findscumulative distribution function of Y .

Proposition 2.1 [2]. Let E[ξi] = µ and Y = ξ1 + ξ2 + ... + ξN . ThenY ∼ Exp(p/µ).

Proof. We apply the Laplace transform technique, i.e.

L(Y ) = E[e−zY ] =∞∑

k=1

P (N = k) · E[e−z(ξ1+ξ2+...+ξn) | N = k]

=∞∑

k=1

(1− p)k−1p · [ϕ(z)]k =p ϕ(z)

1− (1− p)ϕ(z), (1)

where ϕ(z) = E[e−zξi ]. As E[ξi] = µ, then ξi ∼ Exp(1/µ), so that

E[e−zξi ] =∞∫0

e−ztd(1 − e−t/µ) = − e−(z+ 1

µ )t

z+1/µ |∞0 = 11+zµ . Substituting it into

(2.0), we obtain that L(Y ) =(1 + zµ

p

)−1which means that Y ∼ Exp(p/µ),

or P (Y ≤ t) = 1 − e−pt/µ. Therefore, the sum of a geometrically distributednumber of exponential sumands has an exact exponential distribution. Weprove that if the summands ξi have an arbitrary distribution then under someappropriate condition, including p → 0, the sum ξ1 + ξ2 + ... + ξn converges indistribution to an exponential random variable. 2

We need the following lemma:

Lemma 2.1 Let Y ∼ G(p) and Z = pY . Then Zd→ Exp(1) as p → 0.

Proof. Denote by [t/p] the integer part of t/p.

limp→0

P (Z > t) = limp→0

P (pY > t) = P (Y ≥ [t/p])

= limp→0

(1− p)[t/p] = limp→0

[(1− p)−1/p

]−p[t/p]= e− limp→0 p[t/p] = e−t

Regenerative process with a rare event 155

2

Theorem 2.1 Let:

i. Xi, i = 1, 2, ... be a random sequence satisfying the strong law of large num-bers: P

(lim

n→∞ Xn =Pn

i=1 Xi

n = µ)

= 1, 0 < µ < ∞;

ii. the random variable N ∼ G(p), N is independent on Xi, Y = X1 + X2 +... + XN .

Then pYµ

d→ Exp(1) as p → 0.

Proof. We have pYµ = p

µN ·PN

i=1 Xi

N = pN · XNµ . By Lemma 2.1, we have

pNd→ Exp(1). Further, P

(XNµ → 1

)= 1 as N → ∞ (the strong law of

large numbers). Thus, by Slutsky’s theorem in [6], pYµ = pN · XN

µd→ Exp(1).

Remark that Theorem 2.1 does not require the existence of a finite secondmoment for random variables Xi. 2

3. EXAMPLE-TWO UNIT SYSTEM WITHSTANDBY AND UNRELIABLEREPLACEMENT OF FAILED UNIT

A system has two identical units; one is operating and the second is incold standby. The average lifetime of the operating unit is E[τ ] = µ andE[τ2] = µ2. When the operating unit fails, its place is taken immediately bythe standby unit and the failed unit goes to repair. The repair is, in fact,an instantaneous replacement of the failed unit by a new one taken from thestorage. There is a small probability p that the unit is not available at thestorage and then the repair will be delayed for a long time. In that case, thesystem fails at the instant of the failure of the operating unit. Assume thatsystem operation began with one unit being sent for replacement. It is easyto see that the system lifetime is:

τ∗ = τ1 + τ2 + ... + τN

where N ∼ G(p), that is P (N = k) = (1− p)k−1p , k ≥ 1. If p is small, thenone can assume that, by Theorem 2.1, the exponential approximation is validfor the system lifetime P

(τ∗pµ ≤ t

)≈ 1− e−t, or P (τ∗ > t′) ≈ e

− pµ

t.Example. Let p = 0, 01, µ = 10 hr , µ2 = 175 hr , t = 500 hr. ThenP (τ∗ > 500) = P

(τ∗pµ > 500p

µ

)≈ e−0.5 ≈ 0.606. In order to see how accu-

rate the exponential approximation is, let us make use of the following result

156 Bogdan-Gh. Munteanu

established by Brown in [1]

supt≥0

∣∣∣∣∣P(

N∑

i=1

τi > t

)− e−tp/µ

∣∣∣∣∣ ≤µ2p

µ2. (2)

For our data we obtain that µ2p/µ2 = 0.0175, which means that the approxi-mation (2) is quite accurate.On the other side, in [3], Theorem 2.1, show that

supx≥0

∣∣∣∣P(

τ∗

E(τ∗)> x

)− e−x

∣∣∣∣ <1−√1− 4a2

1 +√

1− 4a2(3)

The right-hand side to be approximated with a2, if a2 → 0. The Remark 2.5.in [3] clarifies the probabilistic meaning of the quantity a2 expressed by therelation a2 = 1 − E(τ∗2)

2E2(τ∗) , or a2 = 1 − µ2

2µ2 . In the our example, a2 = 0.125.Thus, the approximation (3) is very accurate.

Remark 3.1 The relation (3) establishes the deviation from standard expo-nential distribution of the distribution of normalized system lifetime, as soonas the approximation (2) provides that the cumulative distribution function ofsystem lifetime τ∗ is rather close to 1−e

pµ

x, that is the exponential distributionwith the parameter p

µ .

References[1] M. BROWN, Bounds for exponential approximation on geometric convolution, City

College, CUNY Report No MB84-03, 1986.

[2] I. B. GERTSBAKH, Statistical reliability theory, Marcel Dekker, New-York, 1989.

[3] B. Gh. MUNTEANU, Reliability of a system with renewable components and fast repair,Carpathian J. Math., 20, 1 (2004), 81-86.

[4] G. V. ORMAN, Handbook of limit theorems and stochastic approximation, TransilvaniaUniversity Press, Brasov, 2003.

[5] N. RAVICHANDRAN, Stochastic methods in reliability theory, Wiley Eastern Lt., NewDehi, 1989.

[6] R. J. SERFLING, Approximation theorems of mathematical statistics, Wiley, New York,1980.

[7] A. D. WENTZELL, Limit theorems on large deviations for Markov stochastic processes,

Kulwer, Dordrecht,1990.

STABILITY OF LIMIT CYCLES IN ACALCIUM OSCILLATIONS DYNAMICS MODEL

ROMAI J., 2, 1(2006),157–168

Ileana Rodica NicolaDepartment of Mathematics I, University Politehnica of Bucharestnicola [email protected]

Abstract The stability of the limit cycles of a nonlinear delay system of ordinary differ-ential equations is studied. The system is a model proposed by Borghans in1997 [2] describing calcium oscillations in nonexcitable cells. The unique posi-tive equilibrium point of the associated dynamical system can loose its stabilityon the account of a Hopf bifurcation. We then investigate the stability of thebifurcating limit cycles by using the center manifold theorem.

Keywords: dynamical system, time delay, bifurcation, limit cycle.

2000 MSC: 34K18, 37G10, 37M20, 37N25, 92C35.

1. INTRODUCTIONThis paper is devoted to the analysis of the nonlinear delay system of ordi-

nary differential equations (SODE)

dZ

dt= −kZ(t) + V0 + βV1 + kf (Y (t))− VM2

(Z(t))2

k22 + (Z(t))2

+

+ VM3

(Z(t))m

k2Z(t)m + (Z(t))m

(Y (t))2

k2Y + (Y (t))2

(A(t− τ0))4

k4A + (A(t− τ0))4

dY

dt= −kfY (t) + VM2

(Z(t))2

k22 + (Z(t))2

−

− VM3

(Z(t))m

kZ(t)m + (Z(t))m

(Y (t))2

k2Y + (Y (t))2

(A(t− τ0))4

k4A + (A(t− τ0))4

dA

dt= βVM4 − VM5

(A(t− τ0))p

kp5 + (A(t− τ0))p

(Z(t))n

knd + (Z(t))n

− εA(t− τ0).

(1)

With the autonomous SODE (1.1) we associate the initial condition

Z(0) = Z0, Y (0) = Y0, A(θ) = ϕ(θ), θ ∈ [−τ, 0], τ ≥ 0,

where ϕ : [−τ, 0] → R is a differentiable function which describes the behaviorof the flow in the O direction.

157

158 Ileana Rodica Nicola

These equations arise from a model of calcium oscillations dynamics. Thenon-delayed model was previously proposed and investigated by Houart et al.in [6]; it is based on the mechanism of Ca2+ -induced Ca2+ -release, that takesinto account the Ca2+ -stimulated degradation of inositol 1,4,5 -trisphosphate(InsP3) by a 3-kinase. In the vast majority of cells, an external stimulusinitiates the synthesis of InsP3, starting an intracellular chain reaction, whichculminates with the release of Ca2+ from an internal store of the cell, inthe cytosol. Two mechanisms are responsable for calcium oscillation: theautocatalitic nature of Ca2+ release in the cytosol and the increased InsP3

degradation, due to the Ca2+-stimulated of the InsP3 3-kinase.The variables involved in the process are Z, Y and A, representing the

concentration of free Ca2+ in the cytosol, in a certain internal pool of the cell,respectively the InsP3 concentration.

The biological interpretation of the parameters is as follows: V0 refers to aconstant input of Ca2+ from the extracellular medium and V1 is the maximumrate of stimulus-induced influx of Ca2+ from the extracellular medium. Para-meter β reflects the degree of stimulation of the cell by an agonist. The ratesV2 and V3 refer, respectively, to pumping of cytosolic Ca2+ into the internalstores and to the release of Ca2+ from these stores into the cytosol in a processactivated by citosolic calcium (CICR); VM2 and VM3 denote the maximum val-ues of these rates. Parameters k2, kY , kz and kA are treshold constants forpumping, release, and activation of release by Ca2+ and by InsP3; kf is a rateconstant measuring the passive, linear leak of Y into Z; k relates to the as-sumed linear transport of citosolic calcium into the extracellular medium; VM4

is the maximum rate of stimulus-induced synthesis of InsP3. V5 is the rate ofphosphorylation of InsP3 by the 3-kinase; VM5 is the maximum value of thisrate, while k5 is a half-saturation constant. m, n and p are Hill coefficientsrelated to the cooperative processes and ε is the phosphorilation rate of InsP3

by the 5-phosphatase.The study of the Hopf bifurcation was previously performed in [13] for three

sets of values of the parameters, namely the set for which the oscillations exibitchaotic behaviour, bursting behaviour, respectively birhitmicity. We proovedthat in the case when the perturbation A(t − τ) is obtained via the Diracdistribution, the calcium oscillations model may undergo a Hopf bifurcationfor some critical value of the time delay τ = τ0. We further develop only the”bursting” case, that is, we consider the following values of the parameters

β = 0.46, n = 2, m = 4, p = 1, K2 = 0.1µM, k5 = 1µM,kA = 0.1µM, kd = 0.6µM, kY = 0.2µM,kz = 0.3µM, k = 0.1667s−1, kf = 0.0167s−1, ε = 0.0167s−1,V0 = 0.0333µMs−1, V1 = 0.0333µMs−1, VM2 = 0.1µMs−1,VM3 = 0.3333µMs−1, VM4 = 0.0417µMs−1, VM5 = 0.5µMs−1.

Stability of limit cycles in a calcium oscillations dynamics model 159

We mention that, in this case, using the Maple 8 software package, we find theequilibrium point (Z∗, Y ∗, A∗)=(0.2916496701; 0.2344675015; 0.1989819160),the eigenvalues ←= ±iω = ±0.08289718923 and the bifurcation parameterτ0 = 21.25439515. The existence of a Hopf bifurcation leads to the existenceof a limit cycle when the bifurcation occurs; herein our aim is to show thestability of this limit cycle following the approach in [7].

2. STABILITY OF LIMIT CYCLESDenote by Λ = ±iΩ0 the simple eigenvalues of the characteristic equation

associated with the delayed SODE (for detaills, see [13]). Denote x(t) =(z(t), y(t), a(t)). By the translation Z = z + Z∗, Y = y + Y ∗, A = a + A∗ theautonomous delay differential system (1) changes to

xi(t) = Xi(z(t), y(t), a(t− τ)), (2)

z(0) = Z0 − Z∗, y0 = Y0 − Y ∗, ϕ(θ) = ϕ(0)−A∗, τ ≥ 0, with the equilibriumpoint (0, 0, 0). In order to characterize the orbits of the linearized delay-differential system, we can use the standard form

xt(θ) = Lxt =

d

dtxt(θ), θ ∈ [−τ, 0)

∫ 0

−τ[dη(θ)]xt(θ), θ = 0

wherext ∈ B = C ′([−τ, 0],R3], xt(θ) = x(t + θ), θ ∈ [−τ, 0).

The mapping η : [−τ, 0] → R3 ×R3 is a matrix whose elements are functionswith bounded variation and the Lebesgue-Stieltjes integral can be written as

∫ 0

−τ[dη(θ)]xt(θ) = N1x(0) + N2x(−τ).

The spectrum of the linear operator L coincides with the set of eigenvalues ofthe characteristic equation associated to the equilibrium point of the SODE.

Consider the linear operator L : B → B, Lϕ = N1ϕ(0) + N2ϕ(−τ), ϕ ∈ B,where

N1 =

n11 n4

1 0

n21 −n4

1 0

n31 0 0

, N2 =

0 0 n12

0 0 −n12

0 0 n22

(3)

andn1

1 = 0.38860527, n12 = −0.55530527, n3

2 = −0.08796881 ,


n14 = 0.324091, n1

2 = −0.10314090, n22 = −0.08317366

are the constituitive matrices of the linearized system. The operator A : B →B∗

Aϕ(θ) =

dϕ(θ)dθ

, θ ∈ [−τ, 0]

N1ϕ(0) + N2ϕ(−τ0), θ = 0, ϕ ∈ B (4)

determines the adjoint operator

A∗ψ(s) =

dψ(s)ds

, s ∈ [0, τ ]

ψ(0)N1 + ψ(τ0)N2, s = 0, ψ ∈ B∗, (5)

with respect to the bilinear form (”scalar product”)

(φ∗(s), ϕ(θ)) = φ∗(0)ϕ(0)−φ∗(−τ)N2

∫ −τ

0e−←ξϕ(ξ)dξ, θ ∈ [−τ, 0], s ∈ [0, τ ].

Proposition 2.1 The following statements are true:

(i) the generalized eigenvectors of the infinitesimal generators A and A∗,corresponding to the proper eigen Λ, are

φ(θ) = (f1, f2, f3)T eiΩ0θ, φ(θ) =

(f1, f2, f3

)Te−iΩ0θ, θ ∈ [0, τ0] (6)

f1 = 0.151551− i 0.172126, f2 = 0.497684− i 0.132631,

f3 = −0.763963− i 1.018165

respectively,

φ∗(s) = (f∗1 , f∗2 , f∗3 )eiΩ0s, φ∗(s) = (f∗1 , f∗2 , f∗3 )e−iΩ0s, s ∈ [0, τ0], (7)

f∗1 = −0.536353 + i 0.373245, f∗2 = −0.413810 + i 0.479090,

f∗3 = 0.594553− i 0.870017;

(ii) the bilinear form (·, ·) : MΛ(A∗)× B → C, with Λ = ±i Ω0 is definedby

(φ∗(s), ϕ(θ)) = f∗1 ϕ1(0) + f∗2 ϕ2(0) + f∗3 ϕ3(0)− α0F (ϕ3)(−τ0),

(φ∗(s), ϕ(θ)) = f∗1 ϕ1(0) + f∗2 ϕ2(0) + f∗3 ϕ3(0)− α0F (ϕ3)(−τ0),

s ∈ [0, τ0], ϕ ∈ B, θ ∈ [−τ0, 0), α0 = 0.072121− i 0.049286, (8)

where MΛ(A∗) is the generalized vector space of A∗ and

F (ϕ3)(−τ0) =∫ −τ0

0e−i Ω0ξϕ(ξ)dξ; (9)


(iii) the generalized eigenvectors of φ∗(s), φ∗(s) and φ(θ), φ(θ) satisfy therelations

(φ∗(s), φ(θ)) = r1,

(φ∗(s), φ(θ)) = r2,

(φ∗(s), φ(θ)) = r2,

(φ∗(s), φ(θ)) = r1, s ∈ [0, τ0], θ ∈ [−τ0, 0],

(10)

where r1 = −2.058522− i 2.065137, r2 = −1.441408− i 1.972671;

(iv) the vectorsψ(s) = f11φ

∗(s) + f12φ∗(s)

ψ(s) = f12φ∗(s) + f11φ

∗(s), s ∈ [0, τ0], (11)

are generalized eigenvectors of the operator A∗. They satisfy the relations

(ψ(s), φ(θ)) = 1, (ψ(s), φ(θ)) = 0

(ψ(s), φ(θ)) = 0, (ψ(s), φ(θ)) = 1, θ ∈ [−τ, 0], s ∈ [0, τ0]. (12)

The bilinear form is associated to the linearized delay differential system.The numbers fij are the entries of the matrix F, the inverse of the matrixE = (eij)i,j=1,2, where

e11 = (φ∗(s), φ(θ)), e12 = (φ∗(s), φ(θ)), e21 = e12, e22 = e11.

Proof. (i) The generalized eigenvectors of A, corresponding to Λ are

φ(θ) = φ(0)eiΩ0θ, φ(θ) = φ(0)e−iΩ0θ, (13)

where φ(0) is a nonzero solution of the following linear system

(iΩ0I −N1 − e−iΩ0τ0N2)φ(0) = 0. (14)

Replacing N1 and N2 by their expressions (2) and using the software packageMaple 8, we find

φ(0) = (f1, f2, f3)T , (15)

where f1 = −0.151551 − i 0.172126, f2 = 0.497684 − i 0.132631 and f3 =−0.763963− i 1.018165.

From (14) and (12) it follows (5). For the operator A∗, the generalizedeigenvectors corresponding to Λ are

φ∗(s) = φ∗(0)eiΩ0s, φ∗(s) = φ∗(0)e−iΩ0s, s ∈ [0, τ0], (16)


where φ∗(0) is a nonzero solution of the following linear system

φ∗(0)(iΩ0I −N1 − eiΩ0τ0N2) = 0. (17)

After computations we get

φ∗(0) = (f∗1 , f∗2 , f∗3 ), (18)

where f∗1 = −0.536353 + i 0.373245, f∗2 = −0.413810 + i 0.479090 and f∗3 =0.594553− i 0.870017.

(ii) The definition of the bilinear form implies

(φ∗(s), ϕ(θ)) = f∗1 ϕ1(0) + f∗2 ϕ2(0) + f∗3 ϕ3(0)−−eiΩ0τ0 [(f∗1 − f∗2 )n1

2 + f∗3 n22]F (ϕ3)(−τ0)

= f∗1 ϕ1(0) + f∗2 ϕ2(0) + f∗3 ϕ3(0)− α0F (ϕ3)(−τ0),

α0 = 0.072121− i 0.049286,

(φ∗(s), ϕ(θ)) = f∗1 ϕ1(0) + f∗2 ϕ2(0) + f∗3 ϕ3(0)−−e−iΩ0τ0 [(f∗1 − f∗2 )n1

2 + f∗3 n22]F (ϕ3)(−τ0)

= f∗1 ϕ1(0) + f∗2 ϕ2(0) + f∗3 ϕ3(0)− α0F (ϕ3)(−τ0).

In conclusion, we get the relation (7).(iii) From (7), for ϕ(θ) = φ(θ), we have

(φ∗(s), φ(θ)) = f∗1 φ1(0) + f∗2 φ2(0) + f∗3 φ3(0)− α0

∫ −τ0

0e−iΩ0ξf3e

iΩ0ξdξ =

= f∗1 f1 + f∗2 f2 + f∗3 f3 + α0τ0f3 = r1, r1 = −2.058522− i 2.065137.

(φ∗(s), φ(θ)) = f∗1 f1 +f∗2 f2 +f∗3 f3 + α0τ0f3 = r2, r2 = −1.441408− i1.972671.

(φ∗(s), φ(θ)) = f∗1 f1 + f∗2 f2 + f∗3 f3 + α0τ0f3 = (φ∗(s), φ(θ))(φ∗(s), φ(θ)) = f∗1 f1 + f∗2 f2 + f∗3 f3 + α0τ0f3 = (φ∗(s), φ(θ))These lead to the relation (9).

(iv) Because the function (·, ·) is bilinear and EF = I, we have the relations

(ψ∗(s), φ(θ)) = f11(φ∗(s), φ(θ)) + f12(φ∗(s), φ(θ)) = f11e11 + f21e12 = 1

(ψ∗(s), φ(θ)) = f11(φ∗(s), φ(θ)) + f12(φ∗(s), φ(θ)) = f11e21 + f21e22 = 0

(ψ∗(s), φ(θ)) = (ψ∗(s), φ(θ)) = 0

(ψ∗(s), φ(θ)) = (ψ∗(s), φ(θ)) = 1.

2


Let N = MΛ(A) be the vector space generated by the generalized eigenvec-tors Φ, Φ.

Definition 2.1 A submanifold W cloc(0) = W c(0, V ) in the Banach space B,

tangent at 0 ∈ B to the vector space N and invariant with respect to thesemigroup of operators T (t) of the delayed system (1), where V is an open setwith 0 ∈ V ⊂ B, is called stable-unstable local centre manifold of the system(1).

In the sequel, let us denote the nonlinear part of the system (1) byF (x(t), x(t− τ)) = X(x(t), x(t− τ))−N1x(t)−N2x(t− τ),

where x = X(x), x = (z, y, a)T is the vector field given by the relation (1).Using the Taylor formula of order three at the point (0,0,0), the autonomousdelay differential system (1) can be written as

x(t) = N1x(t) + N2x(t− τ0) + P (x(t), x(t− τ0)) + O1(|x(t)|4),x(t) = (z(t), y(t), a(t)),

(19)

where P is the vector whose components are Taylor polynomials of degrees twoor three of the vector F and has the form P (x(t), x(t−τ)) = (P1(z(t), y(t), a(t−τ)), P2(z(t), y(t), a(t− τ)), P3(z(t), a(t− τ)))T , where

P1(z(t), y(t), a(t− τ)) = −0.537512z(t)2 + 2.226838z(t)y(t)+

+0.747183a(t− τ)z(t)− 0.862223y(t)2 + 0.3708534a(t− τ)y(t)−−1.171530a(t− τ)2 − 9.088956z(t3)− 2.956420z(t)2y(t)−−0.991983a(t− τ)z(t)2 − 6.246203z(t)y(t)2 + 2.684263a(t− τ)z(t)y(t)−−8.486912a(t− τ)2z(t) + 1.020635y(t)3 − 1.039337a(t− τ)y(t)2−−4.208734a(t− τ)2y(t) + 10.050654a(t− τ)3

P2(z(t), y(t), a(t− τ)) = −P1(z(t), y(t), a(t− τ)),

P3(z(t), a(t− τ)) = −0.035519z(t)− 0.368724a(t− τ)z(t)+

+0.055441a(t− τ)2 = 0.244208z(t)3 − 0.148882a(t− τ)z(t)+

+0.307531a(t− τ)2z(t)− 0.046240a(t− τ)3

Introduce the function

w(θ, zc, zc) = w20(θ)z2c

2+ w11(θ)zczc + w02(θ)

z2c

2+ O1(|zc|3),

w02 = w20(θ), w11(θ) ∈ R, θ ∈ [−τ, 0].Replacing, in P (x(t), x(t−τ)), x(t) by zcφ(0)+zcφ(0)+w(0, zc, zc) and x(t−τ)


by zcφ(−τ) + zcφ(−τ) + w(−τ, zc, zc), we find

F (zc, zc) = F20z2c

2+ F11zczc + F02

z2c

2+ F21

z2c z2

c

2.

where

F20 =

F 120

−F 220

F 320

, F11 =

F 111

−F 111

F 311

, F02 =

F 102

−F 202

F 302

, F21 =

F 121

−F 221

F 321

.

By extracting from expansion of F the coefficients of the termsz2c

2, zczc,

z2c

2

respectivelyz2c z2

c

2, we find

F 120 = −1.137957 + i 3.464395, F 3

20 = −0.010181 + i 0.041143,

F 111 = −3.981737− i 1.600830, F 3

11 = 0.035900− i 0.014433,

F 102 = 1.576785− i 3.287968, F 3

02 = 0.021130− i 0.036741,

F 121 = −108.851161 + i 11.064876− (1.444304 + i 1.706172)w1

11−− (1.606180− i 2.701638)w2

11 + (5.042470 + i 2.815720)w311−

− (0.988250− i 0.522131)w120 − (0.241235 + i 1.552891)w2

20+

+ (1.814090 + i 2.246725)w320

F 321 = −0.068091 + i 0.020966 + (0.872300− i 0.426980)w1

11−− (0.108014− i 0.037829)w3

11 + (0.325032− i 0.360775)w120

+ (0.043053 + i 0.037695)w320.

Proposition 2.2 The following statements are true:

(i) the central manifold W cloc(0) of the system (1) has the elements ϕ ∈ B,

where

ϕ(θ) = zcφ(θ)+zcφ(θ)+w20(θ)z2c

2+w11(θ)zczc+w02(θ)

z2c

2+. . . , θ ∈ [−τ, 0]

(20)where zc = x1 + iy1, (x1, y1) ∈ V1 ⊂ R2, V1 is a neighborhood of zeroand

w20(θ) = −g20

iΩφ(0)eiΩθ − g02

3iΩφ(0)e−iΩθ + E1e

2iΩθ

w11(θ) =g11

iΩφ(0)eiΩθ − g11

iΩφ(0)e−iΩθ + E2, θ ∈ [−τ0, 0]

w02(θ) = w20(θ),

(21)


g20 = 0.302874 + i 0.912785

g11 = −1.079822 + i 0.272609

g02 = −0.171742− i 0.947895

g21 = −40.208575 + i 35.475448 + (0.032165− i 1.7317770)w111+

+0.064066 + i 0.764149)w211 + (0.514234− i 1.094785)w3

11+

+(0.722945 + i 0.294077)w120 − (0.268046 + i 0.274150)w2

20+

+(0.579122 + i 0.194816)w320, (22)

wherew20 = (w1

20, w220, w

320)

T , w11 = (w111, w

211, w

311)

T ,

E1 =

−22.833534− i 15.188009

38.104506− i 7.7702492.614298− i 11.178749

,

E2 =

012.423192 + i 4.994660−0.431636− i 0.173536

;

(23)

(ii) if the initial condition of (1) is

ϕ(θ) = (ψ, ϕ1)φ(θ) + (ψ, ϕ1)φ(θ) + w20(θ)(ψ, ϕ1)2

2+

+ w11(θ)(ψ,ϕ1)(ψ,ϕ1) + w02(ψ,ϕ1)

2

2,

(24)

then the solution of autonomous delay differential system (1) in a neigh-borhood of the equilibrium point (0,0,0) is (25):

z(ϕ)(t) = 2x1Re (f1)− 2y1Im (f1) + r120(x

21 − y2

1)− i120x1y1 + r111(x

21 + y2

1)

y(ϕ)(t) = 2x1Re (f2)− 2y1Im (f2) + r220(x

21 − y2

1)− i220x1y1 + r211(x

21 + y2

1)

a(ϕ)(t) = 2x1Re (f3)− 2y1Im (f3) + r320(x

21 − y2

1)− i320x1y1 + r311(x

21 + y2

1)

where

rk20 = Re (wk

20(0)), ik20 = Im (wk20(0)), rk

11 = wk11(0), ∀k = 1, 3 (26)

and (x1(t), y1(t)) is the solution of the differential equation

zc(t) = iΩ0zc(t) +g20

2zc(t)2 + g11zc(t)zc(t) +

g02

2z(t)2 +

g21

2z(t)2z(t)

zc(0) = x1(0) + iy1(0), (27)


wherex1(0) = Re (ψ, ϕ1), y1(0) = Im (ψ, ϕ1)

ϕ1(θ) = (Z0 − Z∗, Y − Y ∗, ϕ(θ)−A∗)T .(28)

Proof. (i) We use the following relations

E1 = −(N1 + e−2iΩ0τ0N2 − 2iΩ0I)−1F20, E2 = −(N1 + N2)−1F11.

g20 = ψ(0)F20, g11 = ψ(0)F11, g02 = ψ(0)F02, g21 = ψ(0)F21,

whereψ(0) = f12φ

∗(0) + f22φ∗(0)

is one of the eigenvectors of A∗ (see Proposition 2.1 (iv))(ii) The solution of autonomous delay differential system (1) in a neighbor-

hood of the equilibrium point (0,0,0) is

z(ϕ)(t) = zc(t)f1 + zc(t)f1 +12zc(t)2w1

20(0) + zc(t)zc(t)w111(0) +

12zc(t)2w1

02(0)

y(ϕ)(t) = zc(t)f2 + zc(t)f2 +12zc(t)w2

20(0) + zc(t)zc(t)w211(0) +

12zc(t)w2

02(0)

a(ϕ)(t) = zc(t)f3 + zc(t)f3 +12zc(t)w3

20(0) + zc(t)zc(t)w311(0) +

12zc(t)w3

02(0),

where fi, i = 1, 3 are given by (5). Replacing zc by x1 + iy1, we find (24).Remark 2.1. Let R20 = Re (g20), I20 = Im (g20), R11 = Re (g11), I11 =

Im (g21), R21 = Re (g21), I21 = Im (g21). The autonomous delay differentialsystem (1) can be rewritten in the form

x1(t) = −Ω0y1(t) +12(R20 + 2R11 + R02)x2

1(t)−

− 12(R20 − 2R11 + R02)y2

1(t) + (I02 − I20)x1(t)y1(t)+

+ R21x1(t)(x1(t)2 + y1(t)2)− I21y1(t)(x1(t)2 + y1(t)2),

y1(t) = −Ω0x1(t) +12(I20 + 2I11 + I02)y1(t)2−

− 12(I20 − 2I11 + I02)x1(t)2 + (R20 −R02)x1(t)y1(t)+

+ R21y1(t)(x1(t)2 + y1(t)2)− I21x1(t)(x1(t)2 + y1(t)2),

x1(0) = Re (ψ, ϕ1), y1(0) = Im (ψ, ϕ1),

(29)

where ϕ1 is given by (27).


Remark 2.2. The term g21 depends explicitly on w111(0), w2

11(0), w311(−τ),

w120(0), w2

20(0) and w320(−τ); using this and the relations (20) we find g21 =

−55.129357 + i 4.070772.Remark 2.3. The limit cycle can be characterized by the following numbers

µ2 = −Re(C1)Re(M)

, T2 = − Im(C1) + µ2Im(M)Ω0

, β2 = 2Re(C1), (30)

where

C1 =i

2Ω0

(g20g11 − 2|g11|2 − 1

3|g02|2

)+

g21

2

M =(

d ←dτ

)

τ=τ0,←=iΩ0

.

(31)

One knows from [5] that the following properties hold: if µ2 > 0 (< 0) thenthe Hopf bifurcation is supercritical (subcritical) and the bifurcating periodicsolution exists for τ > τ0 (τ < τ0); the solutions are orbitally stable (unstable)if β2 > 0 (< 0), and the bifurcating periodic solution increases (decreases) ifT2 > 0 (< 0).

Consider the values of the parameters as in the case of ”bursting”. UsingMaple 8 programming tehniques, we find µ2 = 10587.6533, β2 = −44.2353and T2 = −180.9126. Hence, the unique Hopf bifurcation of the equation (1.1)seems to be supercritical and the solutions orbitally stable, with decreasingperiod.

Remark 2.4. The cycle of the dynamics of calcium oscillations with time-delay around the equilibrium point (Z∗, Y ∗, A∗) is given by

Z(ϕ)(t) = z(ϕ)(t) + Z∗, Y (ϕ)(t) = y(ϕ)(t) + Y ∗, A(ϕ)(t) = a(ϕ)(t) + A∗,

where z(ϕ)(t), y(ϕ)(t), a(ϕ)(t) are given by Proposition 2.2.

3. AKNOWLEDGMENTThe present work was partially supported by the Grant CNCSIS MEN

A1478/2005.

References

[1] M. Adimy, F. Crauste, A. Halanay, M. Neamtu, D. Opris, Stability of limit cyclesin a pluripotent stem cell dynamics model, Chaos, Solitons and Fractals, 27,4 (2006),1091-1107.

[2] J. A. M. Borghans, G. Dupont, A. Goldbeter, Complex intracellular calcium oscilla-tions: A theoretical exploration of possible mechanisms, Biophys. Chem., 66 (1997),25-41.


[3] E. Hopf, Abzweigung einer periodischen Losung von einer stationaren Losung einesDifferentialsystems, Akad. Wiss. Leipzig, 94 (1942), 3-22.(Translated and commentedby M. Golubitsky and P.H. Rabinowitz in Appl. Math. Sci. 19, Springer, 1976.)

[4] J.Hale, Theory of functional differential equations, Springer, New York, 1977.

[5] B. D. Hassard, N. D. Kazarinoff, Y.-H. Wan, Theory and applications of Hopf bifurca-tion, Cambridge University Press, Cambridge, 1981.

[6] G. Houart, G. Dupont, A. Goldbeter, Bursting, chaos, birhytmicity originating fromself-modulation of the inositol 1,4,5-triphosphate signal in a model for intracellularCa2+ oscillations, Bull. of Math. Biology, 61 (1999), 507-530.

[7] G. Mircea, M. Neamtu, D. Opris, Dynamical systems in economy, mechanics, bi-ology described by differential equations with time-delay (Romanian), Mirton Press,Timisoara, 2003.

[8] I. R. Nicola, C. Udriste, V. Balan, Time-delayed flow of hepatocyte physiology, Ann. ofthe Univ. of Bucuresti, 55, 1(2005), 97-104.

[9] D. Opris, C. Udriste, Pole shifts explained by a Dirac delay in a Stefanescu magneticflow, Ann. Univ. Bucuresti, 53, 1 (2004), 115-144.

[10] C. Robinson, Dynamical systems: stability, symbolic dynamics and chaos, CRC Press,Boca Raton, 1995.

[11] G. Stepan, Retarded dynamical systems. Stability and characteristic functions, Long-man Sci & Technical, England 1989.

[12] N. V. Tu, Introductory optimization dynamics, Springer, Berlin, 1991.

[13] C. Udriste, V. Balan, I. R. Nicola, Linear stability and Hopf bifurcations for time-delayed intra-cell calcium variation models, Proc. of the 3-rd International Colloquium”Mathematics in Engineering and Numerical Physics” (MENP-3), October 7-9, 2004,BSG Proceedings 12, Geometry Balkan Press, Bucharest 2005, 273-284.

ESTIMATION OF HAUSDORFF AND PACKINGLINEAR MEASURE AND DIMENSIONS FORSOME PLANE SETS OF POINTS

ROMAI J., 2, 1(2006),169–174

Antonio NuicaUniversity of Pitestiantonio [email protected]

Abstract Mainly, this paper is a survey of an example of a few classical concepts andresults of fractal geometry.

In Section 1, it is emphasized the fact that the box counting dimension,more easy to calculate than any other dimension, has a key role in understand-ing the relationship between Hausdorff and packing measure and dimension.The section ends reminding two important results: the equality of packingdimension and upper box counting dimension on compact ”self-similar” sets[2] and the density theorem [4]. In Section 2, an example of a plane set Ewith 0 < 1 = H1(E) < 5/4 ≤ c1(E) ≤ 2 < p1(E) = 4 < ∞, and there-fore dimH(E) = dimP (E) = 1, is shown to be theoretically important. InSection 2.2, a plane set with zero linear Hausdorff measure (H1) and infi-nite packing linear measure (p1) is presented following [7]. More precisely,we show that dimH(E) < 1 < dimP (E). Then, by generalizing this examplewe construct a set with specific different Hausdorff and packing dimension:0 = dimH(E) < 1 < dimP (E) = log 7/ log 3.

1. A FEW CLASSICAL RESULTS IN FRACTALGEOMETRY IN RN

Let A ⊂ Rn be a set. A s-dimensional Hausdorff measure Hs of E (s ≥ 0) is

Hs(E) := supδ>0

Hsδ(E), where Hs

δ(E) := inf ∞∑

i=1|Ei|s, (Ei)i δ − covering of E

.

(the diameter d(E) of E is denoted by |E|; a δ-covering of a set E is a family(Ei)i≥1 ⊂ P(Ω) with E ⊂ ⋃

i≥1Ei, |Ei| ≤ δ).

For every E ⊂ Rn there is an unique value associated with E:dimH(E) := inf s ≥ 0 |Hs(E) = 0 = sup s ≥ 0 |Hs(E) = ∞ , (1)

called the Hausdorff dimension of the set E. It is monotone, countably stable,invariant to Lipschitz transforms and behaves ”well” on smooth manifolds, i.ewe get the usual topological dimension.

The packing premeasure is defined as follows: for every E ∈ Pb(Rn) (familyof bounded sets in Rn), P s(E) := inf

δ>0P s

δ (E), where, for all δ > 0, P sδ (E) =

169

170 Antonio Nuica

sup ∞∑

i=1|Bi|s

∣∣∣ (Bi)i≥1 packing of E, |Bi| ≤ δ

(A packing of a set E is a mu-

tually disjoint sequence of balls with centers at points of E).We refer to P s as a ”premeasure” because it is monotone, additive but not

countably additive [6]. Another disadvantage of it is the fact that a set hasthe same packing premeasure as its closure.

By the classical Caratheodory method we construct the set function

ps(E) := inf

∞∑

i=1

P s(Ei)∣∣∣ (Ei)i ⊂ Pb(Rn), E ⊂

⋃

i≥1

Ei

, E ∈ P(Rn),

which is a measure on Rn, called packing measure; it is a metric measure,Borel regular and ps(E) ≤ P s(E), ∀E ∈ P(Rn) (we can extend them on Rn)[6]. For E ⊂ Rn, the packing dimension is similar to Hausdorff dimension, i.e.dimP (E) := sup

s∣∣∣ ps(E) = ∞

= inf

s∣∣∣ ps(E) = 0

. The unique number

determined as above is called packing dimension of the set E. The packingdimension is monotone and countably stable.

There is a set function that brings together Hausdorff and packing measure,namely centered covering measure.

Define the mapping Cs : P(Rn) −→ [0,∞], Cs(E) := supρ>0

Csδ (E), where

Csδ (E) = inf

∞∑i=1

|Bi|s∣∣∣ (Bi)i≥1 Bi balls centered in E, E ⊂ ⋃

i≥1Bi

. Ac-

cording to [4], Cs is not monotone, so, in order to become monotone onedefine cs(E) = sup

Cs(E)

∣∣∣F ⊂ E

, which is a measure on Rn, indeed. It iscalled the centered covering measure.

The relationships between the three measures are [4](

12s

)cs(E) ≤ Hs(E) ≤ Cs(E) ≤ cs(E) ≤ ps(E) ≤ P s(E), E ⊂ Rn,

whence 0 ≤ dimH(E) ≤ dimP (E) ≤ n.The main result used in this paper is

Theorem 1.1 (Raymond-Tricot ”density theorem”) [4] For every pos-itive, finite, Borelian measure µ and x ∈ Rn we have

ps(E) < ∞ =⇒ ps(E) infx∈E

dsµ(x) ≤ µ(E) ≤ ps(E) sup

x∈Eds

µ(x),

cs(E) < ∞ =⇒ cs(E) infx∈E

dsµ(x) ≤ µ(E) ≤ cs(E) sup

x∈Ed

sµ(x),

where dsµ(x) := lim inf

r→0

µ(Br(x))(2r)s

, dsµ(x) := lim sup

r→0

µ(Br(x))(2r)s

.

Estimation of Hausdorff and packing linear measure and dimensions for some plane sets of points 171

Corollary 1.1 If Hs(E) < ∞ and µ(F ) = Hs(E ∩ F ), then dsµ(x) and d

sµ(x)

become the ”Besicovitch” densities, denoted by Ds(x,E), Ds(x,E), and we

have

ps(E) < ∞ =⇒ ps(E) infx∈E

Ds(x, E) ≤ Hs(E) ≤ ps(E) supx∈E

Ds(x,E),

cs(E) < ∞ =⇒ cs(E) infx∈E

Ds(x,E) ≤ Hs(E) ≤ cs(E) sup

x∈ED

s(x,E).

The oldest notion of ”fractal” dimension is the so called box-counting di-mension. For E ⊂ Rn bounded and δ > 0, let Nδ(E) the smallest numberof sets of diameter at most δ which can cover E. The lower and upper box-counting dimensions of E are defined as follows: dimB(E) := lim inf

δ→0

log Nδ(E)− log δ ,

dimB(E) := lim supδ→0

log Nδ(E)− log δ . If these are equal we refer to the common value

as the box-counting dimension of E and write dimB(E) := limδ→0

log Nδ(E)− log δ .

We get the same previous numbers if we denote by Nδ(E) the smallestnumber of closed balls of radius δ that cover E, or the smallest number ofcubes of side δ that cover E, or the number of δ-mesh cubes that intersect E,or the largest number of disjoint balls of radius δ with centers at the points ofE.

The relations between Hausdorff, packing and box-counting dimensions are

0 ≤ dimH(E) ≤ dimP (E) ≤ dimB(E) ≤ n, ∀E ⊂ Rn,0 ≤ dimH(E) ≤ dimB(E) ≤ dimB(E) ≤ n, ∀E ⊂ Rn.

The second important result used in this paper is

Theorem 1.2 (Equality between upper box-counting dimension and packingdimension of self similar sets ) Let E ⊂ Rn be compact with the propertythat for every open set V that intersects E, dimB(V ∩ E) = dimB(E). ThendimP (E) = dimB(E).

2. ESTIMATION OF HAUSDORFF ANDPACKING LINEAR MEASURE ANDDIMENSIONS FOR SOME PLANE SETS OFPOINTS

2.1. AN EXAMPLE OF PLANE SET WITH0 < H1(E) < P 1(E) < ∞

Let E0 be a closed equilateral triangle of side 1. Following [7], we constructthe set E1 by replacing E0 by 3 equal triangles of side 1/3, as shown in fig.1.

172 Antonio Nuica

Fig. 1. Construction of En

Supposing that En is constructed (it consists of 3n equal triangles of side1/3n), we obtain En+1 by replacing each triangle of En by 3 equal trianglesas in fig. 1. Finally we define E := ∩n≥1En. Let Cn be the cover of E by thetriangles of En. Because Cn is a 1/3n-cover of E, we have H1(E) ≤ ∑

T∈Cn

|T | =3n ·(1/3n) = 1. But the projection of E of one side of E0 is a segment of length1, so linear Hausdorff measure is equal to 1 and we know the behaviour of theHausdorff measure with respect to contractions: H1(pr(E)) = 1 ≤ H1(E).

So H1(E) = 1. On the other hand, ın [7] it is proved that D1(x,E) =14

H1-a.e. on E. From Corollary 1.1 we get p1(E) = 4. Using C1(E) ≥ 5/4 (thecomplete algebraic computations can be found in [4]) and again by Corollary1.1 we get C1(E) ≤ c1(E) ≤ 2H1(E), so

0 < 1 = H1(E) < 5/4 ≤ c1(E) < p1(E) = 4 < ∞.

This example has two major consequences. First, there exists a plane setwith finite distinct values for linear Hausdorff measure, linear packing measureand linear centered covering measure. Second, the set function C1 is notmonotone: if we consider F to be the set of all centers of the triangles of eachEn, for all n, then E ∪ F can be covered by 3n centered balls of diameter(2/√

3)3−n, so C1(E ∪ F ) ≤ 2/√

3 < 5/4 ≤ C1(E).

2.2. EXAMPLES OF SETS WITHdimH(E) < 1 < dimP (E)

I. Consider a sequence of positive integers an > 0, n ≥ 1 and define cn =n∑

k=1

ak, bn =n∑

k=1

a2k. Let the initial set E0 be a closed disc of diameter 1. The

operation of replacing a closed disc of diameter d with another concentric discof diameter d/3 is called the ”operation” O1. The operation of replacing aclosed disc of diameter d with 7 discs of diameter d/3 ”packed” in the initialdisc (as in fig.2) is called the ”operation” O2. At the first step, we apply tothe disc E0 operation O1 a1 times and get E1, the concentric disc of diameter(1/3)a1 = (1/3)c1 . Then, to E1 we apply the operation O2 a2 times to getE2, formed with 7a2 = 7b2 discs of diameter (1/3)a1+a2 = (1/3)c2 ”packed”

Estimation of Hausdorff and packing linear measure and dimensions for some plane sets of points 173

Fig. 2. The ”operation” O2

into E1. In general, if E2n is formed with 7bn discs of diameter (1/3)c2n , weget E2n+1 applying O1 a2n+1 times to each disc from E2n and obtain 7bn

discs of diameter (1/3)c2n+1 , and then E2n+2 applying O2 a2n+2 times to eachdisc of E2n+1 obtaining 7bn+1 discs of diameter (1/3)c2n+2 . Finally, we defineE := ∩n≥0En.

For each n ≥ 0, E is covered by E2n+1 and ”packed” by E2n, so H1(E) ≤lim infn→∞ 7bn · 3−c2n+1 , P 1(E) ≥ lim sup

n→∞7bn · 3−c2n . Let an = Nn−1, n ≥ 1, so

bn =N2n+2 − 1

N2 − 1, cn =

Nn+1 − 1N − 1

, 7bn · 3−c2n+1 =

(7

1N2−1

31

N−1

)N2n+2−1

, 7bn ·

3−c2n =3

1N−1

71

N2−1

·(

71

N2−1

31

N(N−1)

)N2n+2

. But7

1N2−1

31

N−1

< 1 and7

1N2−1

31

N(N−1)

> 1, for all

N ≥ 2, soH1(E) ≤ lim inf

n→∞ 7bn · 3−c2n+1 = 0,

P 1(E) ≥ lim supn→∞

7bn · 3−c2n = ∞.

In this case let us prove dimH(E) < 1 < dimP (E). This is equivalent to∃t < 1, such that Ht(E) = 0, ∃s > 1, such that P s(E) = ∞, or, in other

words, ∃t < 1, such that

(7

1N2−1

3t

N−1

)N2n+2−1

−→ 0 and ∃s > 1, such that

3s

N−1

71

N2−1

·(

71

N2−1

3s

N(N−1)

)N2n+2

−→ ∞. But this is true, because ∃t < 1, such that

71

N2−1

3t

N−1

< 1, ∃s > 1, such that7

1N2−1

3s

N(N−1)

> 1, for example t ∈(

1, 953 · 1, 09

, 1)

and

s ∈(

1,1110

).

174 Antonio Nuica

II. Put an = n! in the previous example to get 0 = dimH(E) < 1 <dimP (E) = ln 7/ ln 3.

In order to prove that dimH(E) = 0, we need to show that ∀t < 1, Ht(E) =

0, that is ∀t < 1,72!+4!+...+(2n)!

3t(1!+2!+...+(2n+1)!)−→ 0. With some effort, this fact can

be proved directly by elementary computations. We give an alternative proof:remark that E2n+1 is a cover for E by 7bn balls of diameter δn := (1/3)c2n+1 ,so Nδn(E) ≤ 7bn (Nδ(E) is the smallest number of sets of diameter at most δthat can cover E), so

dimB(E) ≤ lim infn→∞

log Nδn(E)− log δn

≤ lim infn→∞

log 7bn

log 3c2n+1=

log 7log 3

limn→∞

bn

c2n+1= 0,

using definition of ”lim inf” and Stolz-Cesaro criterion. Then 0 ≤ dimH(E) ≤dimB(E) ≤ 0, so dimH(E) = 0.

On the other hand, since E is self-similar, by Theorem 1.2, dimP (E) =dimB(E). But, again, E2n is a packing for E by 7bn balls of diameter δn :=(1/3)c2n , so Nδn(E) ≥ 7bn (here Nδ(E) is considered to be the largest numberof disjoint balls of diameter δ with centers in E) and

dimB(E) ≥ lim supn→∞

log Nδn(E)− log δn

≥ lim supn→∞

log 7bn

log 3c2n=

log 7log 3

limn→∞

bn

c2n=

log 7log 3

,

using definition of ”lim sup” and Stolz-Cesaro criterion. Then dimP (E) =dimB(E) ≥ log 7/ log 3. But dimB(E) = log 7/ log 3 (supposing that dimB(E) >log 7/ log 3, it is easy to see that we contradict the definition of upper box-counting dimension). So, finally

0 = dimH(E) < dimP (E) = ln 7/ ln 3.

References[1] K. J. Falconer, The geometry of fractal sets, Cambridge University Press, 1985.

[2] K. J. Falconer, Fractal geometry. Mathematical foundations and applications, JohnWiley & Sons, New York, 2003.

[3] L. Olsen, Integral, probability, and fractal measures, by G. Edgar, Springer, New York,1998, Bull. Amer. Math. Soc., 37 (2000), 481-498.

[4] X. S. Raymond, C. Tricot, Packing regularity of sets in n-space, Math. Proc. Camb.Phil. Soc., 103 (1988), 133-145.

[5] C. A. Rogers, Hausdorff measures, Cambridge University Press, 1970.

[6] S. J. Taylor, C. Tricot, Packing measure and its evaluation for a brownian path, Trans.Amer. Math. Soc., 288, 2(1985).

[7] S. J. Taylor, C. Tricot, The packing measure of rectifiable subsets of the plane, Math.Proc. Camb. Phil. Soc., 99 (1986), 285-296.

ERROR HANDLING IN SOFTWARE SYSTEMS:MODELLING AND TESTING WITH FINITESTATE MACHINES

ROMAI J., 2, 1(2006),175–180

Radu OprisaAbstract Using the W method a test set of input sequences for the implementation of

a software system starting from the Finite State Machine corresponding tospecifications (SFSM) is obtained. To avoid halting, FSM corresponding toimplementation (IFSM) must be completely specified. For the size reasons,usually specifications do not contain error-handling description for each unex-pected input symbol. Due to this reason, for each state the designer must addin SFSM a corresponding error state and transitions to it for all unexpectedinput symbols. With this mechanism we are sure that IFSM is accepting allunexpected input sequences. But this will not prevent the problem of systemrecovery from error. In order to be able to perform a recovery from error wemust allow the system to rich the previous state from each error state by addingfor each error state a transition to the previous state. With this modeling ap-proach we have the advantage of detecting all possible errors and the certitudeof full error recovery in software systems at runtime.

Keywords: Finite State Machine, W method

2000MSC: 68N30

1. INTRODUCTIONBecause the market necessities are growing, the software systems became

more and more complex. Due to this, the effort to verify their correctness isalso severely increased. The cheapest way to check if the implementation of asoftware system with respect to its specifications is an automatic testing is tobe performed. For this purpose several methods had been developed, but allof them have specific limitations. An interesting approach is to model bothsystem specifications and implementation with Finite State Machine and todetermine a set of input sequences which can be used to prove or to invalidatetheir equivalence.

2. FUNDAMENTALSBefore generating the test set we need to introduce some basic notions

about Finite State Machines. The following notation will be used. If A isa finite alphabet, denote by A∗ the set of all finite sequences with membersin A. Let ε denote the empty sequence. For a, b ∈ A the concatenation

175

176 Radu Oprisa

of the sequence a with b is denoted by a • b . For U, V ⊆ A∗ we denoteU • V = a • b | a ∈ U, b ∈ V .Definition 2.1 A Deterministic Finite State Machine (DFSM) is an associ-ation of five elements M = (Q,Li, Lo, h, M0) where:

• Q is a finite set of states;

• Li is a finite set of input symbols;

• Lo is a finite set of output symbols;

• h : Q× Li → Q× Lo is the behaviour (partial) function;

• M0 is the initial state.

Definition 2.2 A DFSM M is said to be completely specified if h is a totalfunction.

Definition 2.3 Let M = (Q,Li, Lo, h,M0) be a DFSM, A ⊆ Li∗ a set ofinput sequences and q, r two states in Q. Then q and r are called A−equivalentif both produce the same responses for each input sequence a ∈ A. Otherwise,they are said to be A− distinguishable. If q and r give identical responses forany input sequence a ∈ Li∗ then q and r are said to be equivalent.

Definition 2.4 Let M = (QM , Li, Lo, hM ,M0) and N = (QN , Li, Lo, hN , N0)be two DFSMs with the same sets of input and output symbols and A ⊆ Li∗.Then M and N are said to be A− equivalent if their initial states M0 and N0

are A− equivalent. Otherwise, M and N are said to be A− distinguishable.If A = Li∗ then M and N are said to be equivalent.

Definition 2.5 A DFSM M is said to be minimal if any other equivalentDFSM has at least the same number of states as M .

Definition 2.6 A finite set S ⊆ Li∗ is called a state cover of a minimalDFSM M = (Q,Li, Lo, h, M0) if S contains the empty sequence and for everystate q ∈ Q, other than M0, S contains an input sequence a that may lead themachine from the initial state M0 to q.

In other words, a state cover is a set of input sequences that enables us toaccess any state in the machine from the initial state.

Definition 2.7 Let M = (Q,Li, Lo, h,M0) be a minimal finite state machine.T ⊆ Li∗ is called a transition cover of M if ∀q ∈ Q,∃a ∈ T so that a leadsthe machine from the initial state M0 to q and ∀x ∈ Li, a • x ∈ T .

In a DFSM M = (Q,Li, Lo, h,M0), for any state q ∈ Q there are sequencesin T that take M from M0 in q and then attempt to exercise transitions with

Error handling in software systems: modeling andtesting with Finite State Machines 177

all input symbols from Li whatever they exists or not. Remark that if S is astate cover of M , then T = S ∪ [S • Li] is a transition cover of M .

Definition 2.8 A finite set W ⊆ Li∗ is called a characterisation set of aDFSM M = (Q, Li, Lo, h, M0) if any two distinct states q, r ∈ Q are W−distinguishable.

Definition 2.9 Let S = (QS , Li, Lo, hS , S0) and I = (QI , Li, Lo, hI , I0) betwo DFSMs having the same sets of input and output symbols. A finite setY ⊆ Li∗ is called a test set for S and I if: S and I are Y− equivalent =⇒ Sand I are equivalent.

The following well-known theorem is determining a test set which can beused to find out if two DFSM are equivalent or not.

Theorem 2.1 Let M = (QM , Li, Lo, hM ,M0) and N = (QN , Li, Lo, hN , N0)be two minimal DFSM, T and W a transition cover and a characterisationset for one of these machines respectively. Then they are isomorphic (behaveidentically) if M0 and N0 are T • Z− equivalent.

In other words, if both DFSMs produce the same outputs when the se-quences from T • Z are applied, then they are equivalent machines.

3. THE W METHODLet S = (QS , Li, Lo, hS , S0) and I = (QI , Li, Lo, hI , I0) be two DFSMs

having the same sets of input and output symbols, where I has at least thesame number of states as S. Let d be the difference between the number ofstates of I and S. If S and I are deterministic and completely specified and Sis in addition minimal, then the W method determinates a test set which canbe used to check if S and I are equivalent.

The test set is determined as follows Y = Sc •Li[d+1]•W , where Sc ⊆ Li∗is a state cover of S, W ⊆ Li∗ is a characterisation set of S, Li[k] = ε∪Li∪. . . ∪ Lik for k ≥ 0.

4. ERROR STATESConsider the DFSM corresponding to specifications of a software system

(SFSM) from fig. 1. As we can see, this SFSM is not completely specified.This is a simple example, but especially for large systems, in practice fre-quently we have combinations of states and input symbols with no descriptionof the behaviour in the specifications. If the implementation is performed withrespect to this SFSM, then for all such combinations of states and input sym-bols, where the behaviour function is not defined, the program will halt. This

178 Radu Oprisa

Fig. 1. Partially specified SFSM.

Fig. 2. SFSM with an error state.

is an undesirable behaviour for any software system especially when a largevolume of user input data is required.

Example 4.1 Consider the specifications of a software system which shouldallow user to fill input data in a form with several fields. At the end, the usershould press ”Submit” button in order to process data. At runtime the userfills the form, but by mistake press the ”ALT+F4” key combination insteadof ”Submit” button. Because nothing is written in the specification about thebehaviour when ”ALT+F4” key combination occurs, the implementation hasan uncontrolled behaviour.

The most easy way for the designer to solve the problem of the uncontrolledbehaviour in SFSM is to add a new state, an error state E, and also transitionsto it for all possible states and input symbols which do not have correspondingtransitions in the initial system. Fig. 2 presents the extended SFSM fromfig. 1 with an error state and transitions to it. Now we have a controlledbehaviour when an input symbol occurs for which we have no description inthe specifications. But this is not enough for the user because the system willreach the E state and no possible action is defined from this state. Actually,even if the implementation will not halt, it is still impossible to recover inputdata from the system.

In order to solve this problem we need a more complex approach. Instead ofdefining only one error state, we will add an error state Ek for each state Sk ofthe initial SFSM and transitions Tke for all unexpected input symbols from theSFSM state to the corresponding error state. In order to be able to performalso a system recovery from the error states we need to add supplementary

Error handling in software systems: modeling andtesting with Finite State Machines 179

Fig. 3. SFSM with many error states.

Fig. 4. SFSM with many error states.

transitions Tkr from error states to the previous state. The set of input symbolswill be extended with a new one ok which will be used for all Tkr transitionswith interpretation that user noticed the error state and allow system recovery.It is possible to avoid the use of a new input symbol and to reuse an existingone. Two output symbols will be added: Err which is used for all transitionswhich ends at Ek states, and Cont which is used for Tkr transitions. Err isthe system response when an error state is reached. To have only one responsein case of any error in the system is not user friendly. Due to this fact, distinctoutput symbols can be added for each transition which ends at an error state.However, for test purposes one output symbol is enough.

The system from fig. 1 is extended as in fig. 3. To keep the figure simpleonly the error state for S3 is represented. This solves the problem of systemrecovery from error but the obtained SFSM still have a weak point. An errorstate is the initial state only for one transition which accepts only the inputsymbol ok (or the reused one). If another input symbol occurs when thesystem is in an error state, we are again in situation to obtain an uncontrolledbehaviour. This can be solved by adding transitions for each error state andeach unexpected input symbol to the error state itself. Such an extension ispresented in fig. 4. To keep the figure simple, only the error state for S3 isrepresented.

5. CONCLUSIONSUsing this modelling technique and W testing method, the designer will

be sure that the implementation will perform all requested operations de-scribed in the specifications. Moreover, all other actions initiated by the user

180 Radu Oprisa

which are not described in the specifications will lead to an error state. An-other big advantage is that the system can be recovered from any error stateand this will eliminate the possibility of loosing data.

References[1] Balanescu, T., Gheorghe, M., Ipate, F., Holcombe, M., Formal black box testing for

partially specified deterministic finite state machines, Fondations of Computing andDecision Sciences, 28, 1 (2003), 17-28.

[2] Balanescu, T. Generalized stream X- Machines with Output Delimited Type, FormalAspects of Computing, 12 (2000), 473-484.

[3] Chow,T.S., Testing software design modelled by finite state machines, IEEE Trans.Softw. Engng, 4, 3 (1978), 178-187.

[4] Fujiwara, S., von Bochmann, G., Khendek, F., Amalou, M., Ghedamsi, A., Test selec-tion based on finite state models, IEEE Trans. Softw. Engng., 17, 6 (1991), 591-603.

[5] Eilenberg, S., Automata machines and languages, Academic, New York, 1974.

[6] Gill, A., Introduction to theory of finite state machines, McGraw-Hill, New York, 1962.

[7] Hoare, C. A. R., Communicating sequential processes, Prentice Hall, Englewood Cliffs,1985.

[8] Holcombe, M., X-machines as a basis for dynamic system specification, Software En-gineering Journal, 3, 2 (1988), 69-76.

[9] Holcombe, M., An integrated methodology for the specification, verification and testingof systems, Software testing, verification and reliability, 3 (1993), 149-163.

[10] Howden, W. E., Functional program testing and analysis, McGraw-Hill, New York,1987.

[11] Ipate, F., Holcombe, M., Correct systems, Springer, London, 1998.

[12] Ipate, F., Holcombe, M., An integration testing method that is proved to find all faults,Int. J. Comput. Math., 63, 3/4 (1997), 159-178.

[13] Ipate, F., Theory of X-machines with applications in specification and testing, PhDthesis, Dept. of Computer Sci., Univ. of Sheffield, 1995.

[14] Ipate, F. and Holcombe, M., Another look at computation, Informatica, 20 (1996),359-372.

[15] Ipate, F., Holcombe, M., A method for refining and testing generalized ma-chine specifications, to appear in Int. Jour. Comp. Math. 1998. [WWW-http://www.dcs.shef.ac.uk/ wmlh/]

[16] Ipate, F., Holcombe, M., A theory of refinement and testing for X-machines, submitted.

[17] Ipate, F., Balanescu, T., Refinement in finite state machine testing, IOS Press, 2004.

[18] Morgan, C., Programming from specifications, Prentice Hall, Englewood Cliffs, 1994.

[19] Selec, B., Gullekson, G., Ward, P., Real-time object-oriented modeling, Wiley, NewYork, 1994.

[20] Sommerville, I., Sawyer, P., Requirements engineering - a good practice guide, Wiley,New York, 1997.

USING STOCHASTIC PETRI NETS INPERFORMANCE ANALYZE OF DISTRIBUTEDSYSTEMS

ROMAI J., 2, 1(2006),181–188

Norocel PetracheAbstract This paper shows how Stochastic Petri Nets (SPNs) (Petri Nets with timed

transitions and random, negative exponentially distributed firing delays) can beused to investigate performances of distributed systems. Because SPN systemsare isomorphic to continuous time Markov chains (CTMCs), their analysis canbe performed by construction of the state transition rate matrix. A model of asimple shared memory system and the software tool GreatSPN are described.

1. PETRI NETS AND TEMPORAL CONCEPTSThe concept of time was intentionally avoided in the original work by C. A.

Petri, because of the effect that timing may have on the behaviour of PNs: theassociation of timing constraints with the activities represented in PN mod-els or systems may prevent certain transitions from firing, thus invalidatingthe important assumption that all possible behaviours of a real system arerepresented by the structure of the PN.

Temporal concepts were introduced in Petri nets models in the mid 1970s.This initiated a hot debate about the suitability of such an extension, andabout the most appropriate ways to make it. Both Petri nets with timedplaces and timed transitions were proposed; in the latter case there was eithera time delay (before the atomic occurrence of a transition) or a time interval(between the consumption of input tokens and the generation of output to-kens). Whether deterministic or stochastic timing should be used was also anissue of the discussion.

The firing of a transition in a PN model corresponds to the event thatchanges the state of the real system. This change of state can be due to thecompletion of some activity. Transitions can be used to model activities, sothat transition enabling periods correspond to activity executions and transi-tion firings correspond to activity completions. Hence, time can be naturallyassociated with transitions. Transitions with associated temporal specifica-tions are called timed transitions; they are graphically represented by boxesor thick bars.

A timed transition T can be associated with a local clock or timer. WhenT is enabled the associated timer is set to an initial value. The timer is then

181

182 Norocel Petrache

decremented at constant speed and the transition fires when the timer reachesthe value zero. The timer can thus be used to model the duration of an activitywhose completion induces the state change that is represented by the changeof marking produced by the firing of T.

It is important to note that the activity is assumed to be in progress whilethe transition is enabled. This means that in the evolution of complex nets, aninterruption of the activity may take place if the transition loses its enablingcondition before it can actually fire. The activity may be resumed later on,during the evolution of the system in the case of a new enabling of the associ-ated transition. This may happen several times until the timer goes down tozero and the transition finally fires.

When introducing time into PN models and systems, it would be extremelyuseful not to modify the basic behaviour of the underlying untimed model.By so doing, it is possible to study the timed PNs exploiting the propertiesof the basic model as well as the available theoretical results. The additionof temporal specifications therefore must not modify the unique and originalway of expressing synchronization and parallelism that is peculiar to PNs. Inuntimed PN systems, the choice of which transition to fire in a free-choiceconflict is completely nondeterministic. In the case of timed PN systems, theconflict resolution depends on the delays associated with transitions and isobtained through the so-called race policy : when several timed transitions areenabled in a given marking M, the transition with the shortest associated delayfires first.

An important issue that arises at every transition firing when timed tran-sitions are used in a model is how to manage the timers of all the transitionsthat do not fire. From the modelling point of view, the different policies thatcan be adopted link the past history of the systems to its future evolutionconsidering various ways of retaining memory of the time already spent on ac-tivities. The question concerns the memory policy of transitions, and defineshow to set the transition timers when a state change occurs, possibly modify-ing the enabling of transitions. Two basic mechanisms can be considered fora timed transition at each state change:- continue: the timer associated with the transition holds the present valueand will continue later on the count-down;- restart: the timer associated with the transition is restarted, i.e., its presentvalue is discarded and a new value will be generated when needed.

The main reason for the introduction of temporal specifications into PNmodels is the interest for the computation of performance indexes. The intro-duction of temporal specifications in PN models must not reduce the modellingcapabilities with respect to the untimed case (it must be done so as not tomodify the basic behaviour and specifically the non-determinism of the choicesoccurring in the net execution). If the temporal specification is given in a de-

Using stochastic Petri nets in performance analyze of distributed systems 183

terministic way, the behaviour of the model is deterministically specified, andthe choices due to conflicting transitions may be always solved in the sameway; this would make many of the execution sequences in the untimed PNsystem impossible. Instead, if the temporal specifications are defined in a sto-chastic manner, by associating independent continuous random variables withtransitions to specify their firing delays, the non-determinism is preserved (inthe sense that all the execution sequences possible in the untimed PN systemare also possible in the timed version of the PN system).

2. STOCHASTIC PETRI NETSA simple sufficient condition to guarantee that the qualitative behaviour

of PN models with timed transitions is identical to the qualitative behaviourof the underlying untimed PN model (that is, to guarantee that the possibletransition sequences are the same in the two models) is that the delays associ-ated with transitions be random variables whose distributions have an infinitesupport. If this is the case, a probabilistic metrics transforms the nondeter-minism of the untimed PN model into the probabilism of the timed PN model,so that the theory of discrete-state stochastic processes in continuous time canbe applied for the evaluation of the performance of the real system describedwith the timed PN model.

Discrete-state stochastic processes in continuous time may be very diffi-cult to characterize from a probabilistic viewpoint and virtually impossibleto analyse. Their characterization and analysis become reasonably simplein some special cases that are often used just because of their mathematicaltractability. The simplicity in the characterization and analysis of discrete-state stochastic processes in continuous time can be obtained by eliminatingthe amount of memory in the dynamics of the process. This is the reasonfor the adoption of the negative exponential probability density function (pdf)for the specification of random delays. The negative exponential is the onlycontinuous pdf that enjoys the memoryless property, i.e., the only continuouspdf for which the residual delay after an arbitrary interval has the same pdfas the original delay.

Stochastic Petri Nets (SPNs) were proposed, simultaneously, by S. Natkin[Nat80] and M. K. Molloy [Mol82]. The new net class paved the way to theutilization of Petri nets for performance evaluation. In a SPN, a firing delayis associated with each transition. Firing delays are instances of random vari-ables that have a negative exponential probability distribution. The rates aresufficient to characterize the pdf of the transition delays (the only parameterof the negative exponential pdf is its rate, obtained as the inverse of the mean).The selection of the transition instance to fire among the set of enabled onesfollows a race policy (the transition that has drawn the least delay is the one


that fires). Each timed transition can be used to model the execution of anactivity in a distributed environment; all activities execute in parallel (unlessotherwise specified by the PN structure) until they complete. At completion,activities induce a local change of the system state that is specified with theinterconnection of the transition to input and output places.

No special mechanism is necessary for the resolution of timed conflicts be-cause the probability that two timers expire at the same instant is zero. Thenegative exponential pdf is a continuous function defined in the interval [0,1),that integrates to one. The lack of discontinuities in the function makes theprobability of any specific value x being sampled equal to zero (obviously, theprobability that a value is sampled between two distinct values x1 ≥ 0 andx2 > x1 is positive). Thus, the probability of two timers expiring at the sametime is null (given the value sampled by the first timer, the probability that thesecond one samples the same value is zero). The memoryless property of thenegative exponential pdf, at any time instant, makes the residual time until atimer expires statistically equivalent to the originally sampled timer reading.Thus, whether a new timer value is set or not at every change of markingmakes no difference from the point of view of the probabilistic metrics of theSPN. A formal definition of SPN is the following.

Definition 2.1 SPN = (P, T, A, M, λ ), where P is the set of places, T isthe set of transitions, P

⋂T = θ, P

⋃T 6= θ, A is the set of input and output

arcs, A ⊆ (P × T )⋃

(T × P ), M is the initial marking and λ is the set oftransition rates.A marking of a stochastic Petri net is a distribution of tokens in its places;a marking may be viewed as a mapping from the set of places to the naturalnumbers. A stochastic Petri net is said to be k-bounded if there exists a finitenonnegative integer k such that for every marking M in the reachability set,and for every place Pi M(Pi) < k. With each marking we can associate astate of the system and in the following the terms state and marking are usedwith essentially the same meaning.

Molloy [Mol81] has proved that there is an isomorphism between k-boundedSPNs and finite Markov processes. Two stochastic systems are isomorphic if- there are one-to-one mappings between the state space of the two systems,and between the set of state transitions of the two systems,- the probability of a transition from one state to another in one system equalsthe probability of a transition between the corresponding states of the othersystem.

The SPNs are isomorphic to continuous time Markov chains due to thememoryless property of the negative exponential distribution of firing times.The SPN markings correspond to states of the corresponding Markov chain.Since the size of the state space of a SPN is equal to the size of Markov processspace, the complexity of solving an SPN model is the same as in the case of


the model based upon the Markov process; the methodology used to find thesteady-state solution for a Markov chain can be used for an SPN system inwhich each marking corresponds to a Markov state. Although the SPNs donot provide more modelling power than Markov processes, they can be usedas a convenient description of the system being modelled.

3. THE STOCHASTIC PROCESS ASSOCIATEDWITH A SPN

SPN systems are isomorphic to continuous time Markov chains (CTMCs); k-bounded SPN systems are isomorphic to finite CTMCs. The CTMC associatedwith a given SPN system is obtained by applying the following rules:

1. the CTMC state space S = si corresponds to the reachability setRS(M0) of the PN associated with the SPN (si ↔ Mi)

2. the transition rate from state si (corresponding to marking Mi) to statesj (Mj) is obtained as the sum of the firing rates of the transitions that areenabled in Mi and whose firings generate marking Mj .

It is possible to develop algorithms for the automatic construction of theinfinitesimal generator (also called the state transition rate matrix) of theisomorphic CTMC, starting with the SPN description. Denoting this matrixby Q, with wk the firing rate of Tk, and with Ej(Mi) = Th ∈ E(Mi) :Mi[Th > Mj the set of transitions whose firings bring the net from markingMi to marking Mj , the components of the infinitesimal generator are

qij =

− ∑k:Tk∈E(Mi)

wk, i = j

∑k:Tk∈Ej(Mi)

wk, i 6= j

In SPN analysis, as in Markov analysis, ergodic (irreductible) systems are ofspecial interest. A k-bounded SPN system is said to be ergodic if it generatesan ergodic CTMC; in [FN85] is showed that a SPN system is ergodic if M0,the initial marking, is a home state (a marking M ∈ RS(M0) is a home stateif and only if ∀N ∈ RS(M0), M ∈ RS(N)). For ergodic SPN systems, thesteady-state probability of the system being in any state always exists and isindependent of the initial state.

Let the row vector η represent the steady-state probability distribution onmarkings of the SPN; to compute η we solve the linear system expressed inmatrix form as

ηQ = 0,

η1T = 1,


where 0 is row vector with all its components equal to zero and 1T is a columnvector with all its components equal to one, used to enforce the normalizationcondition. We will use ηi instead of η(Mi) to denote the steady state prob-ability of marking Mi. The sojourn time is the time spent by the system ina given marking. Because a CTMC can be associated with the SPN system,the sojourn time in marking Mi is exponentially distributed with rate qi. Thepdf of the sojourn time in a marking corresponds to the pdf of the minimumamong the firing times of the transitions enabled in the same marking. Theprobability that a given transition Tk ∈ E(Mi) fires first in marking Mi hasthe expression: PTk/Mi = wk

qiand the average sojourn time in marking Mi

is given by E[SJi] = 1qi

.To allow the efficiency of the SPN system to be evaluated, performance

indices can be computed by using reward functions with different interpreta-tions: the probability of a particular condition, the mean number of firings perunit of time of a given transition, the expected value of the number of tokensin a given place etc. If r(M) is a reward function, the average reward can becomputed with the following weighted sum

R =∑

i:Mi∈RS(M0)

r(Mi)ηi.

In order to illustrate this analysis step, the following example [MBCDF94] ispresented.

Fig. 1. A shared memory system

In fig. 1, two processors try to access a common shared memory; in theinitial marking, the two processors are both in a locally active state (pact1 +pact2+pidle). Processor 1 works locally for an exponentially distributed randomamount of time with average 1

λ1, and then requests an access to the common

memory (Treq1 fires). If common memory is available, (place pidle is marked),


the acquisition of the memory starts and takes an average of 1α1

units of timeto complete. After transition Tstr1 fires, processor 1 uses the common memory(which is not available for processor 2) for 1

µ1units of time (on average) and

when transition Tend1 fires, the net is returning to its initial state. A similarprocessing cycle is possible for processor 2.

The evolution of the net describes the interleavings between the activitiesof the processors. A conflict exists when transitions Tstr1 and Tstr2 are bothenabled (both processors want to simultaneously access the common memory).Transition Tstr1 fires with probability PTstr1 = α1

α1+α2, whereas transition

Tstr2 fires with probability PTstr2 = α2α1+α2

. The conflict is resolved whenthe first transition fires and the speed at which the PN model exits fromthis marking is the sum of the individual speeds of the two transitions. Thereachability set is M0 = pact1 + pidle + pact2, M1 = preq1 + pidle + pact2,M2 = pacc1 + Pact2, M3 = pacc1 + preq2, M4 = preq1 + pidle + preq2, M5 =pact1+pidle+preq2, M6 = pact1+pacc2, M7 = preq1+pacc2 and the infinitesimalgenerator of the Markov chain is Q=

−λ1 − λ2 λ1 0 0 0 λ2 0 00 −α1 − λ2 α1 0 λ2 0 0 0µ1 0 −λ2 − µ1 λ2 0 0 0 00 0 0 −µ1 0 µ1 0 00 0 0 α1 −α1 − α2 0 0 α2

0 0 0 0 λ1 −α2 − λ1 α2 0µ2 0 0 0 0 0 −λ1 − µ2 λ1

0 µ2 0 0 0 0 0 −µ2

The system of linear equations whose solution yields the steady-state distrib-ution over the CTMC states is

(η0, η1, η2, η3, η4, η5, η6, η7)Q = 0,7∑

i=0ηi = 1.

Assuming λ1 = 1, λ2 = 2, α1 = α2 = 100, µ1 = 10 and µ2 = 5, we obtainη0 = 0.61471, η1 = 0.00842, η2 = 07014, η3 = 0.01556, η4 = 0.00015, η5 =0.01371, η6 = 0.22854, η7 = 0.04876.

Since one token is present in place pidle in markings M0,M1,M4 and M5,the average number of tokens in pidle is E[M(pidle)] = η0 + η1 + η4 + η5 =0.637 and the utilization of the shared memory is U [sharedmemory] = 1.0 −E[M(pidle)] = 0.363.

Since Treq1 is enabled only in M0,M5 and M6, the rate of access to theshared memory from the first processor (the throughout of Treq1) is freq1 =(η0 + η5 + η6)λ1 = 0.8570.


These results show that Stochastic Petri Nets can be used as a tool forcomputing performance indices that allow the efficiency of modelled systemsto be evaluated.

Developed at Universita di Torino, GRaphical Editor and Analyzer forTimed and Stochastic Petri Nets (GreatSPN) [Chi91] is a software package forthe modeling, validation, and performance evaluation of distributed systemsusing Stochastic Petri Nets. It runs on Linux (gcc, Motif 1.2 and X11R6 are re-quired) and it offers graphical model editing, definition of timing and stochas-tic specifications, parameters, and performance measures, Markovian solversfor steady-state, graphical representation of performance results and graphicalinteractive simulation of stochastic models. The package is available for freefor universities and non-profit organizations (http://www.di.unito.it/ great-spn/index.html).

References[Chi91] G. Chiola , GreatSPN 1.5 software architecture, in Proc. 5th Int. Conf. Modeling

Techniques and Tools for Computer Performance Evaluation, Torino, Italy, 1991.

[FN85] G. Florin, S. Natkin, Les reseaux de Petri stochastiques, Technique et Science Infor-matiques 4, 1, 143-160.

[MBBCCC85] M. A. Marsan, G. Balbo, A. Bobbio, G. Chiola, G. Conte, A.C. Cumani, OnPetri nets with stochastic timing, in Proc. Int. Workshop Timed Petri Nets, Torino,Italy, July 1985, 80-87.

[MBCDF94] M. A. Marsan, G. Balbo, G. Conte, S. Donatelli, G. Franceschinis, Modellingwith generalised stochastic Petri nets, Universita degli studi di Torino, Dipartimentodi Informatica, Torino, Italy, 1994.

[Mol81] M.K. Molloy, On the integration of delay and throughout measures in distributedprocessing models, Ph. D. Thesis, Univ. of California, Los Angeles, Sept. 1981.

[Mol82] M. K. Molloy, Performance analysis using stochastic Petri nets, IEEE Trans. Com-put., vol. C-31, Sept. 1982, 913-917, .

[Nat80] S. Natkin, Le reseaux de Petri stochastiques et leur application a l’evaluation des

systems informatiques, These de Docteur Ingenieur, CNAM, Paris, 1980.

SOLVING METHOD FOR MULTIPLEOBJECTIVE FUZZY CONSTRAINTS MIXEDVARIABLES OPTIMIZATION

ROMAI J., 2, 1(2006),189–192

Bogdana Pop, Silviu Dumitrescu, Vlad Monescu”Transilvania” University of Brasov, Dept. of Computer Science

Abstract In the classical problems of mathematical programming the coefficients of theproblems are assumed to be exactly known. This assumption is seldom satisfiedby the great majority of real-life problems. Taking into account the multiplecriteria optimization in the environment of fuzzy constraints and mixed vari-ables, a solving method is developed here.

Keywords: fuzzy constraints, mixed variables, multiple criteria.

1. INTRODUCTIONIn the real life the certainty in and the solidity of data accuracy are illu-

sory. The possibility to obtain an optimal solution is also under the influencesof some data missing. Uncertainty has to be taken into consideration whenreal systems are analyzed. The compromise to accept the uncertainty intothe mathematical models must be done working with complex systems. Inorder to build and to solve real life problem it is natural to interpret the in-formation in fuzzy manner. The philosophy of fuzzy sets theory is related tothe model of human thinking and making decisions. Concepts of fuzzy setstheory ”crowded” into a lot of research fields since 1980, when fuzzy logichas had a great success in the control systems theory. Real advantages of afuzzy approach to solve optimization problems could be highlighted when acomparison is made with the stochastic methods to deal with imprecision ([4, 9]).

”The best compromise” (or Pareto-optimality, efficiency, non-domination)is the central concept in multiple criteria optimization only because an optimalsolution for one objective function is not necessary an optimal solution for oth-ers. In the classical theory weak-efficiency, strong-efficiency, proper-efficiencyare gradual definitions for the same essence. Starting with classical definitions,the following concepts appear in the fuzzy multiple objective optimization:M-Pareto optimality (related to the membership functions of ”equal” goals),alpha-Pareto optimality (related to the multiple objective programming prob-lems associated with each alpha-level set), M-alpha-Pareto optimality (related

189

190 Bogdana Pop, Silviu Dumitrescu, Vlad Monescu

to the multiple objective programming problems associated with each alpha-level set after that ”fuzzy equal” goals were defined) [1, 2].

In [6], Perkgoz et al. focused on multiobjective integer programming prob-lems with random variable coefficients in objective functions and/or con-straints. In [11], Sakawa and Kato presented an interactive fuzzy satisfy-ing method for nonlinear integer programming problems through a geneticalgorithm. In [10] α-Pareto optimal solutions were determined for the mul-tiobjective integer nonlinear programming problems having fuzzy parametersin the constraints together with the corresponding stability set of the firstkind. The paper [5] presented a method useful in solving a special class oflarge-scale multiobjective integer problems based upon a combination of thedecomposition algorithm coupled with the weighting method together withthe branch-and-bound method.

Taking into account the multiple criteria optimization in the environmentof fuzzy constraints and mixed variables, a solving method is developed here.In what follows a multiple objective fuzzy constraints mixed variables opti-mization problem (MOFCMVOP) will be taken into account.

2. THE MOFCMVOP MODELConsider the multiple objective integer programming problem with fuzzy

constraints and mixed variables

”min ”z (x) = (z1(x), z2(x), ..., zp(x)) | x ∈ X

, (1)

where

(i) X =x ∈ Rn | Ax≤b, x ∈ Zk ×Rr

is the feasible set of the problem,

(ii) A is an m× n constraint matrix, x is an n-dimensional vector of mixed(k−integer, r−real) decision variable and b ∈ Rm,

(iii) p ≥ 2,

(iv) zi (x) , i = 1, p are the objective functions which could be linear, lin-ear fractional or convex functions (in order to make the new methodworkable).

The term ”min” used in Problem (1) is for finding all efficient solutionsin a minimization sense in terms of the Pareto optimality. A possible way tohandle constraints imprecision is to consider the following parametric problem

” min ” z (x) = (z1(x), z2(x), ..., zp(x)) | x ∈ X(θ) (2)

where X(θ) =x ∈ Rn | Ax ≤ b + θb′, x ∈ Zk ×Rr

is the feasible set of the

problem, θ ∈ [0, 1] and b′ is a given perturbation vector [13].

Solving method for multiple objective fuzzy constraints mixed variables optimization 191

3. THE SOLVING METHODIn [7] an algorithm to solving multiobjective linear fractional programming

problem (MOLFPP) is presented. In [8] the algorithm is used to solve afuzzy multiple objective integer programming problem. We will use the abovealgorithm (under the name MultiObjAlg(θ)) to solve Problem (2) meaningthe deterministic problem with the feasible set X(θ). In order to improve theinteractivity of the method different values for θ will be considered.

Wang and Horng [14] proposed an approach to perform complete parametricanalysis in integer programming by considering all possible candidates of θ.They defined the principal candidates of θ as being that θ which makes bi +θb

′i

an integer. We also work with these θ’s. In [13], Wang and Liao proposeda heuristic algorithm to analyze the same fuzzy problem. We will use theirmethod to solving multiple objective problem with integer variables and fuzzyconstraints.

For a fixed value θ we start defining Problem (3) as Problem (2) withoutthe integrity restriction of variables.

”min ”z (x) = (z1(x), z2(x), ..., zp(x)) | Ax ≤ b + θb′

. (3)

We obtain an efficient solution for Problem (3) using MultiObjAlg(1). It isa solution for Problem (2) if and only if its components are integer numbers.In this case it is also solution for Problem (1) but with minimal degree in fuzzyenvironment. Consequently, our next goals are to transform the solution intoa mixed (k−integer, r−real) one and also to improve its fuzzy degree. Thesegoals could be attend modifying values xi in [xi] or [xi]+1 for i = 1, k such thatthe deviation of the perturbation vector decreases using a modified back(p, r)procedure described in [8].

Taking into account the above remarks the solving algorithm could be de-scribed as follows.

Step 1. Define the thresholds 0 = θ1 < θ2 < ... < θq = 1 using theprincipal candidates of parameters θ. Put q = 1.

Step 2. For j = q down to 1

– Compute values xj = (x1, x2, ..., xn) using MultiObjAlg(θj).– if xj ∈ Zk ×Rr then favorable STOP with xj the θj−fuzzy degree

acceptable solution of the problem.– otherwise call back(1, j, yj) and obtain xj′ = (x′1, x

′2, ..., x

′n). Iden-

tify jmin such that Ax′ < b+θjminb′. Then xjmin is the θjmin−fuzzydegree acceptable solution of the problem.

Step 3. Describe the problem solution as (yj)j=1,q.

192 Bogdana Pop, Silviu Dumitrescu, Vlad Monescu

References[1] N. M. Badra, Stability of multiciteria nonlinear programming problems using fuzziness ,

Fuzzy Sets and Systems, 125(2002), 343-354.

[2] M. Chakraborty, S. Gupta, Fuzzy mathematical programming for multi objective linearfractional programming problem, Fuzzy Sets and Systems, 125(2002), 335-342.

[3] C. Fabian, M. Stoica, Fuzzy integer programming, TIMS /Studies in the ManagementSciences, 20, H. J. Zimmermann, L. A. Zadah, B. R. Gaines, (eds.), 1984, 123-131.

[4] M. Inuiguchi, J. Ramık, Possibilistic linear programming: a brief review of fuzzy math-ematical programming and a comparison with stochastic programming in portfolio se-lection problem, Fuzzy Sets and Systems, 111(2000), 3-28.

[5] M. S. Osman, O. M. Saad, A. G. Hasan, Solving a special class of large-scale fuzzymultiobjective integer linear programming problems , Fuzzy Sets Syst., 107, 3(1999),289-297.

[6] C. Perkgoz, K. Kato, H. Katagiri, M. Sakawa, An interactive fuzzy satisfying method formultiobjective stochastic integer programming problems through variance minimizationmodel , Sci. Math. Jp., 60, 2(2004), 327-336.

[7] B. Pop, Algorithm to solving multiple criteria linear fractional optimization , An. Univ.Bucuresti Mat. Inform., 1(2004), 105-114.

[8] B. Pop, Solving method for fuzzy multiple objective integer optimization , Presented atthe 7th Balkan Conference on Operational Research, May 25-28, Constanta, Romania,2005.

[9] H. Rommelfanger, The advantages of fuzzy optimization models in practical use , FuzzyOptimization and Decision Making, 3(2004), 295-309.

[10] O.M. Saad, On the solution of fuzzy multiobjective integer nonlinear programming prob-lems with a parametric study , J. Fuzzy Math. 7, 4(1999), 835-843.

[11] M. Sakawa, K. Kato, An interactive fuzzy satisfying method for multiobjective nonlinearinteger programming problems through genetic algorithms , Lect. Notes Comput. Sci.,2715(2003), 710-717.

[12] I. M. Stancu-Minasian, Fractional programming: Theory, methods and applications ,Kluwer, Dordrecht, 1997.

[13] H.-F. Wang, Y.-C. Liao, Fuzzy non-linear integer program by parametric programmingapproach, Fuzzy Sets Syst. 122, 3(2001), 245-251.

[14] H.-F. Wang, J.S. Horng, Structural approach to parametric analysis of an IP on the

case of the right-hand-side, Europ. J. Oper. Res. 92(1996), 148-156.

ON THE APPLICABILITY OF THELINDSTEDT-POINCARE METHOD IN THETWO-DIMENSIONAL CASE

ROMAI J., 2, 1(2006),193–196

Nicolae - Doru StanescuDepartment of Applied Mechanics, University of Pitesti

Abstract In this note we give sufficient conditions for a system of two second orderordinary differential equations (ode’s) be transformed into the canonical form,i.e. to which the Lindstedt-Poincare method applies. We also exemplify thetheory by an application.

Keywords: Lindstedt-Poincare method, canonical form

2000 MSC: 74B20, 70K42

1. ONE-DIMENSIONAL CASEThe Lindstedt-Poincare method is applied to the canonical form of ode

u + f (u) = 0, (1)

where f is a nonlinear function [1]. In addition, f is assumed to be analyti-

cal at the origin, therefore there exists the following expansion f =∞∑i=1

aiui,

with ai = 1i! f

(i) (0). If the critical point is not u0 = 0, the translation

x = u − u0 transforms f and ai into f =∞∑i=1

aixi and ai = 1

i! f(i) (u0), re-

spectively. Correspondingly, in the classical Lindstedt-Poincare method, the

solution of the equation x +∞∑i=1

aixi = 0 is written as a formal asymptotic

expansion, x (t, ε) = εx1 (t) + ε2x2 (t) + ε3x3 (t) + . . . With the notationω0 =

√a1 =

√f’ (u0) and performing a change of the time variable τ = ωt,

where ω = ω0 + εω1 + ε2ω2 + . . ., ε being a small dimensionless parameter,the equation becomes

(ω0 + εω1 + ε2ω2 + . . .

) d2

dτ 2

(εx1 + ε2x2 + ε3x3 + . . .

)+

∞∑i=1

(εx1 + ε2x2 + ε3x3 + . . .

)= 0.

193

194 Nicolae - Doru Stanescu

2. TWO-DIMENSIONAL CASEConsider now the two-dimensional case, i.e. of two nonlinear second order

ode’s in two unknown functions

x + f1 (x, y) = 0, y + f2 (x, y) = 0, (2)

where the functions f1 and f2 are analytical and the following expansions hold

f1 (x, y) = ∂f1∂x

∣∣∣(0,0)

x + ∂f1∂y

∣∣∣(0,0)

y+

12

(∂2f1∂x2

∣∣∣(0,0)

x2 + 2 ∂2f1∂x∂y

∣∣∣(0,0)

xy + ∂2f1∂y2

∣∣∣∣(0,0)

y2

)+ . . . ,

f2 (x, y) = ∂f2∂x

∣∣∣(0,0)

x + ∂f2∂y

∣∣∣(0,0)

y+

12

(∂2f2∂x2

∣∣∣(0,0)

x2 + 2 ∂2f2∂x∂y

∣∣∣(0,0)

xy + ∂2f2∂y2

∣∣∣∣(0,0)

y2

)+ . . . ,

where the critical point is assumed to be (x, y) = (0, 0). Then (1) becomes

x + ∂f1∂x

∣∣∣(0,0)

x + ∂f1∂y

∣∣∣(0,0)

y + f1 (x, y) = 0,

x + ∂f2∂x

∣∣∣(0,0)

x + ∂f2∂y

∣∣∣(0,0)

y + f2 (x, y) = 0,(3)

with the obvious notation. Let us find a linear transformation

x = αξ1 + βξ2, y = γξ1 + δξ2, (4)

i.e. deduce the coefficients α, β, γ and δ, such that the equation in ξ1 (respec-tively ξ2) has no term in ξ2 (respectively ξ1). We have

αξ1 + βξ2 + ∂f1∂x

∣∣∣(0,0)

(αξ1 + βξ2) + ∂f1∂y

∣∣∣(0,0)

(γξ1 + δξ2) + f∗1 (ξ1, ξ2) = 0

γξ1 + δξ2 + ∂f2∂x

∣∣∣(0,0)

(γξ1 + δξ2) + ∂f2∂y

∣∣∣(0,0)

(αξ1 + βξ2) + f∗2 (ξ1, ξ2) = 0,

(5)where f∗1 (ξ1, ξ2) = f1 (αξ1 + βξ2, γξ1 + δξ2), f∗2 (ξ1, ξ2) = f2 (αξ1 + βξ2, γξ1 + δξ2).Multiplying (4-1) by −δ, (4-2) by β and adding the obtained equations, wefind

(−αδ + βγ) ξ1 + ξ1

(−αδ ∂f1

∂x

∣∣∣(0,0)

− γδ ∂f1∂y

∣∣∣(0,0)

+ αβ ∂f2∂x

∣∣∣(0,0)

+ γβ ∂f2∂y

∣∣∣(0,0)

)+

+ξ2

(−βδ ∂f1

∂x

∣∣∣(0,0)

− δ2 ∂f1∂y

∣∣∣(0,0)

+ β2 ∂f2∂x

∣∣∣(0,0)

+ βδ ∂f2∂y

∣∣∣(0,0)

)−

−δf∗1 (ξ1, ξ2) + βf∗2 (ξ1, ξ2) = 0.(6)

On the applicability of the Lindstedt-Poincare method in the twodimensional case 195

Multiplying (4-1) by −γ, (4-2) by α and adding the results, we have

(−βγ + αδ) ξ2 + ξ1

(−αγ ∂f1

∂x

∣∣∣(0,0)

− γ2 ∂f1∂y

∣∣∣(0,0)

+ α2 ∂f2∂x

∣∣∣(0,0)

+ αγ ∂f2∂y

∣∣∣(0,0)

)+

+ξ2

(−βγ ∂f1

∂x

∣∣∣(0,0)

− αγ ∂f1∂y

∣∣∣(0,0)

+ αβ ∂f2∂x

∣∣∣(0,0)

+ αδ ∂f2∂y

∣∣∣(0,0)

)−

−γf∗1 (ξ1, ξ2) + αf∗2 (ξ1, ξ2) = 0.(7)

Imposing to the term in ξ2 in (5) and term in ξ1 in (6) to vanish it follows

−βδ ∂f1∂x

∣∣∣(0,0)

− δ2 ∂f1∂y

∣∣∣(0,0)

+ β2 ∂f2∂x

∣∣∣(0,0)

+ βγ ∂f2∂y

∣∣∣(0,0)

,

−αγ ∂f1∂x

∣∣∣(0,0)

− γ2 ∂f1∂y

∣∣∣(0,0)

+ α2 ∂f2∂x

∣∣∣(0,0)

+ αγ ∂f2∂y

∣∣∣(0,0)

.(8)

Note that the two equations (7) coincide if α → β and γ → δ.We are lead to the general problem: which are the necessary and sufficient

conditions for the equation Auv+Bu2 +Cv2 = 0 have real solutions for u andv? Looking at this equation as a second degree equation in the unknown u ifB 6= 0, its discriminant is

∆ =(A2 − 4BC

)v2. (9)

For B = 0, one real solution always exists, so further on it is understoodthat B 6= 0. Therefore, the equation has real solution if

A2 − 4BC ≥ 0. (10)

With the notation

A = −∂f1∂x

∣∣∣(0,0)

+ ∂f2∂y

∣∣∣(0,0)

, B = − ∂f1∂y

∣∣∣(0,0)

, C = ∂f2∂x

∣∣∣(0,0)

, (11)

the condition (9) for the equation (7) reads(

∂f2∂y

∣∣∣∣(0,0)

)2

+

(∂f1∂x

∣∣∣∣(0,0)

)2

− 2∂f2∂y

∣∣∣∣(0,0)

∂f1∂x

∣∣∣∣(0,0)

+ 4∂f12

∂y

∣∣∣∣(0,0)

∂f2∂x

∣∣∣∣(0,0)

≥ 0.

(12)Assuming that the condition (11) holds, the system (2) can be written as

A1ξ1 + B1ξ1 +¯f1 (ξ1, ξ2) = 0, A21ξ2 + B21ξ2 +¯f2 (ξ1, ξ2) = 0, (13)

with A1 = −αδ + βγ, A2 = −βγ + αδ, B1 = −αδ ∂f1∂x

∣∣∣(0,0)

− γδ ∂f1∂y

∣∣∣(0,0)

+

αβ ∂f2∂x

∣∣∣(0,0)

+ γβ ∂f2∂y

∣∣∣(0,0)

, B2 = −βγ ∂f1∂x

∣∣∣(0,0)

− αγ ∂f1∂y

∣∣∣(0,0)

+ αβ ∂f2∂x

∣∣∣(0,0)

+

196 Nicolae - Doru Stanescu

αδ ∂f2∂y

∣∣∣(0,0)

, ¯f1 (ξ1, ξ2) = −δf∗1 (ξ1, ξ2)+βf∗2 (ξ1, ξ2) = 0, ¯f2 (ξ1, ξ2) = −γf∗1 (ξ1, ξ2)+

αf∗2 (ξ1, ξ2) = 0. Since the condition D ≡∣∣∣∣

α βγ δ

∣∣∣∣ 6= 0, always holds because

(3) is a transformation, the system (12) becomes

ξ1 + a11ξ1 + g1 (ξ1, ξ2) = 0, ξ2 + a22ξ2 + g2 (ξ1, ξ2) = 0, (14)

where a11 = B1

A1, a22 = B2

A2, g1 =

¯f1(ξ1, ξ2)A1

, g2 =¯f2(ξ1, ξ2)

A2, and it is referred

to as the canonical form. Consequently, a sufficient condition for (1) to betransformed into the canonical form is (11) and B 6= 0 and C 6= 0.

Similarly, several conditions can be written in the case B = 0 (e.g. A, C 6=0) or C = 0.

3. APPLICATIONLet us transform the system

x + 3x + 2y + x3 = 0, y + 2x + 3y + y3 = 0. (15)

into the canonical form.The functions f1 and f2 are f2 = 3x + 2y + x3 and f1 = 2x + 3y + y3. The

unique critical point is (x, y) = (0, 0). We have ∂f1∂x

= 3 + 3x2, ∂f1∂y

= 2,∂f2∂x

= 2, ∂f2∂y

= 3 + 3y2, such that (10) imply A = 0, B = −2, C = 2, and

the discriminant in the relation (8) is ∆ = 16v2. We obtain u = v or u = −v.Choosing α = 1, β = 1, γ = −1, δ = 1, (3) becomes x = ξ1 + ξ2, y = −ξ1 + ξ2,and the canonical form is

ξ1 + ξ1 + ξ31 + 3ξ1ξ

22 = 0; ξ2 + 5ξ2 + 3ξ2

1ξ2 + ξ32 = 0. (16)

Throughout the article, it is assumed that the conditions ensuring the ex-istence of periodic solutions are fulfilled.

References[1] Minorsky, N., Nonlinear Oscillations, Van Nostrand Reinhold, New York, 1962.

[2] Nayfeh, A., H., Mook, D., T., Nonlinear Oscillations, John Wiley & Sons, New York,1979.

THE GENERALIZED ALGORITHM FORSOLVING THE FRACTIONAL MULTI-OBJECTIVE TRANSPORTATION PROBLEM

ROMAI J., 2, 1(2006),197–202

Alexandra I. TkacenkoDept. of Appl. Math., Moldova State [email protected]

Abstract An algorithm is proposed to solve a multicriterial ”bottleneck” transportationmodel.

Keywords: nonlinear programming, transportation model.

The transportation problem dealing with the total cost minimizing criterion,considered as a classical one, is well-known and sufficiently analyzed in therespective sources.

The transportation model of a ”bottleneck” type is a specific problem withinthe transportation classical issue, the objective function of which is a nonlinearone. Special cases of these types of problems are investigated in many papers,e.g. [1], [2], [5], [6], where concrete algorithms used to solve them are carriedout. The transportation model of the ”bottleneck” type with two criteria,where the first one is providing the total transportation cost minimization andthe second one, that is nonlinear, is strangling in time, is studied in article[7], where the authors propose the concrete algorithm to solve it. The specialalgorithm for solving transportation model of the ”bottleneck” type with 3criteria is presented in paper [8], where it is tested on a concrete example.

In our daily life the multiobjective fractional programming models are ofgreat interest. We are often concerned about the optimization of the ratioslike the summary cost of the total transportation expenditures to the maximalnecessary time to satisfy the demands, the total benefits or production valuesinto time unit, the total depreciation into time unit and many other importantsimilar criteria, which may appear in order to evaluate the economical activ-ities and make the correct managerial decisions. These problems led to the”bottleneck” transportation model with multiple fractional criteria, where the”bottleneck” criteria appear as a ”minmax” time constraining. The commoncharacteristic of these objective ratios is the identical denominators. Concretealgorithms for solving special models of transportation type with one criterion,where the objective function is a fractional one, are proposed in [3],[4].

The multicriterial transportation model of ”bottleneck” type with two frac-tional criteria is defined as follows

197

198 Alexandra I. Tkacenko

min z1 =

m∑

i=1

n∑

j=1

cijxij

maxi,jtij |xij > 0 (1)

min z2 =

m∑

i=1

n∑

j=1

dijxij

maxi,jtij |xij > 0 (2)

min z3 = maxi,jtij |xij > 0 (3)

in the conditions n∑

j=1

xij = ai, ∀i = 1,m (4)

m∑

i=1

xij = bj ,∀j = 1, n (5)

m∑

i=1

ai =n∑

j=1

bj (6)

xij ≥ 0, i = 1, m, j = 1, n (7)

where cij - cost of transporting a unit from the source i to destination j, dij

- deterioration of a unit while transporting from the source i to destinationj, ai - availability at source i, bj - requirement at destination j, xij - amounttransported from source i to destination j, tij - time of transporting a unitfrom source i to destination j.

A non traditional algorithm of building numerous efficient solutions of themodels is carried out here. It is useless to look for an optimal solution to settlethe multicriterial mathematical models. Indeed, as it often occurs, there areno solutions at all.

That is why, one should better determine the multitude of non-dominantsolutions, which are known as efficient solutions or optimal in the sense ofPareto.

In order to solve the multicriteria model the notion of an efficient solutionhas been introduced.

Definition 0.1 The feasible solution for the multicriterial model is consideredto be efficient iff there exists no other feasible solution, for which we obtaina better value at least for one criterion while the values of the rest criteriaremain unmodified.

In order to solve the problem (1)- (7) by finding the set of the efficient basicsolutions, we reduce it to the following model

The generalized algorithm for solving the fractional multiobjectivetransportation problem 199

min z1 =m∑

i=1

n∑

j=1

cijxij , min z2 =m∑

i=1

n∑

j=1

dijxij , min z3 = maxi,jtij |xij > 0

(8− 10)in the conditions (4)-(7).

In [1] an algorithm to solve the model (8)-(10) in conditions (4)-(7) is pro-posed. The algorithm determines the multitude of extreme non-dominant so-lutions within the admissible space of solutions. The algorithm is theoreticallyand scientifically tested and proved in a concrete case.

The algorithm of solving the model (1)-(7) develops a procedure of buildinga multitude of all efficient, basic solutions. This set coincides with the set ofthe efficient basic solutions for the model (8)-(10) in conditions (4)-(7). Thatis why, in order to find the set of its basic efficient solutions, we reduce themulticriterial fractional transportation model of ”bottleneck” type (1)-(7) tothe problem (8)-(10) with the constraints (4)-(7).

Theorem 0.1 The set of the efficient basic solutions of the model (1)-(7) andthe model (8)-(10) coincide.

Proof. Let X1 be an efficient basic solution for the model (1)-(7), and T 1 =max

i,jtij/x1

ij > 0. Taking into account the definition of the efficient solution,

we state that for each available solution X2 of this model and correspondingT 2, where T 2 = max

i,jtij/x2

ij > 0, the following inequalities

Z1(X1) < Z1(X2) and Z2(X1) ≤ Z2(X2), orZ1(X1) ≤ Z1(X2) and Z2(X1) < Z2(X2), (11)

where T 2 ≤ T 1, T 1 ≥ 0, T 2 ≥ 0, hold.Suppose that the solution X1 is not efficient for the model (8)-(10) in the

conditions (4)-(7). Similarly to the previous reasoning, using the definition ofthe efficient solution, it follows that there exists the available solution X2 ofthis model and corresponding T 2, for which the following inequalities

Z1(X2)T 2 < Z1(X1)

T 1 and Z2(X2)T 2 ≤ Z2(X1)

T 1 ,or

Z1(X2)T 2 ≤ Z1(X1)

T 1 and Z2(X2)T 2 < Z2(X1)

T 1

(12)

hold, where T 2 ≤ T 1, T 1 ≥ 0, T 2 ≥ 0.Multiplying inequalities (12) by T 1 and supposing k = T 1

T 2 , we obtain thatthe following inequalities

kZ1(X2) < Z1(X1) and kZ2(X2) ≤ Z2(X1),or

kZ1(X2) ≤ Z1(X1) and kZ2(X2) < Z2(X1)(13)


hold, where T 2 ≤ T 1, T 1 ≥ 0, T 2 ≥ 0.Obviously k ≥ 1, therefore from (13) we conclude that for the solution X2

the following inequalities

Z1(X2) < Z1(X1) and Z2(X2) ≤ Z2(X1),or

Z1(X2) ≤ Z1(X1) and Z2(X2) < Z2(X1)(14)

hold, where T 2 ≤ T 1, T 1 ≥ 0, T 2 ≥ 0, that contradicts (11).Similarly, it can be proved that each efficient solution of the model (8)-(10)

is also an efficient solution for the model (1)-(7).The theorem is proved.Generalizing this idea for the model with multiple number of fractional

criteria with the ”bottleneck” constraining criterion, we conclude that, in orderto find the set of its efficient basic solutions, it may be reduced to the model

min z1 =m∑

i=1

n∑

j=1

c1ijxij , min z2 =

m∑

i=1

n∑

j=1

c2ijxij , ... ,

min zr =m∑

i=1

n∑

j=1

crijxij , min zr+1 = max

i,jtij |xij > 0, (15)

in the conditions (4)-(7). The values Ckij , k = 1, ..., r, i = 1, ..., m, j = 1, ..., n

correspond to the concrete interpretation of the corresponding criteria.If among the set of criteria from the model (15), there are some criteria of

”max” type, it is not difficult to reduce this case to the initial one. Obviously,the model (8)-(10) is a particular case of the model (15). Therefore the algo-rithm to solve the model (15) in the conditions (4)-(7) can be used to solvethe model (8)-(10).

The truthfulness of the above theorem for the model (15) can be provedsimilarly.

The algorithm of finding the set of the efficient basic solutions for the model(15) is an interactive one. Initially, in order to find the first efficient basicsolution of model (15), we consider at least (m + n− 1) cells from the tablesCk, k = 1, 2, . . . , r. The indexes’ order is maintained the same as in thetable T , where the cells are numbered according to the respective time valueswell arranged in the increasing order. Each iteration supposes a deep levels’exploration and a completion of the multitude of efficient basic solutions for anew unblocked stochastic time-variable.

In the case when the same solutions have been found at upper level of otherbranch or when all possibilities of improvement have been spent at this level,the exploration procedure of each time instant chain is finite in depth andends on every branch.

The generalized algorithm for solving the fractional multiobjectivetransportation problem 201

In the case when the solution of a certain configuration detains the formrecorded in another link, which has been investigated earlier, its depth explo-ration has no justification, that is why it is eventually stopped.

We propose the logic scheme to construct the algorithm for solving themulticriterial transportation models of ”bottleneck” type with a finite numberof criteria, where ∆ij = (ui + vj)− cij , ni < p (p is defined by the dimensionof the problem, ni is an index of ordering the cells by data from the table T ).

ALGORITHM1. Table T with the increasing order of time values which uses the k indexis being well arranged. The index order is maintained for the respective cellsfrom the tables Ck, k = 1, ..., r.

2. The adoption of an initial, basic solution in the first p = (m+n−1) cellsfrom the table C is performed. The other cells are considered to be blocked.

3. All configurations of basic solutions can be recorded at the level l = 0,using only the non-blocked cells and providing the dozing in all those cellswith (i, j|xij > 0), for which the relation ∆ij ≥ 0 is to be true at least for onecriterion.

Each configuration of the solution is iteratively investigated, in this wayobtaining the following records at the next level: l = l + 1.

If a certain level of a basic solution, which was previously obtained, isfound, the latter will be not further studied. Since the problem covers a finitedimension, the multitude, of all basic solutions for the unblocked cells will beobtained by exploring a finite number of levels in depth.

4. If p < m ∗ n, the following p = p + 1 cell is unblocked, and for thispurpose the exploration of the basic solutions is revived, then we start withthe level 0. The 4th step will be repeated until we get p = m ∗ n.

The basic efficient solution set is selected out of the multitude of the basicsolutions.

Theorem 0.2 The set of all efficient basic solutions for the multiple crite-ria transportation problem of ”bottleneck” type is found by applying the abovealgorithm.

Proof. Let L be a list of efficient basic solutions of model (15) being foundby applying the above algorithm. We suppose that the efficient basic solutionS1, that was not found using the above algorithm, exists and S1 /∈ L. LetS1 correspond to T1. We will fix it on the branch that corresponds to theT1 beginning with the level 0, when the corresponding cells from table T arecleared. An wide exploration of the fixed branch leads to the registration of allbasic solutions of branch T1. Thus, all basic solutions corresponding to timeT1 are contained in this set. We will separate from the set LT1 the efficientbasic solutions corresponding to time T1. Obviously LT1 ⊂ L. As a result,


if S1 ∈ LT1 , then S1 is a basic efficient solution found by applying the abovealgorithm or if S1 /∈ LT1 , then S1 is not a basic solution and moreover it isnot a basic efficient solution. Therefore, either S1 is not a basic solution or itis contained in list L. The theorem is proved.

Example. Consider the following 3-criteria problem.Time, Supply, Demand= Cost 1,2=

10 95 73 52 8

68 66 30 21 19

37 63 19 23 17

11 3 14 16 bj |ai

1 2 7 74 4 3 4

1 9 3 45 8 9 10

8 9 4 66 2 5 1

Using the above proposed algorithm we have found the following 11 efficientbasic solutions:

S1=(176,207,68); S2=(164,276,68); S3=(178,203,68);S4=(172,213,68); S5=(158,283,68); S6=(208,167,73);S7=(202,173,73); S8=(156,200,95); S9=(176,175,95);S10=(143,265,95); S11=(186,171,95).The authors of the article [1], using their own algorithm for this example,

have obtained 9 efficient extreme solutions.

References[1] Y. P. Aneja, K. P. K. Nair, Bicriteria transportation problem Management Sci., 25,1

(1979), 73–79.

[2] S. Chandra, P. K. Saxena, Time minimizing transportation problem with impurities,Asia-Pacific J. Op. Res. 4 (1979), 19–27.

[3] G. R. Bitran, A. G. Novaes, Linear programming with a fractional objective function,N.R.L.Q. 3(1972), 22–29.

[4] S. R. Sharma, Kanti Swarup, Transportation fractional programming with respect totime, Ricerca Operativa 7, 3 (1977), 49–58.

[5] R. S. Garfinkel, N. R. Rao, The bottleneck transportation problems, N.R.L.Q. 18 (1971),465–472.

[6] T. Glickman, P. D. Berger, Cost-completion-date tradeoffs in the transportation prob-lem, Op.Res. 25, 1 (1977), 163–168.

[7] A. Tkacenko, I. M. Stancu-Minassian, Metoda de rezolvare a problemei de transport detip ”bottleneck” cu doua criterii, Studii si Cercetari de Calcul Economic si CiberneticaEconomica, 4 (1996), 77–85.

[8] A. Tkacenko, A. Alhazov, The multiobjective bottleneck transportation problem, Com-puter Science Journal of Moldova, Kishinev, 9, 3 (2001), 321–335.

LISC – A MATLAB PACKAGE FOR LINEARDIFFERENTIAL PROBLEMS

ROMAI J., 2, 1(2006),203–208

Damian TrifBabes-Bolyai University of [email protected]

Abstract A MATLAB package, based on the Lanczos tau method, useful in the Lyapunov-Schmidt (LS) method for nonlinear second order differential boundary valueproblems of the type Lu = Nu is presented. Future applications to evolutionproblems or bifurcation studies are also taken into account.

1. INTRODUCTIONThe Lyapunov-Schmidt (LS) method, elaborated in the years 1906-1908 and

reformulated in a modern mathematical language by L. Cesari [3] after 1963applies to some nonlinear equations of the type Lu = Nu, in the presence ofboundary conditions, considered on the domain of the linear operator L.

This method could be easily extended to the case of a nonlinear evolution

equation on a Hilbert space H (usually a L2 space) of the formdu

dt= F (u) ≡

Lu+Nu, where the domain of F is dense in H. We can also use the Lyapunov-Schmidt method to discuss the bifurcation problem x − λAx + G(x, λ) = 0,x ∈ X, where X is a Banach space, A : X → X is a linear compact operatorand G : X × IR → X is a continuous mapping such that G(x, λ) = o(‖x‖) forall λ ∈ IR.

Although it has been used for a long time only for the theoretical demon-stration of the existence and branching of the solutions of such problems, theLS method (or, by Cesari, the alternative method) is also very useful to theeffective approximation of these solutions.

We will present LISC, a MATLAB package based on the Lanczos-tau method,for Sturm-Liouville problems with applications to the Lyapunov-Schmidt methodfor nonlinear equations. The advantage of the LS method consists of theimportant reduction of the dimension of the nonlinear system to be solvedtogether with the possibility to oversee the approximating errors. This advan-tage occurring in some examples, proves that the LS method behaves betterthan other known methods such as bvp4c or sbvp.

203

204 Damian Trif

2. THE LS METHODWe assume that the linear part L of the equation Lu = Nu is a Sturm-

Liouville operator

Ly ≡ 1r(x)

[−(p(x)y′)′ + q(x)y], x ∈ [a, b]

y(a) cos α + (py′)(a) sin α = 0,

y(b) cos β + (py′)(b) sinβ = 0,

where 1/p, q, r are real-valued functions on [a, b], p(x) > 0, r(x) > 0 on [a, b],p ∈ C1[a, b], q, r ∈ C[a, b]. It is well known that the eigenvalues of L form anincreasing sequence λ0 < λ1 < ... converging to infinity and the correspondingeigenfunctions ϕn form an orthogonal (orthonormal) basis of the Hilbert spaceL2

r(a, b). The asymptotic behaviour of the eigenvalues is λn ∈ O(n2).A theoretical constructive variant of the LS method can be found in [7], [8].

Summarizing, the approximating algorithm is:a) we are looking for an approximate solution of the equation Lu = Nu of

the form

u =m∑

k=1

ckϕk +N∑

k=m+1

ckϕk,

where 0 ≤ m ≤ N ;b) by fixing u∗ =

∑mk=1 ckϕk, we generate the associate function y(u∗)

performing the iterations

y0 = u∗, ys+1 = u∗ + HmNys = u∗ +N∑

k=m+1

Cskϕk, s = 0, 1, ..., S;

c) with y = yS+1 as an approximation of the associated function, we canwrite the system Lu∗ = PmNy of the determining equations, with the un-knowns c1, ..., cm. This system of the form F (c1, ..., cm) = 0 is then numeri-cally solved, by a suitable method, for instance by the Newton’s method. Everyevaluation of the function F means the reiteration of the b) step. Finally, thusdetermined u∗ generates, also by the b) iterations, an approximation of thesolution of the equation Lu = Nu.

We remark that in the case of the Galerkin’s method, the approximatingsolutions are being looked for under the form u∗ =

∑Nk=1 ckϕk, where the co-

efficients ck, k = 1, ..., N are determined from the equations (Lu∗−Nu∗, ϕk) =0, k = 1, ..., N i.e.

(λku∗ −Nu∗, ϕk) = 0, k = 1, ..., N.

These equations are got from the determining equations for m = N . If m = 0the system of the determining equations disappears. The function associated

LISC – A MATLAB package for linear differential problems 205

with a certain u∗ satisfies the equation y = L−1Ny. Therefore, in this case,the algorithm is reduced to the transformation of the equation Lu = Nu intoa fixed point problem. Obviously, this case arises only when the inverse L−1

exists and L−1N is a contraction.The first version of our package applies only to the Sturm-Liouville case for

the linear operator L, in the form

Lu =1

r(x)

[d

dx

(p(x)

du

dx

)+ g(x)u

],

au′(0) + bu(0) = 0, cu′(1) + du(1) = 0.

There exists a Matlab package MATSLISE of V. Ledoux (2004), based onthe works of L. Ixaru which uses the so called CP methods to calculate theeigenfunctions of Sturm-Liouville or Schrodinger operators but this packageworks slowly. A more interesting package is MWRtools of R. A. Adomaitis(1998-2001) [1] which uses spectral methods to calculate the eigenfunctionsof the Sturm-Liouville operator in order to solve some linear boundary valueproblems.

In the next section we propose a Chebyshev-tau method to solve the Sturm-Liouville problem in order to get a good basis ϕi, and the corresponding Matlabpackage.

3. CHEBYSHEV-TAU METHOD

3.1. THE PROBLEMThe module Chebyshev of our package solves problems of the type

p2(x)u′′ + p1(x)u′ + p0(x)u = g(x), x ∈ (a, b) (1)

α11u(x11) + α12u′(x12) = β1,

α21u(x12) + α22u′(x22) = β2 (2)

using the Chebyshev-tau method.For the moment let us suppose that a = −1, b = 1. A powerful method to

solve (1) is to express u as a Chebyshev series u(x) = c0T0(x)

2+ c1T1(x) + ...,

where Ti(x) = cos(i cos−1(x)) is the standard Chebyshev polynomial of orderi [5], [4].

For the practical implementation, we define the vectors c and t by cT =

(c0, c1, c2, ...), tT = (T0

2, T1, T2, ...) so that u(x) = cT t = tT c. By using the

properties of Chebyshev polynomials we have x · Ti =Ti−1

2+

Ti+1

2, hence,

x · u(x) = cT XT t = (Xc)T t, where X0,1 = 1, Xii = 0, Xi,i−1 = Xi,i+1 = 1/2.

206 Damian Trif

Then, in general, xmu(x) = (Xmc)T t and f(x)u(x) = (f(X)c)T t for analyt-

ical functions f. Moreover,u(x)xm

= (X−mc)Tt if the l.h.s. has no singularity

at the origin. Of course, X is a tri-diagonal matrix, X2 is a penta-diagonalmatrix and so on but, in general, the matrix version funm(X)of the scalarfunction f(x) or X−m = [inv(X)]m are no longer sparse matrices.

Similarly, let D be the differentiation matrix definingdmu

dxm= (Dmc)T t. The

matrix D is a upper triangular and its entries are

Dii = 0, Dij = 0 for (j − i) even and Dij = 2j otherwise.

Applying these formulae to equation (1), we get

(p2(X)D2 + p1(X)D + p0(X))c = g,

where G are the coefficients of the r.h.s. function

g(x) = g0T0(x)

2+ g1T1(x) + ...

The condition (2) can be formulated in a similar manner. We define tij =tT (xij) so that it can be written in the form hT

i c = βi, i = 1, 2, where

hTi =

2∑

j=1

αijtTijD

j−1, i = 1, 2.

Define the matrices A =∑2

i=0 Pi(X)Di and H = (h1, h2)T . Then the vectorc satisfies (

HA

)c =

(βq

)(3)

of the form Ac = b, where β = (β1, β2)T .Of course, in reality we cannot work with infinite matrices but only with

finite portions (N ×N) of them.If instead of [−1, 1] we have another interval [a, b] for x, we use the change

of coordinates x = αξ +β where α =b− a

2and β =

b + a

2, so that ξ ∈ [−1, 1].

We must change X to αX + βI and D to D/α.

4. EXAMPLES FOR LS METHODThe package LISC is under construction. Here we present some test prob-

lems for nonlinear equations, using 50 grid points and 21 eigenfunctions. Thetutorials and the m-files of the package can be obtained by e-mail from theauthor ([email protected]) or from Matlab Central - File Exchange.

LISC – A MATLAB package for linear differential problems 207

For the moment, it is based on a Sturm-Liouville solver LiScEig for a boundedinterval (for the ϕi basis) and the files as.m and lisc.m which implement theLS method.

Example 1. Solve the boundary value problem

x′′ + x = 0.5t− 0.5x3,

x(0) = 0,x(1) + x′(1) = 0.

The eigenvalues of the linear part are λi = 1 − l2i , where sin li = −li cos li,ϕi = ci sin (lit). The errors between our solution and the solution calculatedby bvp4c of Matlab is less than 14× 10−6.

Example 2. Consider the two-point problem for the Emden equation (witha singularity)

y′′ +2x

y′ + y5 = 0,

y′(0) = 0, y(1) =√

32

.

The general Sturm-Liouville operator now has the form1x2

(x2u′

)′ = λu. The

closed-form solution reads y(x) =(

1 +x2

3

)−12

and the error is of order 1.e−5.

Example 3. If comes from the combustion theory [6],

−u′′ + Mu′ = DYoYfe−

θ

T0 + u ,

Mu(0)− u′(0) = 0, u(1) = 0,

where Yo = −u + Yo1eM(x−1), Yf = −u + 1 − eM (x−1), T0 = 0.118, Yo1 =

0.21, θ = 2.6. The computed solution for M = 6 and D = 5.5× 108 (with ahigh gradient at the burning front) agrees well with those of [6].

5. CONCLUSIONSThe comparison between LISC and SBVP 1.0 of Auzinger [2] and bvp4c of

Matlab (see Matlab help) shows a computing time for LISC about 1 -2 timeslarger. The Matlab profile reports show that about 75% of the computingtime was spent on computation of the eigenfunctions and only about 6% onthe effective computation of the numerical solution. But we have good reasonsto use LS method.

208 Damian Trif

We can build a database with known eigenfunctions. In the problems withparameters, where we have (for example) bifurcations, or in evolution prob-lems, we can use repeatedly the same eigenfunctions. The eigenfunctions carryphysical information, so that our LS solution has a better structure for stud-ies. LS method could be easily extended to 2D or 3D (evolution) problems,with non-invertible linear part. In all the cases, we finally have to solve avery small nonlinear system, usually with m = 0, 1, 2 values, which also carryinformation about bifurcation behaviour.

References[1] Adomaitis, R. A., Lin Yi-hung, A collocation/quadrature based Sturm-Liouville problem

solver, ISR Technical Research Report TR 99-01.

[2] Auzinger, W., Kneisl, G., Koch, O., Weinmuller E. B., SBVP 1.0 A Matlab solver forsingular boundary value problems, Technische Universitat Wien, ANUM preprint no.02/02.

[3] Cesari L., Functional analysis and Galerkin’s method, Mich. Math. J., 11, 3(1964),383-414.

[4] Liefvendahl, M., A Chebyshev tau spectral method for the calculation of eigenvalues andpseudospectra (http://www.nada.kth.se/~mli/research.html)

[5] MacLeod, A. J., An automatic matrix approach to the Chebyshev series solution ofdifferential equations (http://maths.paisley.ac.uk/allanm/amcltech.htm)

[6] Rameau, J. F., Schmidt-Laine, C., Perturbed bifurcation phenomenon in a diffusionflame problem, Appl. Numer. Math., 5,3(1989), 237-255.

[7] Petrila T., Trif D., Basics of fluid mechanics and introduction to computational fluiddynamics, Springer, Berlin, 2005.

[8] Trif, D., The Lyapunov-Schmidt method for two-point boundary value problems, Fixed

Point Theory, 6,1(2005), 119-132.

OPTIMAL CONTROL AND STOCHASTICUNIFORM OBSERVABILITY

ROMAI J., 2, 1(2006),209–218

Viorica Mariela Ungureanu”Constantin Brancusi” Univ., Targu [email protected]

Abstract G.Da Prato and I. Ichikawa solved in [3] the quadratic control problem (1), (2),under stabilizability and detectability conditions. We replace the detectabilitycondition with the uniform observability property and we obtain an optimalcontrol and the minimal value of the cost functional (2). So, we generalize theresults obtained by T.Morozan in [7] for finite dimensional case and we alsoconclude that our result is distinct from the one of G.Da Prato and I. Ichikawa.We use the above results to give a method for the numerical computation ofthe optimal cost. Under the same hypotheses we solve the tracking problem(1), (3).

Keywords: quadratic control, tracking, Riccati equation, detectability, stabilizability and

observability

2000 MSC: 93E20, 49K30, 93B07.

1. NOTATION AND STATEMENT OF THEPROBLEM

Let H,U, V be separable real Hilbert spaces. Denote by L(H, V ) the Banachspace of all bounded linear operators from H into V ( if H = V we putL(H, V ) = L(H)). Denote by H the subspace of L(H) formed by all self-adjoint operators. Write 〈., .〉 for the inner product and ‖.‖ for norms ofelements and operators . Then S ∈ L(H) is called nonnegative (S ≥ 0) if S isself-adjoint and 〈Sx, x〉 ≥ 0 for all x ∈ H. We set L+(H) = S ∈ L(H), S ≥0. For each interval J ⊂ R+, we denote by Cs(J, L(H)) the space of allstrongly continuous mappings G(t) : J ⊂ R+ → L(H) and by Cb(J, L(H))the subspace of Cs(J, L(H)), which consist of all mappings G(t) such thatsupt∈J

‖G(t)‖ < ∞. Let E is a Banach space; denote by C(J,E) the space of all

continuous mappings G(t) : J ⊂ R+ → E. We need the following assumption:P1: a) A(t), t ∈ [0,∞) is a closed linear operator on H with constant domainD dense in H;

b) there exist M > 0 , η ∈ ( 12π, π) and δ ∈ (−∞, 0) such that Sδ,η = λ ∈

C; |arg(λ− δ)| < η ⊂ ρ(A(t)), for all t ≥ 0 and ‖R(λ, A(t))‖ ≤ M|λ−δ| for all

λ ∈ Sδ,η;

c) there exist numbers α ∈ (0, 1) and eN > 0 such that

209

210 Viorica Mariela Ungureanu

A(t)A−1(s)− I ≤ eN |t− s|α , t ≥ s ≥ 0, where ρ(A) , R(λ, A) are the resolventset of A and the resolvent of A respectively.

It is known [4] that if P1 holds, then the family A(t), t ≥ 0 generates anevolution operator U(t, s). For any n ∈ N we have n ∈ ρ(A(t)). The operatorsAn(t) = n2R(n, A(t))−nI are called the Yosida approximations of A(t). If wedenote by Un(t, s) the evolution operator relative to An(t), for each x ∈ Hone has lim

n→∞Un(t, s)x = U(t, s)x uniformly on any bounded subset of (t, s);t ≥ s ≥ 0.

Let ( Ω, F, Ft, t ∈ [0,∞), P ) be a stochastic basis and assume that P1 holds.Consider the following stochastic equation with control

dy(t) = A(t)y(t) + B(t)u(t)dt +m∑

i=1

Gi(t)y(t)dwi(t), y(s) = x ∈ H, (1)

denoted by A,B;Gi, where u ∈ Uad = u ∈ L2(R+ × Ω, U), u is Ft−adapted, wi’s are independent real Wiener processes relative to Ft. Supposethat the hypotheses

P2 : B ∈ Cb([0,∞), L(U, H)), B∗ ∈ Cb([0,∞), L(H, U)), C ∈ Cb([0,∞), L(H, V )), C∗C, Gi ∈Cb([0,∞), L(H)), K(t) ∈ Cb([0,∞), L+(U)) and there exists δ0 > 0 such thatK(t) ≥ δ0I for all t ∈ [0,∞)

are fulfiled. Denote Z = supr∈[0,∞]

‖Z(r)‖ for Z = B, C, Gi, K.

Assume that P1, P2 hold. It is known that under uniform observabilityand stabilizability conditions the Riccati equation of stochastic control (3)has a unique, uniformly positive, bounded on R+ and stabilizing solution (seeTheorem 3.2). We use this result to solve both a quadratic control problemand a tracking problem associated with (1).

Quadratic control problem. We look for an optimal control u ∈ Uad, whichminimize the following quadratic cost

Is(u) = E

∞∫

s

‖C(t)y(t)‖2 + < K(t)u(t), u(t) > dt, (2)

where Uad = u ∈ Uad such as E ‖y(t)‖2 → 0 as t →∞.Tracking problem. Given a signal r ∈ Cb(R+,H) we want to minimize the

cost

J(s, u) = limt→∞

1t− s

E

t∫

s

‖C(σ) (y(σ)− r(σ))‖2 + 〈K(σ)u(σ), u(σ)〉 dσ (3)

in a suitable class of controls u subject to the equation A,B; Gi.

Optimal control and stochastic uniform observability 211

2. BOUNDED SOLUTIONS OF LINEARSTOCHASTIC EQUATIONS

Assume that P1, P2 hold. With (1) associated the equation

dy(t) = A(t)y(t)dt +m∑

i=1

Gi(t)y(t)dwi(t), y(s) = x ∈ H, (4)

denoted by A;Gi.It is known [3] that A; Gi has a unique mild solution in C([s, T ],

L2(Ω;H)) that is adapted to Ft; namely the solution of

y(t) = U(t, s)x +m∑

i=1

t∫

s

U(t, r)Gi(r)y(r)dwi(r). (5)

Let y(t, s; x) be the mild solution of A; Gi.We have the following definition (see [5] for the autonomous case) :

Definition 2.1 We say that A; Gi is uniformly exponentially stable if thereexist constants M ≥ 1, ω > 0 such that

E ‖y(t, s; x)‖2 ≤ Me−ω(t−s) ‖x‖2 for all t ≥ s ≥ 0 and x ∈ H.

Definition 2.2 [3] We say that A,B; Gi is stabilizable if there exists F ∈Cb([0,∞), L(H, U)) such that A + BF ;Gi is uniformly exponentially stable.

Lemma 2.1 Assume P1 and F ∈ Cb([0,∞), L(H, U)). If h ∈ Cb(R+,H),then the equation

g′n(t) = − (A∗n + F ∗) gn(t)− h(t), g(T ) = x0 ∈ H (6)

where An, n ∈ N are the Yosida approximations of A and the weak dif-ferentiability is considered, has a unique solution. The functions (t, x) →〈g′n(t), x〉 , n ∈ N are continuous on [0,∞) × H. Moreover if Vn(t, s) (resp.V (t, s)) is the evolution operator generated by An + F (resp. A + F ), then wehave

〈gn(t), y〉 →n→∞

⟨V ∗

A+F (T, t)x0, y⟩

+

⟨ T∫

t

V ∗A+F (σ, t)h(σ)dσ, y

⟩, (7)

uniformly with respect to t ∈ [0, T ].

Proof. Since ∂VAn+F (t,s)x∂s = −VAn+F (t, s)(An(s) + F (s))x, it is easy to

see that gn(t) = V ∗An+F (T, t)x +

T∫t

V ∗An+F (σ, t)h(σ)dσ is the unique solution


of the equation (6). Using Lemma 3 in [9] it follows that for each y ∈ H,lim

n→∞VAn+F (t, s)y = VA+F (t, s)y uniformly with respect to 0 ≤ s ≤ t ≤ T and

(7) holds. 2

Remark 2.1 Assume that P1 holds and A,B;Gi is stabilizable with thestabilizing sequence F ∈ Cb([0,∞), L(H, U)) and let h ∈ Cb(R+,H). SinceA + BF, Gi is uniformly exponentially stable it follows that there exist con-stants M ≥ 1, ω > 0 such as ‖VA+BF (t, s)‖ ≤ Me−ω(t−s) for all t ≥ s ≥ 0.Hence, the integral

g(s) =

∞∫

s

V ∗A+BF (σ, s)h(σ)dσ (8)

is convergent in H and g(s) is bounded on R+. Moreover⟨T∫t

V ∗A+BF (σ, t)h(σ)dσ, x

⟩→

T→∞

⟨∞∫s

V ∗A+BF (σ, s)h(σ)dσ, x

⟩. If we consider

the solution of (6) with the initial condition gn(T ) = g(T ) it is not difficult tosee that 〈gn(s), y〉 →

n→∞ 〈g(s), y〉 for all y ∈ H.

3. THE RICCATI EQUATION OF STOCHASTICCONTROL AND THE UNIFORMOBSERVABILITY

Consider the system A;Gi; C formed by equation A; Gi and the obser-vation relation z(t) = C(t)y(t, s; x).

Definition 3.1 [7] The system A; Gi; C is uniformly observable if there ex-

ist τ > 0 and γ > 0 such that Es+τ∫s‖C(t)y(t, s;x)‖2 dt ≥ γ ‖x‖2 for all s ∈ R+

and x ∈ H.

In the deterministic case it is known (see [6] for the autonomous case) thatuniform observability implies detectability. In [9] we proved that this assertionis not true in the stochastic case.

Let us consider the Riccati equation

P ′(s) + A∗(s)P (s) + P (s)A(s) +m∑

i=1

G∗i (s)P (s)Gi(s)+

+C∗(s)C(s)− P (s)B(s)(K(s))−1B∗(s)P (s) = 0. (9)


If An(t), n ∈ N are the Yosida approximations of A(t), then we introduce theapproximate equation

P ′n(s) + A∗n(s)Pn(s) + Pn(s)An(s) +

m∑

i=1

G∗i (s)Pn(s)Gi(s)+

+C∗(s)C(s)− Pn(s)B(s)(K(s))−1B∗(s)Pn(s) = 0. (10)

Lemma 3.1 [3] Let 0 < T < ∞ and let R ∈ L+(H). Then there exists aunique mild (resp. classical) solution P (resp. Pn) of (3) (resp. (3)) on [0, T ]such that P (T ) = R (resp. Pn(T ) = R). They are given by

P (s)x = U∗(t, s)RU(t, s)x +

t∫

s

U∗(r, s)[m∑

i=1

G∗i (r)Q(r)Gi(r)+

+ C∗(r)C(r)− P (r)B(r)(K(r))−1B∗(r)P (r)]U(r, s)xdr (11)

Pn(s)x = U∗n(T, s)RUn(T, s)x +

T∫

s

U∗n(r, s)[

m∑

i=1

G∗i (r)Pn(r)Gi(r)+

+C∗(r)C(r)− Pn(r)B(r)(K(r))−1B∗(r)Pn(r)]]Un(r, s)xdr (12)

and for each x ∈ H, Pn(s)x → P (s)x uniformly on any bounded subset of[0, T ]. Moreover, if we denote these solutions by P (s, T ; R) and Pn(s, T ; R) re-spectively, then they are monotone in the sense that P (s, T ; R1) ≤ P (s, T ;R2)if R1 ≤ R2.

We say [3] that P is a mild solution on an interval J of (3, if P ∈ Cs(J, L+(H))and if P (s, T ; P (s)) satisfies (11) for all s ≤ t, s, t ∈ J . Moreover, if P is a mildsolution on R+ of (3) and sup

s∈R+

‖P (s)‖ < ∞, then P is said to be a bounded

solution.

Remark 3.1 From the above lemma it is easy to deduce that for all α, β ∈R+, α ≤ β we have

P (s, α, 0) ≤ P (s, β, 0) (13)

for all s ∈ [0, α].

Definition 3.2 A mild solution of (3) is called stabilizing for A,B; Gi ifL not= A−BS; Gi is uniformly exponentially stable, where S(t) = K−1(t)B∗(t)P (t).

We introduce the following hypothesis:


P3 : The evolution operator U(t, s) has an exponentially growth, i.e. thereexist M0 and ω positive constants such that ‖U(t, s)‖ ≤ M0e

ω(t−s).

Theorem 3.1 [9] Assume that A,Gi; C is uniformly observable and P3

holds. If P (t) is a nonnegative bounded solution of (3) thena) there exists δ > 0 such that P (t) ≥ δI for all t ∈ [0,∞),b) P is a stabilizing solution (for the stochastic system A,B;Gi).

Proposition 3.1 Under the assumptions of Theorem 9, the Riccati equation(3) has at most one nonnegative and bounded solution.

Proof. Since it is not very difficult to prove (see Corollary 3.2 in [3]) thatany bounded and stabilizing solution of (3) is maximal (see [3]) in the class ofbounded solutions, the conclusion follows from the above theorem . 2

The following theorem is a consequence of the above results.

Theorem 3.2 Assume that P3 holds, A,Gi; B is stabilizable and A,Gi; Cisuniformly observable. Then the Riccati equation (3) has a unique nonnegativebounded on R+ solution P (t), which is a stabilizing solution and there existsδ > 0 such that P (t) ≥ δI for all t ∈ [0,∞) ( P is uniformly positive on R+).

Proof. From Theorem 4.1 in [3] it follows that under stabilizability condi-tions the equation (3) has a bounded solution on R+. From Theorem 3.1 andProposition 3.1 we deduce the conclusion. 2

Remark 3.2 Assume that the hypotheses of the above theorem hold. a) FromLemma 4.1 and the stochastic version of the Theorem 3.1 from [3], it is easyto see that the unique uniformly positive solution P (.) of the Riccati equa-tion (3) is the strong limit of the monotone sequence of nonnegative operatorsP (., α, 0), α ∈ R+ (see (13)). Consequently, if M = lim

α→∞ supt∈[0,α]

‖P (., α, 0)‖ ,

then P (t) ≤ MI for all t ∈ [0,∞).b) Using the same type of proof as in [9] we can see that if δ = (1/2)γ−r1C

2,

where r1 = (m + 1)(M20 e2ωτ B21/δ0) exp(τM2

0 e2ωτ (m + 1)m∑

i=1G2

i ), τ, γ, M0,

ω and δ0 are those introduced by the Definition 3.1, P3 and P2 hold, thenP (t) ≥ δI for all t ∈ [0,∞).

c) From the proof of the statement b) of the Theorem 3.1 (see [9]) it followsthat the mild solution z(t, s; x) of L = A−BS; Gi satisfies the inequality

E ‖z(t, s; x)‖2 ≤ βe−α(t−s) ‖x‖2 for all t ≥ s ≥ 0 and x ∈ H,

where α = − ln(1− δM )

τ and β depends only on M0, ω, τ (which are introducedabove) and m. Hence the evolution operator UL, associated with the operatorL, satisfies the inequality ‖UL(t, s)‖2 ≤ βe−α(t−s)too.


Assume that the hypotheses of the Theorem 11 holds. Theorem 13 givesan estimate for the convergence rate of the sequence P (., T, 0) to the uniquesolution of the Riccati equation (3).

Theorem 3.3 Assume that P3 holds, A,Gi; B is stabilizable and A,Gi; Cisuniformly observable. If P (.) is the unique nonnegative and stabilizing solutionof the Riccati equation (8), then

‖P (t)− P (t, T, 0)‖ ≤ ε

for all t ≤ T ≤ T and T ≥ Tε,T = −α ln ε

bCebKα

.

Proof. If P (.) is the solution of the Riccati equation (8) such as the systemL = A−BS;Gi is uniformly exponentially stable, then [3] P (.)−P (., T, 0)is a mild solution of the equation

Z ′ + L∗Z + ZL +m∑

i=1

G∗i ZGi + ZB(K)−1B∗Z = 0

with the final condition Z(T ) = P (T ). Denote K = β

(2M eB2

δ0+

m∑i=1

G2i

)eαT

and C = MβeαT, where T > 0 is fixed, we use the integral equation (see(11)) satisfied by Z, the statement a) of the above remark and the Gronwall’sinequality to get

‖P (t)− P (t, T, 0)‖ ≤ Ce−αT +

T∫

t

Ke−αu ‖P (u)− P (u, T, 0)‖ du and

‖P (t)− P (t, T, 0)‖ ≤ Ce−αT ebKα for all t ≤ T ≤ T. (14)

Now, obviously, for all t < T, ε > 0 and T ≥ Tε,T = −α ln ε

bCebKα

we have

‖P (t)− P (t, T, 0)‖ ≤ ε. 2

4. OPTIMAL QUADRATIC CONTROL

A consequence of Theorem 3.2, Theorem 4.2 and the stochastic version ofTheorem 3.1 from [3] is the following theorem:

Theorem 4.1 Assume that the hypotheses of the Theorem 11 are fulfilledand consider the control problem (1), (2). The optimal control is given by thefeedback law

u (t) = −K (t)−1 B∗ (t) P (t) y (t) ,


where P is the unique bounded positive solution of (3) (y(t) is the correspondingsolution of (1)) and the optimal cost is

Is(u) = 〈P (s)x, x〉 . (15)

If all operators in (1) and (2) are time invariant, then P (.) is constant.

The following conclusion provides a numerical method for the computationof the optimal cost.

Conclusion 1 Assume that the hypotheses of the above theorem hold and letH =Rn. From Remark 3.2 we deduce an algorithm which allows us to obtainthe optimal cost (15) in the finite dimensional case. Let ε > 0 and t ∈ [0,T]be fixed.

1. We can use the Runge Kutta method to obtain the solutions P (t, α, 0),α ∈ R+ of the Riccati equation (3). We compute the constant M from thestatement a) of Remark 12 (as a limit of an monotonously increasing sequenceof real numbers). Consequently we can obtain all the constants from relation(14).

3. Using the constant Tε,T, introduced in Section 3, we see that P (t, Tε,T, 0)is a good approximation (with the error ε) of the solution P (t), t ∈ [0,T] ofthe Riccati equation (3). Hence

⟨P (t, Tε,T, 0)x, x

⟩is an approximation of the

optimal cost (15)

5. TRACKING PROBLEMConsider the set of admissible controls Uad = u is an U - valued random

variable, Fs−measurable such as limt→∞

1t−sE

t∫s‖u(σ)‖2 dσ < ∞ and sup

t≥sE ‖y(t)‖2 <

∞, where y(t) is the solution of (1).Theorem 5.1 Assume that the hypotheses of Theorem 3.2 hold. Let P be theunique and bounded on R+ solution of the Riccati equation (3) and g(t) begiven by (8), where F (t) = −K−1(t)B∗(t)P (t) and h(t) = C∗(t)C(t)r(t).Thenthe optimal control is

u(σ) = K−1(σ)B∗(σ)[g(σ)− P (σ)y(σ)]

and the optimal cost is

J(s) = infu∈Uad

J(s, u) =

limt→∞

1t− s

[

t∫

s

‖C(σ)r(σ)‖2 dσ −t∫

s

||K−1/2B∗(σ)g(σ)||2dσ].


Proof. Let Pn(s) = Pn(s, t1;P (t1)) and let gn(s) be the solution of (6)with the final condition gn (t1) = g(t1), where Fn(t) = −B(t)Sn(t),Sn(t) = K−1(t)B∗(t)Pn(t) and h(t) = C∗(t)C(t)r(t). Consider the functionFn(t, x) = 〈Pn(t)x, x〉 − 2 〈gn(t), x〉, which is continuous together its partialderivatives Ft, Fx, Fxx on [0,∞)×H, according to Lemmas 3.1 and 2.1. Letu ∈ Uad and let yn(t) be its response, where the approximate system associatedwith (1) is considered [3]. Using the Ito’s formula for Fn(t, x) and yn(t) we get

E 〈Pn(t1)yn(t1), yn(t1)〉 − 2E 〈gn(t1), yn(t1)〉 − 〈P (s)x, x〉+ 2 〈gn(s), x〉 =

−Et1∫s‖C(σ) [yn(σ)− r(σ)]‖2 + 〈K(σ)u(σ), u(σ)〉 dσ+

Et1∫s

∥∥K1/2Sn(σ)yn(σ) + u(σ)−K−1B∗(σ)gn(σ)∥∥2dσ

+t1∫s‖C(σ)r(σ)‖2 dσ −

t1∫s||K−1/2B∗(σ)gn(σ)||2dσ

+2Et1∫s〈(Sn(σ)− S(σ)) yn (σ) , B∗gn(σ)〉 dσ.

Passing to the limit as n →∞ we obtainE 〈P (t1)y(t1), y(t1)〉 − 2E 〈g(t1), y(t1)〉 − 〈P (s)x, x〉+ 2 〈g(s), x〉 =

−Et1∫s‖C(σ) [y(σ)− r(σ)]‖2 + 〈K(σ)u(σ), u(σ)〉 dσ+

Et1∫s

∥∥K1/2S(σ)y(σ) + u(σ)−K−1B∗(σ)g(σ)∥∥2dσ

+t1∫s‖C(σ)r(σ)‖2 dσ −

t1∫s||K−1/2B∗(σ)g(σ)||2dσ.

Since P (t) and g(t) are bounded on R+, we multiply the last relation by 1t1−s

and taking the limit as t1 →∞ and, then, the infimum we get the conclusion.2

References[1] W. Grecksch, C. Tudor, Stochastic evolution equations, A Hilbert space approach, Math.

Res. 75, Akademie, Berlin, 1995

[2] A. Pazy , Semigroups of linear operators and applications to partial differential equa-tions, Applied Mathematical Sciences 44, Springer, Berlin, 1983.

[3] G. DaPrato, A. Ichikawa, Quadratic control for linear time-varying systems,SIAM.J.Control and Optimization, 28, 2 (1990), 359-381.

[4] G. DaPrato, A. Ichikawa, Lyapunov equations for time-varying linear systems, Systemsand Control Letters 9 (1987), 165-172.

[5] G. DaPrato, J. Zabczyc, Stochastic equations in infinite dimensions, Cambridge Uni-versity Press , 1992.

[6] A. J. Pritchard, J. Zabczyc, Stability and stabilizability of infinite dimensional systems,SIAM Review, 23, 1 (1981).


[7] T. Morozan, Stochastic uniform observability and Riccati equations of stochastic con-trol, Rev. Roum. Math. Pures Appl., 38, 9 (1993), 771-781.

[8] T. Morozan, On the Riccati equation of stochastic control, International Series on Nu-merical Mathematics, 107, Birkhauser, Basel, 1992.

[9] V. Ungureanu, Riccati equation of stochastic control and stochastic uniform observ-ability in infinite dimensions, Proc. of the Conference ”Analysis and Optimization ofDifferential Systems”, Kluwer, Dorchecht, 2003, 421-423.

AN INVESTIGATION OF MATHEMATICALMODELS OF A PIPELINE - PRESSURESENSOR MECHANICAL SYSTEM

ROMAI J., 2, 1(2006),219–225

Petr A. Velmisov, Yuliya V. PokladovaUlyanovsk State Technical University, [email protected]

Abstract The problem of the dynamics of an elastic element of a sensor, that is thecomponent of the pipeline – pressure sensor mechanical system is studied.Similar problems, but with other position of a sensor (fig.1), are considered in[1]-[3].

Keywords: elastic system, finite difference methods

2000 MSC: 774S20

Consider three system in fig.1.

Fig. 1.

For each of these heuristic models the governing equation was obtained byconnecting the pressure law of the working medium on the input of the pipelineand the deflection function of the plate AB.

The equation for the first model (fig.1) has the form

L(ω) = P0(y, t)− 1y 0

∫ y0

0P (y, t)dy−

−2ρ

y0

∞∑

n=1

cos(λny)tanh(λnx0)

λn

∫ b

aω(y, t) cos(λny)dy−

219

220 Petr A. Velmisov, Yuliya V. Pokladova

− 2y0

∞∑

n=1

cos(λny)cosh(λnx0)

∫ y0

0P (y, t) cos(λny)dy − ρx0

y0

∫ b

aω(y, t)dy.

The equation for the second model (fig.1) reads

L(ω) = −P0(x, t) +1y 0

∫ y0

0P (y, t)dy−

−2ρ

x0

∞∑

n=1

cos(λnx)coth(λny0)

λn

∫ b

aω(x, t) cos(λnx)dx−

−2ρ

y0

∞∑

n=1

cosh(νnx)cos(νny0)

cosh(νnx0)

∫ y0

0P (y, t) cos(νny)dy.

For the third model (fig.1) the mentioned relationship is

L(ω) = P0(y, t)− P (t) +ρ

π

b∫

a

ω(τ, t) ln∣∣∣sin πτ

y0− sin

πy

y0

∣∣∣dτ.

The mathematical model represented in fig. 2 differs from the model in fig.1, by another position of a sensor. Further, this problem is solved in the linearapproximation, relevant for small deformations of an elastic element, smallperturbations of velocity potential for the working medium in the domain Gand small deflections of the plate AB as the composite element of the sensor.The working medium is an ideal incompressible liquid or gas. Velocity fieldis supposed to be planar, the length of a pipeline - infinite. The governingequation follows by, connecting the deflection function ω(x, t) of the plate AB(fig.2) and the law P (t) at the pressure change of the working medium on theinput of the pipeline (x = +∞).

Fig. 2.

Let ϕ(x, y, t) be the velocity potential of the working medium (t - time);ω(x, t) is the deflection (the deformation) of the elastic plate AB. The lin-

Mathematical models of a pipeline - pressure sensor mechanical system 221

earized equation and the boundary conditions satisfied by these functions read

ϕxx + ϕyy = 0, (x, y) : 0 < x < +∞, 0 < y < y0; (1)

ϕy(x, 0, t) = 0, x ∈ (0,+∞); (2)

ϕx(0, y, t) = 0, y ∈ (0, y0); (3)

ϕy(x, y0, t) = 0, x ∈ (0, a) ∪ (b, +∞); (4)

ϕy(x, y0, t) = ω(x, t), x ∈ (a, b); (5)

limx→+∞(ϕ2

x + ϕ2y) = 0; (6)

limx→+∞(P∗ − ρϕt(x0, y, t)) = P (t); (7)

L(w) ≡ Mw + Dw′′′′ + Nw′′ + αw′′′′ + βw + γw =

= P∗ − P0(x, t)− ρϕt(x, y0, t), x ∈ (a, b). (8)

Here P (t) is the expression of the pressure change of a working medium onthe input of the pipeline; P0(x, t) is the distributed external load; P∗ is thepressure of a working medium in the pipeline at the rest state; ρ is the densityof a working medium; M is the mass of a unit length (linear density); Dis the deflection rigidity; N is the compressing (stretching) stress; α, β arecoefficients of internal and external damping; γ is the coefficient of rigidity ofthe base.

By the complex variable functions theory the solving of this problem canbe reduced to the investigation of the equation for deformation of an elasticelement [4]

L(ω) = P (t)− P0(x, t) +ρ

π

b∫

a

ω(τ, t) ln∣∣∣cosh

πτ

y0− cosh

πx

y0

∣∣∣dτ. (9)

We shall apply the Galerkin method for approximations of the m-th order

ω(x, t) =m∑

k=1

ωk(t) sinβk(x− a), βk =πk

b− a.

From equation (8) we have

L∗(ω) ≡ M(m∑

k=1

ωk(t) sin βk(x− a)) + D(m∑

k=1

β4kωk(t) sinβk(x− a))−

−N(m∑

k=1

β2kωk(t) sin βk(x− a)) + α(

m∑

k=1

β4kωk(t) sin βk(x− a))+


+β(m∑

k=1

ωk(t) sin βk(x− a)) + γ(m∑

k=1

ωk(t) sinβk(x− a)) + P0(x, t)−

−P (t)− ρ

π

m∑

k=1

ωk(t)Ik(x),

where Ik(x) =b∫a

sinβk(τ − a)ln | cosh πτy0− cosh πx

y0| dτ . Moreover, we must

haveb∫a

L∗(ω) sin βi(x − a)dy = 0, where i = 1, ...,m. Then ωk(t) satisfy the

system of m ordinary differential equations (i = 1, ...,m)

m∑

k=1

Aikωk(t) + Ciωi(t) + Diωi(t) + Fi(t) = 0, (10)

where

Aik =

M b−a2 − ρ

π

b∫a

Ik(x) sinβk(x− a)dx, i = k

− ρπ

b∫a

Ik(x) sinβk(x− a)dx, i 6= k.

Bi =b− a

2(αβ4

i + β), Ci =b− a

2(Dβ4

i −Nβ2i + γ),

Fi(t) = −b∫

a

P0(x, t) sin βi(x− a)dx− P (t)1− (−1)i

βi.

In order to find ωk(0), ωk(0) the initial conditions w(x, 0) = u(x), w(x, 0) =v(x) are used. Compose the discrepancies

R1(ωk(0), x) =m∑

k=1

ωk(0) sinβk(x− a)− u(x),

R2(ωk(0), x) =m∑

k=1

ωk(0) sinβk(x− a)− v(x).

The initial conditions ωk(0), ωk(0) can be obtained from the orthogonality con-

ditions (i = 1, .., m)b∫a

R1(ωk(0), x) sinβi(x−a)dx = 0,b∫a

R2(ωk(0), x) sin βi(x−a)dx = 0. The system (10) in the matrix form reads Aω+Bω+Cω+F (t) = 0,

Mathematical models of a pipeline - pressure sensor mechanical system 223

or, equivalently, ω = −A−1Bω−A−1Cω−A−1F (t), and reduce it to the nor-mal form

x = y,y = −A−1Cy −A−1Dx−A−1F (t),

where x = w, y = w. Then applying some difference method (the Euler-Cauchy and Runge-Kutta methods) the solution of the system can be found.

Assume now that the liquid in the pipeline be water and the elastic elementbe an aluminium plate.

Example 1. Let P0(x, t) = 0, P (t) = 105(1 + sin(t)), w(x, 0) = 0, w(x, 0) =0. Then using the mathematical system MathCAD 2000 for the values ofparameters m = 2, a = 0.1, b = 0.25, y0 = 0.5, M = 42.4, D = 806.7,N = 103, α = 0.5, β = 0.3, γ = 0.2, ρ = 103, we obtain the graph of thefunction ω(x, t) = ω1(t) sin β1(x− a) + ω2(t) sinβ2(x− a) at x = a+b

2 (fig.3).

Fig. 3.

Example 2. For P (t) = 105 cos(t) and the above particular values of para-meters, by means of the MathCAD 2000, we obtain the graph of the functionω(x, t) = ω1(t) sinβ1(x− a) + ω2(t) sin β2(x− a) at x = a+b

2 (fig.4).

Fig. 4.


Example 3. For P (t) = 105e−10t and the above values of parameters,by using MathCAD 2000, we obtain the graph of the function ω(x, t) =ω1(t) sin β1(x− a) + ω2(t) sin β2(x− a) at x = a+b

2 (fig.5).

Fig. 5.

Example 4. Let P0(x, t) = 0, P (t) = 105e−100t, w(x, 0) = 0, w(x, 0) = 0.Then, by MathCAD 2000, for the values of parameters m = 2, a = 0, b = 0.2,y0 = 1, M = 42.4, D = 806.7, N = 103, α = 0.5, β = 0.5, γ = 0.5, ρ = 103, weobtain the graph of the function ω(x, t) = ω1(t) sinβ1(x−a)+ω2(t) sinβ2(x−a)at x = a+b

2 (fig.6).

Fig. 6.References[1] Vel’misov P. A., Pokladova Yu. V., The investigation of oscillations of an elastic ele-

ment of pressure sensor, Vestnik of ULSTU, Ulyanovsk, 2(2005), 20-22.

[2] Vel’misov P. A., Pokladova Yu. V., Mathematical models of a mechanical system pipeline– pressure sensor, Application of Mathematics in Engineering and Economics, Sofia,STU (2004), 84–89.

[3] Vel’misov P. A., Pokladova Yu. V., The investigation of oscillations of an elastic ele-ment of the pressure sensor, ULSTU, Ulyanovsk (2005), 60–64.

[4] Reshetnikov Yu. A., On dynamics of an elastic element of the pressure sensor, ULSTU,Ulyanovsk (2005), 201–204.

[5] Vel’misov P. A., Garnevska L. V., Gorbokonenko V. D., The investigation of mathe-matical models pipeline – pressure sensor, Application of Mathematics in Engineeringand Economics, Heron Press, Sofia (2002), 542-548.

[6] Vel’misov P. A., Reshetnikov Yu. A., Gorbokonenko V. D., The mathematical modellingof dynamics of an elastic element of pressure sensor, ULSTU, Ulyanovsk (2003), 24–30.

EXTENDING OBJECT-ORIENTED DATABASESFOR FUZZY INFORMATION MODELING

ROMAI J., 2, 1(2006),225–237

Cristina-Maria VladareanS.C. WATERS Romania S.R.L.cristina [email protected]

Abstract In this paper, based on the possibility distribution and semantic measure offuzzy data, an extended object-oriented database model to handle imperfectas well as complex object in the real-world is introduced. Some major notionsin object-oriented databases such as objects, classes, object-classes, relation-ships, subclass/superclass and multiple inheritances are extended under fuzzyinformation environment. A generic model for fuzzy object-oriented databasesand some operations are presented.

Keywords: fuzzy data; object-oriented databases; semantic measure; fuzzy object-oriented

database model.

1. INTRODUCTIONA major goal for database research has been the corporation of additional

semantics into the data model. In real-world applications, information is of-ten vague or ambiguous. Therefore, different kinds of incomplete informationhave been extensively introduced into relational databases. However, classi-cal relational database model in its extension of imprecision and uncertaintydoes not satisfy the need of modeling complex objects with imprecision anduncertainly. So many researches have been concentrated on the developmentof some database models to deal with complex objects and uncertain datatogether.

For practical needs, fuzzy object-oriented databases were developed. Somemajor notions in object-oriented databases such as objects, classes, objects-classes relationships, subclass/superclass, and multiple inheritances are ex-tended under fuzzy information environment. Such a generic model is pre-sented in this paper.

2. BASIC KNOWLEDGEFuzzy set and possibility distribution. Fuzzy data, originally de-

scribed by Zadeh, are defined as follows. Let U be a universe of discourse.Then a fuzzy value on U is characterized by a fuzzy set F in U . A member-ship function µF : U → [0, 1] is defined for the fuzzy set F , where µF (u), for

225

226 Cristina-Maria Vladarean

each u ∈ U , denotes the degree of membership of u in the fuzzy set F . Thus,the fuzzy set F is described as follows:

F = µ(u1)/u1, µ(u2)/u2, . . . , µ(un)/un .When the µF (u) above is explained to be a measure of possibility that a

variable X has the value u in this approach, where X takes values in U , afuzzy value is described by a possibility distribution πX . Let πX and F be thepossibility distribution representation and the fuzzy set representation for afuzzy value respectively. It is apparent that πX = F is true.

In addition, a fuzzy data is represented by similarity relations in domainelements, in which the fuzziness comes from the similarity relations betweentwo values in a universe of discourse, not from the status of an object itself.Similarity relations are thus used to describe the degree similarity of two valuesfrom the same universe of discourse. A similarity relation Sim on the universeof discourse U is a mapping: U × U → [0, 1] such that

(a) for ∀x ∈ U , Sim(x, x) = 1 (reflexivity),(b) for ∀x, y ∈ U , Sim(x, y) = Sim(y, x) (symmetry),(c) for ∀x, y, z ∈ U , Sim(x, z) ≥ maxy(min(Sim(x, y), Sim(y, z))) (transi-

tivity).Semantic measure of fuzzy data. The semantics of fuzzy data repre-

sented by possibility distribution corresponds to the semantic space and thesemantic relationship between two fuzzy data can be described by the rela-tionship between their semantic spaces. The semantic inclusion degree is thenemployed to measure semantic inclusion and thus measure semantic equiva-lence of fuzzy data.

Definition. Let πA and πB be two fuzzy data, and let their semantic spacesbe SS(πA) and SS(πb), respectively. Let SID(πA, πB) denote the degree thatπA semantically includes πB. Then

SID(πA, πB) = (SS(πB) ∩ SS(πA))/SS(πB).For two fuzzy data πA and πB, the meaning of SID(πA, πB) is the percent-

age of the semantic space of πB which is wholly included in the semantic spaceof πA.

Definition. Let πA and πB be two fuzzy data and SID(πA, πB) be thedegree that πA semantically includes πB. Let SE(πA, πB) denote the degreethat πA and πB are equivalent to each other. Then

SE(πA, πB) = min(SID(πA, πB), SID(πB, πA)).Definition. Let U =u1, u2, . . . , un be the universe of discourse. Let πA and

πB be two fuzzy data on U based on possibility distribution and πA(ui), ui ∈ U ,denote the possibility that ui is true. Let Res be a resamblance relation onthe domain U , let α, for 0 ≤ α ≤ 1, be a threshold corresponding to Res.Then SID(πA, πB) is defined by

Extending object-oriented databases for fuzzy information modeling 227

n∑

i=1

minui, uj∈D

(πB (ui) , πA (uj)) /

n∑

i=1

πB (ui) .

The notion of the semantic equivalence degree of attribute values can be ex-tended to the semantic equivalence degree of tuples. Let ti =< ai1, ai2, . . . , ain >and tj =< aj1, aj2, . . . , ajn > be two tuples in fuzzy relational instance r overthe schema R(A1, A2, . . . , An). The semantic equivalence degree of tuples tiand tj is denoted by SE (ti, tj)= min (SE (ti [A1] , tj [A1]) , SE (ti [A2] , tj [A2]) , . . . , SE (ti [An] , tj [An])).

3. FUZZY OBJECTS AND CLASSESThe objects model real-world entities or abstract concepts. The objects

have the properties of being attributes of the object itself or relationships alsoknown as associations between the object and one or more other objects. Anobject is fuzzy because of a lack of information. Formally, objects that haveat least one attribute whose value is a fuzzy set are fuzzy objects.

The objects having the same properties are gathered into classes that areorganized into hierarchies. Theoretically, a class can be considered from twodifferent viewpoints: (a) an extensional class, where the class is defined bythe list of its object instances, and (b) an intensional class, where the class isdefined by a set of attributes and their admissible values.

In addition, a subclass defined from its superclass by means of inheritancemechanism in the OODB can be seen as the special case of (b) above.

Therefore, a class is fuzzy because of the following several reasons. First,some objects are fuzzy, which have similar properties. A class defined by theseobjects may be fuzzy. These objects belong to the class with the membershipdegree of [0, 1]. Second, when a class is intentionally defined, the domain of anattribute may be fuzzy and a fuzzy class is formed. For example, a class Oldequipement is a fuzzy one because the domain of its attribute Using periodis a set of fuzzy values such as long, very long and about 20 years. Third,the subclass produced by a fuzzy class by means of specializations and thesuperclass produced by some classes (in which there is at least one class whois fuzzy) by means of generalizations are also fuzzy.

The main difference between fuzzy classes and crisp classes is that theboundaries of fuzzy classes are imprecise. The imprecision in the class bound-aries is caused by the imprecision of the values in the attribute domain. Inthe fuzzy OODB, classes are fuzzy because their attribute domain are fuzzy.The issue that an object fuzzily belongs to a class occurs since a class or anobject is fuzzy. Similarly, a class is a subclass of another class with mem-bership degree of [0, 1] because of the class fuzziness. In the OODB, theabove mentioned relationships are certain. Therefore, the evaluations of fuzzyobject-class relationships and fuzzy inheritance hierarchies are the core of in-


formation modeling in the fuzzy OODB. In the following discussion, let usassume that the fuzzy attribute values of fuzzy objects and the fuzzy valuesin fuzzy attribute domain are represented by possibility distribution.

Fuzzy object-class relationships. In the fuzzy OODB, the followingfour situations can be distinguished for object-class relationships.

(a) Crisp class and crisp object: this situation is the same as the OODB,where the object belongs or not to the class certainly (such as for example theobjects Car and Computer are for a class Vehicle, respectively).

(b) Crisp class and fuzzy object: although the class is precisely defined andhas the precise boundary, an object is fuzzy since its attribute value(s) may befuzzy. In this situation, the object may be related to the class with the specialdegree in [0, 1] (such as for example the object whose position attribute maybe graduate, reasearch assistant, pr research assistant professor is for the classFaculty).

(c) Fuzzy class and crisp object: being the same as the case in (b), theobject may belong to the class with the membership degree [0, 1] (such as forexample a Ph.D. student is for the Young student class).

(d) Fuzzy class and fuzzy object: in this situation, the object also belongsto the class with the membership degree in [0, 1].

The object-class relationships in (b)-(d) above are called fuzzy object-classrelationships. In fact, the situation in (a) can be seen as the special case offuzzy object-class relationships, where the membership degree of the object tothe class is one. It is clear that estimating the membership of an object to theclass is crucial for the fuzzy object-class relationship when class is instantiated.

In the OODB, determining if an object belongs to a class depends on thefact whether its attribute values are respectively included in the correspondingattribute domains of the class. Similarly, in order to calculate the member-ship degree of an object to the class in a fuzzy object-class relationship, itis necessary to evaluate the degrees that the attribute domains of the classinclude the attribute values of the object. However, it should be noted thatin a fuzzy object-class relationship, only the inclusion degree of object valueswith respect to the class domains is not accurate for the evaluation of themembership degree of an object to the class. The attributes play differentroles in the definition and identification of a class. Some may be dominantand some not. Therefore, a weight w is assigned to each attribute of the classaccording to its importance to the designer. Then the membership degree ofan object to the class in a fuzzy object-class relationship should be calculatedusing the inclusion degree of object values with respect to the class domainsand the weight of attributes.

Let C be a class with attributes A1, A2, . . . , An, o be an object on at-tribute set A1, A2, . . . , An, and let o(Ai) denote the attribute value of oon Ai(1 ≤ i ≤ n). In C, each attribute Ai is connected with a domain de-


noted dom(A1). The inclusion degree of o(Ai) with respect to dom(Ai) isdenoted ID(dom(Ai), o(Ai)). In the following, let us investigate the evalua-tion of ID(dom(Ai), o(Ai)). As we know, dom(Ai) is a set of crisp values in theOODB and may be a set of fuzzy subsets in fuzzy databases. Therefore, in auniform OODB for crisp and fuzzy information modeling, dom(Ai) should bethe union of these two components, dom(Ai) = cdom(Ai) ∪ fdom(Ai), wherecdom(Ai) and fdom(Ai), respectively, denote the sets of crisp values and fuzzysubsets. On the other hand, o(Ai) may be a crisp value or a fuzzy value. Thefollowing cases may be identified for evaluating ID(dom(Ai), o(Ai)):

Case 1: o(Ai) is a fuzzy value. Let fdom(Ai) = f1, f2, . . . , fm, wherefj(1 ≤ j ≤ m) is a fuzzy value, and cdom(Ai) = c1, c2, . . . , ck, where cl(1 ≤l ≤ k) is a crisp value. ThenID(dom(Ai), o(Ai)) = max(ID(cdom(Ai), o(Ai)), ID(fdom(Ai), o(Ai)))

= max(SID( 1c1

, 1c2

, . . . , 1ck, o(Ai)), maxj(SID(fj), o(Ai))),

where SID(x, y) is used to calculate the degree that fuzzy value x includesfuzzy value y.

Case 2: o(Ai) is a crisp value. Then,ID(dom(Ai), o(Ai)) = 1 if o(Ai) ∈ cdom(Ai),

else ID(dom(Ai), o(Ai)) = ID(fdom(Ai), 1o(Ai)

).Now, let us define the formula to calculate the membership degree of the

object o to the class C as follows, where w(Ai(C)) denotes the weight ofattribute Ai to class C:

µC(o) =Pn

i=1 ID(dom(Ai),o(Ai))×w(Ai(C))Pni=1 w(Ai(C))

In the above-given determination that an object belongs to a class fuzzily,it is assumed that the boject and the class have the same attributes, namely,class C is with attributes A1, A2, . . . , An and object o is also on A1, A2, . . . ,An. Such an object-class relationship is called direct object-class relationship.As we know, there exist subclass/superclass relationships in the OODB, wherethe subclass inherits some attributes and methods of the superclass, and de-fines some new attributes and methods. Any object belonging to the subclassmust belong to the superclass since a subclass is the specialization of the super-class. So we have one kind of special object-class relationship: the relationshipbetween superclass and the objects of the subclass. Such an object-class re-lationship is called indirect object-class relationship. Since the object and theclass in indirect object-class relationship have different attributes, in the fol-lowing we show how to calculate the membership degree of an object to theclass in an indirect object-class relationship.

Let C be a class with attributes A1, A2, . . . , Ak, Ak+1, . . . , Am and o be anobject on attributes A1, A2, . . . , Ak, A

′k+1, . . . , A

′m, Am+1, . . . , An. Here at-

tributes A′k+1, . . . , A′m are overridden from Ak+1, . . . , Am and attributes Am+1,

. . . , An are special. Then we have


µC(o) =Pk

i=1 ID(dom(Ai),o(Ai))×w(Ai(C))+Pm

j=k+1 ID(dom(Aj),o(A′j))×w(Aj(C))Pm

i=1 w(Ai(C))

Based on the direct object-class relationship and the indirect object-classrelationship, now we focus on arbitrary object-class relationship. Let C be aclass with attributes A1, A2, . . . , Ak, Ak+1, . . . , Am, Am+1, . . . , An and o bean object on attributes A1, A2, . . . , Ak, A

′k+1, . . . , A

′m, Bm+1, . . . , Bp. Here

the attributes A′k+1, . . . , A′m are overridden from Ak+1, . . . , Am, Am+1, . . . , An.

Attributes Am+1, . . . , An and Bm+1, . . . , Bp are special. Then we have

µC(o) =Pk

i=1 ID(dom(Ai),o(Ai))×w(Ai(C))+Pm

j=k+1 ID(dom(Aj),o(A′j))×w(Aj(C))Pm

i=1 w(Ai(C))

Since an object may belong to a class with membership degree in [0, 1]in fuzzy object-class relationship, it is possible that an object that is in adirect object-class relationship and an indirect object-class relationship simul-taneously belongs to the subclass and superclass with different membershipdegrees. This situation occurs in fuzzy inheritance hierarchies, which will beinvestigated in the next section. Also, for two classes that do not have sub-class/superclass relationship, it is possible that an object may belong to thesetwo classes with different membership degrees simultaneously. This situationonly arises in fuzzy object-oriented databases. In the OODB, an object mayor may not belong to a given class definitely. If it belongs to a given class, itcan only belong to it uniquely (except for the case of subclass/superclass).

The situation where an object belongs to different classes with differentmembership degrees simultaneously in fuzzy object-class relationships is calledmultiple membership of object in this paper. Let us focus on how to handlethe multiple membership of object in fuzzy object-class relationships. Let C1and C2 be (fuzzy) classes and α be a given threshold. Assume there existsan object o. If µC1(o) ≥ α and µC2(o) ≥ α, the conflict of the multiplemembership occurs, namely, o belongs to multiple classes simultaneously. Atthis moment, which one in C1 and C2 is the class of object o depends on thefollowing cases.

Case 1: There exists a direct object-class relationship between object o andone class in C1 and C2. Then the class in the direct object-class relationshipis the class of the object o.

Case 2: There is no direct object-class relationship but only an indirectobject-class relationship between object o and one class in C1 and C2, sayC1. And there exists such a subclass C1′ of C1 that object o and C1′ are ina direct object-class relationship. Then class C1′ is the class of object o.

Case 3: There is neither direct object-class relationship nor indirect object-class relationship between object o and classes C1 and C2. Or there existsonly an indirect object-class relationship between object o and one class in C1and C2, say C1, but there is no such subclass C1′ of C1 that object o and C1′are in a direct object-class relationship. Then class C1 is considered as the


class of object o if µC1(o) > µC2(o), else class C2 is considered as the class ofobject o.

It can be seen that in Cases 1 and 2, the class in direct object-class rela-tionship is always the class of object o and the object and the class have thesame attributes. In Case 3, however, object o and the class that is consideredas the class of object o, say C1, have different attributes. It should be pointedout that class C1 and the object o are definitely defined, respectively, viewedfrom their structures. For the situation in Case 3, the attributes of C1 do notaffect the attributes of o and vice-versa. There should be a class C, and Cand o are in direct object-class relationship. But class C1 is more similar toC than C2. Class C is the class of object o once C is available.

4. FUZZY INHERITANCE HIERARCHIESIn the OODB, a new class, called subclass, is produced from another class,

called superclass by means of inheriting some attributes and methods of thesuperclass, overriding some attributes and methods of superclass, and definingsome new attributes and methods. Since a subclass is the specialization ofthe superclass, any one object belonging to the subclass must belong to thesuperclass. This characteristic can be used to determine if two classes havesubclass/superclass relationship.

In the fuzzy OODB, however, classes may be fuzzy. A class produced froma fuzzy class must be fuzzy. If the former is still called subclass and the latersuperclass, the subclass/superclass relationship is fuzzy. In other words, aclass is a subclass of another class with membership degree of [0, 1] at thismoment. Correspondingly, the method used in the OODB for determinationof subclass/superclass relationship is modified as

(a) for (any) fuzzy object, if the member degree that it belongs to thesubclass is less than or equal to the member degree that it belongs to thesuperclass, and

(b) the member degree that it belongs to the subclass is greater than orequal to the given threshold.

The subclass is then a subclass of the superclass with the membership de-gree, which is the minimum in the membership degrees to which these objectsbelong to the subclass.

Let C1 and C2 be (fuzzy) classes and β be a given threshold. We say C2 isa subclass of C1 if (∀o) (β ≤ µC2(o) ≤ µC1(o)). The membership degree thatC2 is a subclass of C1 should beminµC2(o)≥β(µC2(o)).

It can be seen that by utilizing the inclusion degree of objects to the class,we can asses fuzzy subclass/superclass relationshipd in the fuzzy OODB. Itis clear that such assesment is indirect. If there is no object available, this


method is not used. In fact, the idea used in evaluating the membershipdegree of an object to a class can be used to determine the relationship be-tween fuzzy subclass and superclass. We can calculate the inclusion degreeof a (fuzzy) subclass with respect to the (fuzzy) superclass according to theinclusion degree of the attribute domains of the subclass with respect to theattribute domains of the superclass as well as the weight of attributes. Inthe following, a method for evaluating the inclusion degree of fuzzy attributedomains is given.

Let C1 and C2 be (fuzzy) classes with attributes A1, A2, . . . , Ak, Ak+1, . . . ,Am and A1, A2, . . . , Ak, A

′k+1, . . . , A

′m, Am+1,

. . . , An, respectively. It can be seen that in C2, attributes A1, A2, . . . , Ak

are directly inherited from A1, A2, . . . , Ak in C1, attributes A′k+1, . . . , A′m

are overriden from Ak+1, . . . , Am in C1, and attributes Am+1, . . . , An arespecial. For each attribute in C1 or C2, say Ai, there is a domain, de-noted dom(Ai). As shown above, dom(Ai) should be dom(Ai) = cdom(Ai) ∪fdom(Ai), where cdom(Ai) and fdom(Ai) denote the sets of crisp values andfuzzy subsets, respectively. Let Ai and Aj be attributes of C1 and C2, respec-tively. The inclusion degree of dom(Aj) with respect to dom(Ai) is denotedby ID(dom(Ai), dom(Aj)).

Then we identify the following cases and investigate the evaluation ofID(dom(Ai), dom(Aj)):

(a) when i 6= j and 1 ≤ i, j ≤ k, ID(dom(Ai), dom(Aj)) = 0,(b) when i = j and 1 ≤ i, j ≤ k, ID(dom(Ai), dom(Aj)) = 1, and(c) when i = j and k + 1 ≤ i, j ≤ m, then

ID(dom(Ai), dom(Aj)) = ID(dom(Ai), dom(A′i))= max(ID(dom(Ai), cdom(A′i)), ID(dom(Ai), fdom(A′i))).

Now we respectively define ID(dom(Ai), cdom(A′i)) and ID(dom(Ai),fdom(A′i)). Let fdom(A′i) = f1, f2, . . . , fm, where fj(1 ≤ j ≤ m) is a fuzzyvalue, and cdom(A′i) = c1, c2, . . . , ck, where cl(1 ≤ l ≤ k) is a crisp value.We can consider c1, c2, . . . , ck as a special fuzzy value 1

c1, 1

c2, . . . , 1

ck. Then

we have the following, which can be calculated by using the definition givenabove in this paper:ID(dom(Ai), cdom(A′i)) = ID(dom(Ai), 1

c1, 1

c2, . . . , 1

ck),

ID(dom(Ai), fdom(A′i)) = maxj(ID(dom(Ai), fj)).Based on the inclusion degree of attribute domains of the subclass with

respect to the attribute domains of its superclass as well as the weight ofattributes, we can define the formula to calculate the degree to which a fuzzyclass is a subclass of another fuzzy class. Let C1 and C2 be (fuzzy) classes withattributes A1, A2, . . . , Ak, Ak+1, . . . , Am and A1, A2, . . . , Ak, A

′k+1, . . . , A

′m,

Am+1, . . . , An, respectively, and w(A) denote the weight of attribute A. Thenthe degree that C2 is a subclass of C1, written µ(C1, C2), is defined as follows


µ(C1, C2) =Pm

i=1 ID(dom(Ai(C1)),dom(Ai(C2)))×w(Ai))Pmi=1 w(Ai)

In subclass-superclass hierarchies, a critical issue is multiple inheritance ofclass. Ambiguity arises when more than one of the superclasses have commonattributes and the subclass does not declare explicitly the class from whichthe attribute was inherited.

Let class C be a subclass of classes C1 and C2. Assume that the attributeAi in C1, denoted by Ai(C1), is common to the attribute Ai in C2, denotedby Ai(C2). If dom(Ai(C1)) and dom(Ai(C2)) are identical, there does notexist a conflict in the multiple inheritance hierarchy and C inherits attributeAi directly. If dom(Ai(C1)) and dom(Ai(C2)) are not identical, however, theconflict occurs. At this moment, which among Ai(C1), Ai(C2) is inherited byC depends on the following rule:

if ID(dom(Ai(C1)), dom(Ai(C2)))× w(Ai(C1)) >

ID(dom(Ai(C2)), dom(Ai(C1)))× w(Ai(C2)),

then Ai(C1) is inherited by C, else Ai(C2) is inherited by C.Note that in fuzzy multiple inheritance hierarchy, the subclass has differ-

ent degrees with respect to different superclasses, not being the same as thesituation in classical object-oriented database systems.

5. FUZZY OBJECT-ORIENTED DATABASEMODEL AND OPERATIONS

Based on the discussion above, we have known that the classes in the fuzzyOODB may be fuzzy. Accordingly, in the fuzzy OODB, an object belongsto a class with a membership degree of [0, 1] and a class is the subclass ofanother class with degree of [0, 1] too. In the OODB, the specification ofa class includes the definition of ISA relationships, attributes and methodsimplementations. In order to specify a fuzzy class, some additional definitionsare needed. First, the weights of attributes to the class must be given. Inaddition to these common attributes, a new attribute should be added intothe class to indicate the membership degree to which an object belongs to theclass. If the class is a fuzzy subclass, its superclasses and the degree that theclass is the subclass of the superclasses should be illustrated in the specificationof the class. Finally, in the definition of a fuzzy class, fuzzy attributes may beexplicitely indicated.

Formally, the definition of a fuzzy class is shown as follows:

CLASS class name WITH DEGREE OF degreeINHERITES superclass 1 name WITH DEGREE OF degree 1. . .INHERITES superclass k name WITH DEGREE OF degree k


ATTRIBUTESAttribute 1 name: [FUZZY] DOMAIN dom 1 : TYPE OF type 1

WITH DEGREE OF degree 1. . .Attribute m name: [FUZZY] DOMAIN dom m: TYPE OF type m

WITH DEGREE OF degree mMembership Attribute name: membership degree

WEIGHTw(Attribute 1 name)=w 1. . .w(Attribute m name)=w m

METHODS. . .END

For non-fuzzy attributes, the data types include simple types such as integer,real, Boolean, string, and complex types such as set type and object type. Forfuzzy attributes, however, the data types are fuzzy-type-based on simple orcomplex types, which allow the representation of descriptive form of impreciseinformation. Only fuzzy attributes have fuzzy type and fuzzy attributes areexplicitely indicated in a class definition. Therefore, in the definition above,we declare only the base type (e.g., integer) of fuzzy attributes and the fuzzydomain. A fuzzy domain is a set of possibility distributions or fuzzy linguisticterms (each fuzzy term is associated with a membership function).

The change in database model impacts the operations on the databasemodel. In the following, let us describe some operations based on the pro-posed fuzzy class model above. First, we briefly discuss several combinationoperations of the fuzzy classes. Finally, we investigate the issue of user request-queries based on the fuzzy classes. Depending on the relationships between theattribute sets of the combining classes, three kinds of combination operationscan be identified: fuzzy product(×), fuzzy join(./) and fuzzy union(∪). LetC1 and C2 be fuzzy classes and let Attr(C1) and Attr(C2) be their attributesets, respectively. Assume a new class C is created by combining C1 and C2.Then,C = C1×C2, if Attr′(C1) ∩Attr′(C2) = ∅ ,C = C1 ./ C2, if Attr′(C1) ∩Attr′(C2) 6= ∅ and Attr′(C1) = Attr′(C2),orC = C1∪C2, if Attr′(C1) = Attr′(C2).

Here, Attr′(C1) and Attr′(C2) are obtained from Attr(C1) and Attr(C2)through removing the membership degree attributes from Attr(C1) andAttr(C2), respectively. In the following, µC is used to represent the member-ship degree attribute of C. Assume we have an object o of C. Then µC(o) isused to represent the value of o on µc. For a common attribute in C, say Ai,


o(Ai) is used to represent the value of o on Ai. If we have a set of such commonattributes, say Ai, Aj , . . . , Am, then o(Ai, Aj , . . . , Am) is used to representthe values of o on attributes in Ai, Aj , . . . , Am. Furthermore, o(C) is usedto represent all values of o on the common attributes in C. In the following,the formal definitions of fuzzy product, fuzzy join, and fuzzy union operationsare given.

Fuzzy product: The fuzzy product of C1 and C2 is a new class C, whichconsists of the common attributes of C1 and C2 as well as a membershipdegree attribute. The objects of C are created by the composition of objectsfrom C1 and C2:

C = C1×C2 = o| (∀o′) (∀o′′) (o′ ∈ C1 ∧ o′′ ∈ C2∧ o(Attr′(C1)) = o′(C1))∧ o(Attr′′(C2)) = o′′(C2)∧ µC(o) = op(µC1(o′), µC2(o′′))).

Here, operation op is undefined. Generally, op(µC1(o′), µC2(o′′)) may bemin(µC1(o′), µC2(o′′))

or µC1(o′)× µC2(o′′).Fuzzy join: The fuzzy join of C1 and C2 is a new class C, where Attr′(C1)∩

Attr′(C2) 6= ∅ and Attr′(C1) 6= Attr′(C2). Class C is composed of Attr′(C1)∪(Attr′(C2)− (Attr′(C1) ∩Attr′(C2))) as well as a membership degree at-tribute. The objects of C are created by the composition of objects from C1and C2, which are semantically equivalent on Attr′(C1)∩Attr′(C2) under thegiven thresholds. It should be noted that, however, Attr′(C1)∩Attr′(C2) 6= ∅implies C1 and C2 have the same weights of attributes for the attributes inAttr′(C1)∩Attr′(C2). This is an additional requirement to be met in the caseof the fuzzy join operation. Let α be the given threshold. Then,

C = C1 ./ C2 = o| (∃o′) (∃o′′) (o′ ∈ C1 ∧ o′′ ∈ C2∧ SE (o′(Attr′(C1) ∩Attr′(C2)), o′′(Attr′(C1) ∩Attr′(C2))) ≥ α∧ o(Attr′(C1)) = o′(C1)∧ o(Attr′(C2)− (Attr′(C1) ∩Attr′(C2)))

= o′′(Attr′(C2)− (Attr′(C1) ∩Attr′(C2)))∧ µC(o) = op(µC1(o′), µC2(o′′))).

Here, operation op is also undefined. Generally, op(µC1(o′), µC2(o′′)) maybe min(µC1(o′), µC2(o′′)) or µC1(o′)× µC2(o′′).

Fuzzy union: The fuzzy union of C1 and C2 requires Attr′(C1) = Attr′(C2),which implies that all corresponding attributes in C1 and C2 have the sameweights. Let a new class C be the fuzzy union of C1 and C2. Then the objectsof C are composed of three kinds of objects. The first two kinds of objectsare such objects that directly come from one component class (e.g., C2) underthe given thresholds. The last kind of objects is such that the objects are theresults of merging the redundant objects from two component classes underthe given thresholds. Let α be the given threshold


C = C1∪C2 = o| (∀o′′) (o′′ ∈ C2 ∧ o ∈ C1 ∧ SE(o(C1), o′′(C2)) < α)∨ (∀o′) (o′ ∈ C1 ∧ o ∈ C2 ∧ SE(o(C2), o′(C1)) < α)∨ ((∃o′)) ((∃o′′)) o′ ∈ C1 ∧ (∃o′′) (o′ ∈ C1 ∧ o′′ ∈ C2)

∧ SE(o′(C1) ∧ o′′(C2)) ≥ α ∧ o = merge(o′, o′′)).Here, merge is an operation for merging two redundant objects of the class

to form a new object of the class. Let o′ and o′′ be two objects of class Cand o = merge(o′, o′′). Then o(C) = o′(C) or o(C) = o′′(C) and µC(o) =max(µC1(o′), µC2(o′′)).

Query processing refers to such procedure that the objects satisfying agiven condition are selected and then they are delivered to the user accordingto the required formats. These format requirements include which attributesappear in the result and if the result is grouped and ordered over the givenattribute(s). So a query can be seen as comprising two components, namelya Boolean selection condition and some format requirements. As a simpleillustration, some format requirements are ignored in the following discussion.An SQL (structured query language) like query syntax is represented asSELECT 〈attribute list〉

FROM 〈class names〉WHERE 〈query condition〉,

where attributelist is the list of attributes separated by commas. At least oneattribute name must be specified in this list. Attributes that take place inattributelist are selected from the associated classes which are specified in theFROM clause. classnames contains the class names separated by commas,classes from which the attributes are selected with the SELECT clause.

Classical databases suffer from a lack of flexibility to query. The given querycondition and the contents of the database are all crisps. A query is flexibleif the databases contain imprecise and uncertain information, and the querycondition is imprecise and uncertain. For the fuzzy object-oriented databases,it has been shown above that objects belong to a given class with membershipdegree [0, 1]. In addition, an object satisfies the given query condition alsowith membership degree [0, 1] because fuzzy information occurs in the querycondition and/or the object. Therefore, the query processing based on theproposed fuzzy object-oriented database model refers to such procedure thatthe objects satisfying a given threshold and a given condition under giventhresholds simultaneously are selected from the classes. It is clear that thequeries for the fuzzy object-oriented databases are threshold-based ones, whichare concerned with the number choices of threshold. Therefore, an SQL likequery syntax based on the fuzzy object-oriented database model is representedas follows:SELECT 〈attribute list〉FROM 〈Class1 WITH threshold1, . . . , Classm WITH thresholdm〉WHERE 〈query condition WITH threshold〉.


Here, 〈query condition〉 is a fuzzy condition and all thresholds are crispnumbers in [0, 1]. Utilizing such SQL, one can get such objects that belong tothe class under the given thresholds and also satisfy the query condition underthe given thresholds at the same time. Note that the item WITH thresholdcan be ommitted. The default of the threshold is exactly 1 for such a case.

6. CONCLUSIONIncorporation of imprecise and uncertain information in database model

has been an important topic of database research because such an informationextensively exists in data and knowledge intensive application such as expertsystem, decision making etc. Besides that, these systems are characterized bycomplex object structures. Classical relational database model and its exten-sion of imprecision and uncertainty do not satisfy the need of handling complexobjects with imprecision and uncertainty. Fuzzy object-oriented databases arehereby introduced.

In this paper, based on the possibility distribution and the semantic measuremethod of fuzzy data, a fuzzy object-oriented database model was presentedto cope with imperfect as well as complex objects in the real-world at a logicallevel.

References[1] Buckles, B., Frederick, E., Petry, A., A fuzzy representation of data for relational data-

base, Fuzzy Sets Systems., 7, 3(1982), 213-226.

[2] George, R., Srikanth, R., Petry, F. E., Buckles, B. P., Uncertainty management issuesin the object-oriented data model, IEEE Trans. Fuzzy Systems, 4, 2(1996), 179-1902.

[3] Umano, M., Relational algebra in fuzzy database, IEICE Tech. Rep. DE86-4, 86,192(1986), 1-8.

[4] Zemankova-Leech, M., Kandel, A., Fuzzy relational databases - A key to expert systems,

Verlag TUV Rheiland Gmbh, 1980.

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