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8/4/2019 Abstracts 7

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7-th International Conference on Discrete

Mathematics and Applications (ICDMA7)

June 17-20, 2004, Bansko, Bulgaria

South-West University, Blagoevgrad, Bulgaria

University of Potsdam, Germany


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Contents

Irena Atanassova . . . . . . . . . . . . . . . . . . . . . . . . . . 1Pavel Azalov, Daniela Tuparova . . . . . . . . . . . . . . . . . . 1Dimitar Birov . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Elena Bo jadshieva . . . . . . . . . . . . . . . . . . . . . . . . . 2Adrijan V. Borisov . . . . . . . . . . . . . . . . . . . . . . . . . 3Kiril Chimev . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Svetoslav Christov . . . . . . . . . . . . . . . . . . . . . . . . . 4Ivo Damyanov . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Klaus Denecke, S. L. Wismath . . . . . . . . . . . . . . . . . . 5Ilinka Dimitrova . . . . . . . . . . . . . . . . . . . . . . . . . . 6Ivan Ganchev Donev . . . . . . . . . . . . . . . . . . . . . . . . 7Ivan Georgiev, Svetozar Margenov . . . . . . . . . . . . . . . . 7Kazimierz Glazek . . . . . . . . . . . . . . . . . . . . . . . . . . 8Vassil Grozdanov . . . . . . . . . . . . . . . . . . . . . . . . . . 10Elena Karastranova . . . . . . . . . . . . . . . . . . . . . . . . . 10

Maryana Katsarska . . . . . . . . . . . . . . . . . . . . . . . . . 10P. S. Kenderov . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Nikolay Kirov, Mikhail Krastanov . . . . . . . . . . . . . . . . 11Jorg Koppitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Rumen Kostadinov, P. Milanov, N. Pencheva, I. Trenchev . . . 12Dimiter Stoichkov Kovachev . . . . . . . . . . . . . . . . . . . . 14Krassimir Manev . . . . . . . . . . . . . . . . . . . . . . . . . . 14Smile Markovski, Danilo Gligoroski . . . . . . . . . . . . . . . . 15Nikolay Kitanov, Viktor Plotnikov . . . . . . . . . . . . . . . . 15Marija Mihova, Smile Markovski . . . . . . . . . . . . . . . . . 16Ivan Mirchev, Zoran Aleksov . . . . . . . . . . . . . . . . . . . 16Violeta Nikova . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

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Julia Ninova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Ketty Peeva, Yordan Kyosev . . . . . . . . . . . . . . . . . . . 18D. Schweigert . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Slavcho Shtrakov . . . . . . . . . . . . . . . . . . . . . . . . . . 18Grozio Stanilov . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Stefka Tchincheva, Kostadin Lekov . . . . . . . . . . . . . . . . 20Kalcho Todorov, Iliya Gyudzenov . . . . . . . . . . . . . . . . . 20Margarita Todorova, Nina Siniagina . . . . . . . . . . . . . . . 21Ivan Trenchev, Peter Milanov . . . . . . . . . . . . . . . . . . . 21Ivan Trenchev, Miglena Trencheva . . . . . . . . . . . . . . . . 22Georgi Tuparov . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Krasimir Yordzhev . . . . . . . . . . . . . . . . . . . . . . . . . 23

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How to prove the completeness of a new temporal logic - eRATL

Irena AtanassovaSouth-West [email protected]

This paper is a contribution to the area of modal and temporal logics. Weextend linear discrete temporal logic with several modal operators. In that waywe obtain a unranked tree-like structure. In our logic the new modal operatorswork both along the branches of a tree and along the children of a node. Toprove the completeness theorem of enriched with abstractions of time lineardiscrete temporal logic eRATL we are going to use a method that involvesbuilding a canonical model. In the present paper, we are going to sketch theproof of the completeness theorem.

Discrete Math Module in CS Course: Basic Accents and Implemen-tationPavel Azalov, Daniela TuparovaPennsylvania State University, Hazleton, [email protected]

South-West University, Blagoevgrad, Bulgaria, [email protected]

The standard for inclusion of Computer Science in the high school wasestablished in 2000. One of the fundamental theoretical modules is DiscreteMathematics, which consists of the basic notions and methods used in introduc-tory college courses. The module, however, seems to be too complex, despite

the fact that it is designed for the students with Computer Science concentra-tion. The current paper has as an objective to summarize the characteristicsof the Discrete Mathematics module, including the input and the experiencefrom other countries. An alternative method for teaching the content is pre-sented, which will help to overcome the problems related to the complexity ofthe matter. In addition, a respective approach for testing the knowledge of thestudents is described.

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Overloading of Advices

Dimitar BirovSofia University, [email protected]

Aspect Oriented Programming propose Aspect as a tool for modularize con-cerns. Advices are parts of aspects which specify code, which will be executedat a specific points called jointpoints in the programs. A pointcut specificationdescribes set of joinpoints. With each pointcut more then one advice wichhave a similar signature but one and the same name could be conected. Thisphenomena is recognized as overloading. Actually overloading is a one of thebranches of ad-hoc polymorphism. This way advices became a tool for intro-ducing ad-hoc polymorphism in programs. This paper will discuss intermixingbetween aspectual polymorphism defined by advices and universal F-bounded

polymorphism.

Forschung und Entwicklung im Blinkwinkel der EU-ErweiterungElena BojadshievaSd-West-Universitat Neofit Rilskielena [email protected]

Im Jahre 2000 setzte der Europische Rat ein stratgisches Ziel hinsichtlich derweiteren wissenschaflich-technischen Entwicklung der EU, bis 2010 soll die EUder wettbewerbsfhigsten und dynamischsten Wirtschaftsraum der Welt wer-den. Durch eine Steigerung der Forschungsausgaben auf 3Japan und den USA

berwunden werden. Es wird analysiert, wie sich dieser Prozess im Kontextder EU-Erweiterung vollzieht und welchen Einfluss auf ihn die Beitrittslnderausben.

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Measurability of sets of pairs of nonisotropic and isotropic straight

lines in the Galilean planeAdrijan V. BorisovDept. of Math., South-West [email protected]

In the affine version the Galilean plane 2 is an affine plane with a spe-cial direction [3], [5]. The affine transformations leaving invariant the specialdirection form the group H5 of the general similitudes in 2. We study themeasurability in the sense of M. I. Stoka [4] and G. I. Drinfeld [2] of sets ofpairs nonisotropic + isotropic straight line in 2 with respect to H5 and itssubgroups [1]. Expressions for the corresponding invariant densities are alsogiven.

References

[1] A. V. Borisov. On the subgroups of the similarity group in the Galileanplane. C. R. Acad. Bulg. Sci. 46, (1993), no. 5, 19-21.[2] G. I. Drinfeld. On the measure of the Lie groups. Zap. Mat. Otdel.

Fiz. Mat. Fak. Kharkov. Mat. Obsc., 21, 1949, 47-57, (in Russian).[3] H. Sachs. Ebene Isotrope Geometrie. Vieweg , Braunschweig/Wiesbaden,1987.[4] M. I. Stoka. Geometrie Integrala. Ed. Acad. RSR, Bucuresti, 1967.[5] I. M. Yaglom. A Simple Non-Euclidean Geometry and its Physical Basic.

Springer, Berlin, 1979.

Structural Properties of the FunctionsKiril Chimev

South-West [email protected]

We discuss some structural properties of the functions with respect to theirseparable sets of variables.

With Ess(f) we denote the set of all essential variables for a function f.The set of all functions which depends essentially on n variables is denoted

by F(n).A set M Ess(f) is called separable for f if there are values ci1 , ci2 . . . , cis

for the variables from the set Ess(f) \ M = {xi1 , xi2 . . . , xis} such that thesubfunction f(xi1 = ci1 , xi2 = ci2 . . . , xis = cis) of f essentially depends on allvariables in M.

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The set of all non-empty separable sets for a function f is denoted by

Sep(f).Let f F(n). Unordered hypergraph with vertices the essential variablesof f, and with edges the set of separable sets for f which contain exactly melements is called hypergraph of f with respect to the separable mtuples.

The hypergraph of f with respect to the separable mtuples is denoted byH(f, m).

The hypergraph of a function f with respect to the separable pairs is alsocalled graph of this function.

We will discuss the results obtained as following theorems.Theorem 1. If f F(n), n 2 then for each m, 2 m n, the hyper-

graph H(f, m) is connected, and the distance between any two of its verticesis not greater than 2.

Theorem 2. If the function f(x1, . . . , xm, y1, . . . , yn, z1, . . . , zp, t1, . . . , tq)

F(m + n + p + q) does not form separable pairs of types (xi, zk) and (yj, tl)then all the pairs of types (xi, yj), (yj , zk), (zk, tl) and (xi, tl) for 1 i m,1 j n, 1 k p, and 1 l q are separable for f.

Theorem 3. If f F(n), n 4 and (x1, x2) / Sep(f), (x1, x3) / Sep(f),and (x2, x3) / Sep(f), then there exists a variable xi such that the pairs (xi, xj)are separable for each j = 1, 2, 3.

Theorem 4. There does not exist a function f F(n), n 5 for whichonly the pairs (x1, xi) and (xn, xi) i = 2, 3, . . . , n 1 are separable for f.

IPv6 Address Space Architecture and its effect on Dynamic Routing

Structure PlaningSvetoslav ChristovSouth-West University, [email protected]

The next generation of IP - IPv6 comes with completely new recomenda-tions about how addresses will be allocated world wide. This rises severalquestions about how to plan and implement networks, running dynamic rout-ing protocols. This report presents an overview of these problems and discusessome aspects of planing the usage of dynamic routing with IPv6.

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From Structure to Behavior - Defining Declarative Language Gram-

mars with XMLIvo DamyanovDepartment of Computer Sciences, South-West [email protected]

Most of the modern technologies are mixture of well-known patterns appliedin the right place. Since its introduction, the major players of the software in-dustry adopted XML. As a result of evolution of markup languages, based ontheoretical background of tree automata and regular grammar, XML becomean important building stone for next generation computer systems. In the pa-per some applied aspects of XML technology are concerned. Major benefits indevelopment of loosely coupled code providing better separation of concerns,easiness of implementation of declarative languages and effective data extrac-

tion with schema-driven tools are discussed.

Normalizations of ClonesKlaus Denecke, S. L. WismathUniversitat [email protected]

The clone of a variety is a multi-based algebra whose universes are the setsof n-ary terms of the variety, for each n N \ {0}, and whose operations arethe superpositions of terms. The clone of a variety carries much information

about the variety. We consider the restriction to a type in which all operationsymbols are n-ary, for some fixed n, in which case the set of n-ary termsis the universe of a homogeneous or one-based algebra, using an n + 1-arysuperposition operation. This algebra is a clone-like structure, and we call itthe n-clone of the corresponding type with the difference that the projectionterms are not included a nullary operations, so that the type is (n + 1) ratherthan (n + 1, 0, . . . , 0). Such an algebra is called a Menger algebra of rank n. Acomplexity measuremant or valuation function assigns to each term of a fixedtype a natural number called its complexity. There are several ways to definesuch a complexity measure. For any fixed k 1, we consider the set of all termswhich have complexity k or higher, with respect to a given complexity measure.Under the right assumptions on our complexity measure, this set of terms form

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a subclone of the clone of all terms of a given type. The k-normalization of a

variety V is the model class defined by the set of identities of V for which bothterms in the identity have complexity at least k. In this paper we study theclone of the k-normalization variety of a given variety V. We also show that insome special cases the identities of these clones correspond to M-hyperidentitiesof the variety, for certain monoids M of hypersubstitutions.

On some classification of the maximal idempotent-generated sub-semigroups of the semigroup of all isotone transformationsIlinka DimitrovaBulgaria, Blagoevgrad 2700 South-West University Neofit [email protected]

In this paper we consider the finite set X(


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Ik =

E(Jk)

=

Jk

.

On the role and the place of discrete mathematics in school Mathe-matics courseIvan Ganchev DonevSouth-West [email protected]

In the paper is drawn the attention on the fact, that during the last decadesmore and more convincingly domineers the idea, that with developing the prob-lems of mathematics didactic is necessary to take into account perspicuously

the structure of mathematical knowledge. It occurs that at forming this struc-ture some elements of discrete mathematics play an important role. It is shownin the paper, that the main role has the concept set and related to it opera-tions and relations, especially the concepts relation of equivalence and binaryrelation.

MIC(0) Preconditioning of Rotated Bilinear FEM Elliptic SystemsIvan Georgiev, Svetozar MargenovIPP-BAS,Acad. G. Bonchev Str., [email protected]

New results about preconditioning of non-conforming FEM systems in thecase of mesh anisotropy are presented. This study is focused on the implemen-tations of rotated bilinear elements, where algorithms [MP] and [MV] standfor the variants of the nodal basi functions corresponding to midpoint andintegral mid-value interpolation operators.The model elliptic problem underconsideration is associated with the bilinear form

ah(u, v) =eh

e

a(e)2i=1

uxivxide

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where h is a decomposition of the computational domain into rectangles

denoted by e. To get a stable M IC(0) factorization, in the general case, wefirst substitute the stiffness matrix by an auxiliary M-matrix.Then M IC(0)preconditioner is applied to this matrix.Two such approaches are presentedwhere a local analysis is used to get estimates of the related condition numbers.Construction of the optimal M-matrix is very natural step of this study. Forgiven SPD matrix K we want to find SPD M-matrix B such that the conditionnomber of the generalized eigenvalue problem,

Ku = Bu

is as smal as posible. The presented numerical tests well illustrate the behaviourof the theoretically studied algorithms.

General independence notions and some open problemsKazimierz GlazekUniversity of Zielona Gora, [email protected]

In 1958 E. Marczewski introduced a general notion of independence,which contained as special cases majority of independence notions used in var-ious branches of mathematics. A non-empty set Iof the carrier A of an algebraA = (A; F) is called M-independent if equality of two term operations f and gof the considered algebra on any finite system of different elements of I impliesf = g in A. There are several interesting results on this notion of independence.

However the important scheme of M-independence is not enough wide to coverthe stochastic independence, the independence in projective spaces and someothers. This is why some notions weaker than the M-independence were de-veloped. The notion of independence with respect to family Q of mappings(defined on subsets of A) into A, Q-independence for short, is a common wayof defining almost all known notions of independences. There exists an inter-esting Galois correspondence between families Q of mappings and families ofQ-independent sets. In our talk after a brief survey of these topics we willmainly concentrate on a few easily formulated and interesting results. Thereare several interesting open problems, for example:

Let A = (A;F) be an algebra and C(X) denote a subalgebra ofA generatedby X A.

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1) For which families Q the following property (J IS)Q ofQ-independence

holds?(JIS)Q for arbitrary Q-independent sets I and J (I, J A) , the set I Jis also Q-independent, whenever

C (I) C (J) = C (I J)?

2) For which algebras this property (IJS)Q holds for a well-definedfamily Q?

3) For algebras with (JIS)Q property for Q-independent sets, charac-terize the families Ind (A; Q) of all Q-independent sets in the algebra A.

4) Investigate algebrasA with (JIS)M property for M-independent sets(that is, independent sets in the sense of Marczewski).

References:

[1] K. Glazek, Independence with respect to a family of mappings in abstractalgebras, Dissertationes Math. 81 (1971), 1-55.

[2] K. Glazek, Some old and new problems in the independence theory, Col-loq. Math. 42 (1979), 127-189.

[3] K. Glazek and S. Niwczyk, A new perspective on Q-independence, p.61-69 in the book: General Algebra and Applications (K. Denecke and H.-J.Vogel, eds.), Shaker-Verlag, Aachen 2000.

[4] K. Glazek and S. Niwczyk, Q-independence and weak automorphisms asGalois connections, in the book: Galois Connections (K. Denecke, M. Erneand S. Wismath, eds.), to appear (2002).

[5] E. Marczewski, Independence and homomorhisms in abstract algebras,Fund. Math. 50 (1961), 45-61.

[6] E. Marczewski, Independence in abstract algebras. Results and problems,Colloq. Math. 14 (1966), 169-188.

[7] K. Urbanik, Linear independence in abstract algebras, Colloq. Math. 14(1966), 233-255.

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The generalized badic diaphony of the Zaremba-Halton net over

finite abelian groupsVassil GrozdanovSouth West University N. Rilski, Blagoevgrad, [email protected]

A class of two-dimensional nets, constructed over finite abelian groups withrespect to an arbitrary bijection is proposed. Some well-known two-dimensionalnets are obtained as a special case of the so-called net of Zaremba-Haltonover finite abelian groups. The generalized badic diaphony as a quantitativemeasure of the irregularity of the distribution of nets in [0 , 1)s is considered.The exact order and the exact constant in this order of the generalized badicdiaphony of the nets of this class are found.

Interactive instruction in combinatorics through VBA and MS ExcellElena KarastranovaSouth West University N. Rilski, Blagoevgrad, Bulgariahelen [email protected]

In the paper several ideas for introducing of basic concept of combinatoricsin the math secondary school course through the interactive features of VBAand MS Excell are presented. Also the system of problem solving is proposed.

Solving practical problems connected with graphs in extracurriculareducation in 3rd - 5th classesMaryana KatsarskaNeofit Rilski South-West University

[email protected]

In this research, methodological ideas are presented concerning introductionof the concept of graph (object of discrete mathematics) in extracurricularmathematics education in 3rd - 5th classes. Some problems are consideredwhose solution is connected with elementary level study of concepts connectedwith the graph: complete graph, incomplete graph, oriented graph, graph-tree,etc.

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Dynamical Systems Generated by Plane Convex Compacta

P. S. KenderovBulgarian Academy of [email protected]

Dynamical systems generated by convex subsets of the Euclidean plane willbe considered. As an application a numerical method for the identificationof the best approximation by n-gons to a given convex compact subset in theplane will be described. Another interpretation of the method suggests a newapproach to the approximation of function by special splines (with variablenodes).

Skiba points of optimal investment strategiesNikolay Kirov, Mikhail [email protected]

We consider a simple model in a linear-quadratic control problem for opti-mal investment strategies. The revenue is a maximum of two concave quadraticfunctions and the cost is a convex quadratic function. We determine the exist-ing of a Skiba point and in this case we can calculate numerically this point.

Bijections and hypersubstitutionsJorg KoppitzUniversity of [email protected]

Let be any type. A mapping which assigns to each operation symbol aterm of the same arity is called hypersubstitution of type . Since each of theoperation symbols f is related to its fundamental terms f(x1, . . . , x(f)) ((f)means the arity of f) we can think any hypersubstitution as mapping theterm f(x1, . . . , x(f)) to the term (f). It follows that every hypersubstitutionof type then induces a mapping from the set W(X) of all terms of type

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into W(X). The set of all hypersubstitutions of type together with an asso-

ciative operation forms a monoid which was investigated by various authors.

For example, in the computer science or by the consideration of clone automor-phisms we are interested in such hypersubstitutions for which the extension is a bijection.We characterize for each type the set Bij() of all hypersubstitutions suchthat : W(X) W(X) is a bijection. It turns out that Bij () forms asubmonoid of the monoid of all hypersubstitutions of type .

Further estimation of the affinity and efficacy of partial agonistsRumen Kostadinov, Peter Milanov, Nevena Pencheva, Ivan Trenchev

Department of Informatics, Faculty of Natural and Mathematical Sciences,South-West University, 66 Ivan Mihailov Str., Blagoevgrad 2700, Bulgaria

[email protected]

The approximation of the so-called concentration-response curves in thequantitative pharmacology is important, because allows characterizing the prop-erties of drugs and newly synthesized compounds with respective quantitativeindexes. In our previous investigation we established proper algorithm andexplicit formulas for calculation of following parameters: ETm - a maximal re-sponse of the tissue, EAm -a maximum response of the drugs, EC

A50- an agonist

concentration which produces 0.5EAm, KA - dissociation constant,eAC2

- relativeefficacy, basing on the following well known hyperbolic function:

EA =EAm[A]

P

[A]P + [A50]P,

where P is the slope of the respective curves and C2 - the measure unit ofthe stimulus.

The formulas for KA and relative efficacy are:

KA =[A50]

1 A;

eAC2

=A

1 Awhere A =

EAmETm

.

In this study we explore the role of the slope for data approximation usingconcentration-response curves of the enkephalin (pentapeptide with endogenous

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nature) analogue Dalargin. These curves were obtained by in vitro biological

experiments with isolated tissue (gunea-pig ileum-longitudinal muscles). Thepreparations were electrically stimulated and the responses in the presence ofincreasing concentrations of the drugs were calculated as fraction inhibition

(EA

ETm

where ETm = 1; the values are presented as X+S.E.M ). After approx-

imation of the concentration-response curves, the value of EAm, KA, A50 andrelative efficacy eA

C2calculating with different slopes (range 0.6 - 2 with a step

0.2) are presented in Table 1.

Table 1. Parameters1 of concentration-response curves calculated with dif-ferent slopes Pi.

Pi 0.6 0.8 1 1.2 1.4 1.6 1.8 2EAm 0.9023 0.86 0.8364 0.8211 0.8104 0.8027 0.7988 0.8806A50 0.4511 0.43 0.4182 0.4106 0.4051 0.4013 0.3993 0.4403KA 4.6162 3.0726 2.5562 2.2950 2.1366 2.0337 1.9848 3.6878eAC2

9.2325 6.1452 5.1123 4.59 4.2732 4.0674 3.9696 7.3756

The analysis of the data obtained show that optimal approximation of theconcentration-response curves is with slopes between 0.8 - 1.2 as far as bestfitting is concerned. We need further calculations to precise the criteria forthe optimal slope for approximation of pharmacological experimental data inin vitro investigations.

1

EAm - maximum response of the drugs, A50 - agonist concentration which produces 0.5EAm,

KA - dissociation constant,eA

C2- relative efficacy, C2 - unit to measure stimulus.

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On a Class of Discrete Functions

Dimiter Stoichkov KovachevSouth-West [email protected]

Let m, n, q, k be natural numbers with 1 m n, 1 q k, k 2.We consider the nvariable functions of kvalued logic.The number of different values of a function is called range of this function.We prove some results concerning the class of functions which have sub-

functions of range q obtained when replacing arbitrary n m variables witharbitrary n m constants.

In the particular case q = k, m = 1 we obtain the class of all H functions,considered in [1]. The number of the functions in this class is obtained.

References

[1]K. Chimev, Sur une sorte de dependance de certaines fonctions de Pkde leurs arguments, annuaire des ecoles techniques superieures, mathematique,vol. IV, Livre .1, 1967, p.p. 5-12 (in Bulgarian).

Mathematics and discrete mathematicsKrassimir ManevSofia University St. Kliment Ohridski, American University in Bulgaria

[email protected]

There is a trend to consider the discrete mathematics as a very specificpart of the mathematics, dedicated mostly to serve the computer science. In

this work we try to demonstrate the wrongness of such trend and the negativeconsequences that follow from it. Some elements of the discrete mathematicsare considered that are of significant importance for the general mathemati-cal culture (inductively defined set, Boolean algebra, relation over Cartesiansquare, graph and trees, formal language, formula/expression etc.) and ex-amples of their role in the process of education are given. The necessity ofincluding these elements in traditional school mathematics curriculum as soonas possible is also discussed.

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Construction of quasigroups of huge order

Smile Markovski, Danilo GligoroskiSs Ciryl and Methodius University, Skopje, [email protected]

There are some applications of quasigroups in cryptography that need quasi-groups of huge order (for example of order 21024). Clearly, the applications arepossible only if the quasigroups are effectively defined. We propose severalways of effective definitions of quasigroups of huge order, and possibilities oftheir applications.

Method of Averaging for Impulsive Differential Inclusions and Im-pulsive Optimal Control ProblemsNikolay Kitanov, Viktor PlotnikovBulgarian Academy of Science, Blagoevgrad, [email protected]

Odessa State University, Odessa, Ukraina

In this paper are presented some results conected with applications of theaveraging method for solving of optimal control problems, where the modelsare system differential equations with impulsive effects. We suppose additionalcontrol in the impulses.

Key words: method of averaging, differential inclusion, impulsive differen-tial inclusion, optimal control, small parameter.

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Algorithms for computing n-th derivative of composite function

Marija Mihova, Smile MarkovskiSs Ciryl and Methodius University, Skopje, [email protected]

We give an iterative formula for computing n-th derivative of a compositefunction, different than the formula of Faa de Bruno. Both formulas takesummation on the set K(n) = {(k1, k2, . . . , kn)|ki are non-negative, k1 + 2k2 +. . . n kn = n}. There is not known a formula for computing the sequence ofnumbers |K(n)|. We give an iterative procedure for computing these numbers,as well as some approximation results.

Computer - based model for Teaching and Learning Discrete Math-ematicsIvan Mirchev, Zoran AleksovSWU N. Rilski

[email protected]

An instructional model used by the authors to investigate the benefits fromthe integration of the computers and information technologies in the processesof teaching and learning computer sciences, due to its flexibility, could be easilyadapted to the needs of the processes of teaching an learning Discrete Mathe-matics.

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Logical and mathematical relations equivalence and implication

in the school course in mathematicsVioleta NikovaNeofit Rilski South-West [email protected]

This communication can be considered as a parallel paper to the paperentitled On the Place and Role of Elements of Discrete Mathematics in theSchool Course in Mathematics by Prof. Ivan Ganchev, DSc. It is pointedout in it that it is reasonable to distinguish two types of equivalence relationsin mathematics. Equivalence relations of the first type do not depend on theparticular inferences and inference functions contained in them, whereas equiv-alence relations of the second type depend on these inferences and inferencefunctions. On the basis of the realized differentiation between these two types

of equivalence relations and on Descartess principle of separated overcoming ofdifficulties, some important conclusions are derived for methodology of teachingmathematics.

Modeling of Didactical TasksJulia NinovaSofia [email protected]

Using the set-theory means and the apparatus of propositional calculusfor modelling replaces descriptive prescription for solving didactical tasks with

models. Operating with formal objects is made this way, because the reasoningare moved from the concrete to abstract level. The reasoning are concentratedover the structure, not over the specific content. The profit of this modelingis in revelation the common of actions, the ability of forecasting, the abilityfor determining the actions of the teacher and for organize purposefully theactions of the students. The researches in the methodology of the educationin mathematics are moved to a higher level. The liberating of the empiricismand prescription is possible only in the frames of some theory.

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Fuzzy relational calculus toolbox in MATLAB

Ketty Peeva, Yordan KyosevTechnical University of Sofia. Faculty of Applied Mathematics and Informat-ics, P.O.Box 384, Sofia [email protected]

We develop software for fuzzy relational calculus (maxmin, minmaxand intuitionistic cases) over the fuzzy algebra ([0, 1], max, min) in MATLABworkspace. Solving direct and inverse problems in fuzzy relational calculusare among the main advantages of this software. Applications in finite fuzzymachines and textile engineering are included.

Ordered knapsack problemsD. SchweigertFB Mathematik TU Kaiserslautern [email protected]

Given a graph there are several ways to define an order relation on theset of edges. All of the classical optimization problems such as shortest path,minimal spanning trees, are defined for a linear order.But we will generalize toa poset and present a multiple knapsack problem.

Coloring of Terms by Tree AutomataSlavcho ShtrakovSouth-West [email protected]

Let be a type with set of operation symbols = (0, ..., n). The setof all terms (trees) of type is denoted by W(X). Hypersubstitutions aremappings : W(X) and they form a monoid Hyp(). Any mapping : N Hyp() is called multi-hypersubstitution of type . The set of allmulti-hypersubstitutions is denoted by MHyp(). Let N be the set N :={arity(f) | f }. Let Q, Q N be a non-empty set of natural numberscalled colors. A coloration of type is a sequence C := (ci1 , . . . , cin) where

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cm : m Qm Q for m N, m 1 and c0 : X 0 Q. A term t is

colored if all its subterms are supplied with colors from Q. Let q Q be acolor. Let Col() be the set of all colorations of type . Each coloration C canbe used for coloring of terms as follows:(i) if t 0 X then C(t) := c0(t);(ii) if t = f(t1, . . . , tn) then C(t) := cn(f, C(t1), . . . , C (tn)). This coloring canbe realized by the following tree automaton A := (Q, , Qf, C) with Qf = Qand the assignment := c0.

A pair := (, C) MHyp() Col() is called hypercoloration of type. The set of all hypercolorations is denoted by CMHyp(). The elements = (, C) of CMHyp() can be viewed as mappings : W(X) W(X) asfollows:(i) if t X then [t] := t;(ii) if t = f(t1, . . . , tn) then

[t] := (cn(f, C(t1), . . . , C (tn)))(f)([t1], . . . , [tn]).

The hypercolorations allows us to generalize the concept of hypersubstitu-tions working over colored terms. This concept is important in different fieldsof Computer Science - Graphical User Interface (GUI), XML - technology, Ob-ject Oriented Programming etc. In this paper we obtain some internal resultsconcerning the monoid CMHyp() of all hypercolorations, hypercolored de-rived algebras, hypercolored varieties and hypercolored identities, generatedby hypercolorations.

A triangle transformation and applications in the artGrozio StanilovSofia University St. Kliment [email protected]

If ABC is a triangle and M0(x0, y0) is an arbitrary point,we introduce thefunctions: L1- the length of the section of ABC with the line AM0, L2 - withBM0, L3 - with CM0. The points P,Q,R defined by the conditions determinethe triangle P QR. We investigate the triangle transformation ABC P QR.For example:1. P BC,Q CA,R AB 2.P,Q,R are collinear iff ABCis rectangular triangle; 3. The transformation A P is a product of anorthogonal projection, a symmetry and a translation. Using this transformation

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we give a set of interesting curves and surfaces and by their visualizations -

beautiful bodies, some of them can be used in the applied art. All calculationsare done by Computer Algebra and Computer Graphic.

Statistics methods application in the medico-social work with long-time unemployed personsStefka Tchincheva, Kostadin LekovSSouth-West University Neofit [email protected]

The main topics of the present research study is statistics methods appli-cation in medico-social work with long-time unemployed persons with psychic

disabilities, stress, anxiety and life-crisis. The appropriate dynamic series andtheir characteristics were analyzed. We pay special attention to some featuresdetermining this negative symptoms. We used data concerning unemployedpersons from the state social work cervices.

On the maximal subsemigroups of the finite transformation semi-groupKalcho Todorov, Iliya GyudzenovBulgaria, Blagoevgrad 2700 South-West University Neofit [email protected]

The full transformation of X(


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all isotone transformations of a finite linearly ordered set are given in the work

of Yang Xiuliang, Communications in Algebra, 28(3), (2000), 1503-15132.

Interrelation of Teaching of the Pattern Recognition and DiscreteMathematicsMargarita Todorova, Nina SiniaginaSouth-West [email protected]

Bulgarian Academy of Sciences

The paper surveys the application of discrete mathematics methods lookedout in the elective discipline Pattern Recognition in Information Technology.

The interdependence of disciplines and practical application of the methodsof discrete mathematics is demonstrated important for solving problems fromdifferent fields.

New property of the genetic polytopeIvan Trenchev, Peter MilanovDepartment of Informatics, Faculty of Natural and Mathematical Sciences,South-West [email protected]

The contemporary genetic code is produced from evolutionary forces. Onehypothesis is that the arrangement of amino acid/codon assignments resultsfrom effects of the minimization of the errors of single point mutations fromwhich follows some changes in genetic code information. Which concern theproperties of the peptides sanitizes in the cell. Recently we described explicitlythe set of all theoretical genetic codes as a convex polytope. We call thispolytope a genetic polytope. In this presentation, we show new property ofthis polytope namely, that any two vertices are neighbors. We investigate alsothe errors of point mutations, which are estimated by different criteria.

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Financial Toolbox in MATLAB

Ivan Trenchev, Miglena TrenchevaDepartment of Informatics, Faculty of Natural and Mathematical Sciences,South-West [email protected]

Matlab and the Financial Toolbox provide a complete integrated computingenvironment for financial analysis and engineering. The toolbox has everythingyou need to perform mathematical and statistical analysis of the financial dataand displays the results with state of the arts presentation-quality graphics. Inthis presentation we show how to find a solution of some financial problems asa control risk, analyze or manage a portfolio and others.

Simulation of failure and repair in discrete manufacturing systemsby Colored Petri NetsGeorgi TuparovSouth West University, [email protected]

In the paper the Colored Petri Net models of discrete manufacturing sys-tems are presented. A module for simulation of failure and repair based ondifferent probability distributions is performed. The simulation is realized withCPNtools package.

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On a equivalence relation in the set of the permutation matrices

Krasimir YordzhevSouth-West University, Blagoevgrad, [email protected]

A square matrix having an element equal to 1 in each row and each column,and the other elements being zero, is called permutation matrix. Let Sn bethe set of all n n permutation matrices and let A, B Sn. An equivalencerelation is defined as follows: A B if and only if A can be obtained fromB by a sequential moving of the first row or column to the last place. Thecardinal number of the factor-set Sn/ is found.

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