6th international conference on discrete mathematics and...
TRANSCRIPT
6th International Conference on Discrete
Mathematics and Applications
South-West UniversityBlagoevgrad, Bulgaria
August 31 - September 2, 2001
Contents
Irena Atanasova, Logic and binary trees . . . . . . . . . . . . . 1Verica Bakeva, Hash functions defined by quasigroup enciphering
methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Georgi Bijev, Coregularity In Finite Semigroups . . . . . . . . . 2Kiril Chimev, Separable Sets of Variables for the Functions . . 2Svetoslav Christov, Jordan Denev, Dynamic routing in IP ver-
sion 6 environment . . . . . . . . . . . . . . . . . . . . . 2I. Damyanov, Sl. Strakov, Essential Inputs and Minimal Tree
Automata . . . . . . . . . . . . . . . . . . . . . . . . . . 3K. Denecke, Hypersubstitutions in Computer Science . . . . . . 3Stefan Dodunekov, Kissing numbers—bounds and constructions 4Zoltan Esik, Fixed Point Clones in Computer Science . . . . . 4Lidija Goracinova, Smile Markovski, Approaches to a problem of
constructing free algebras and examples of free groupoids 5Krassimir Iordjev, On a class of binary matrices . . . . . . . . 6Joerg Koppitz, Colored Solid Varieties of Semigroups . . . . . . 6Jelena Kovacevic, Miroslav Ciric, Tatjana Petkovic, Stojan Bog-
danovic, Classes of Automata Defined by Quasi-orderson a Free Monoid . . . . . . . . . . . . . . . . . . . . . . 7
Dimiter Kovachev, Iliya Gyudzhenov, On the number of k-valuedfunctions with given range of their subfunctions . . . . . 7
Boris V. Novikov, On a Characteristic of Subsets of Abelian groups 8Ketty Peeva, Peter Manoilov, Boriana Deliiska Computer Code
for Fuzzy Matrix Operations . . . . . . . . . . . . . . . . 9N. Pencheva, P. Milanov, Artificial neural networks: introduc-
tion and some examples . . . . . . . . . . . . . . . . . . 9
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Anton A. Penzov, Galya Hristova A Multimedia System to Testthe Knowledge Acquired at the Computer Graphics Course 10
Irina Hristova Petrova, Algebraic invariants of knots and theirapplications in Chemistry, Biochemistry anf Phisics . . 11
Vanyo Peychev, Information system for querying structured doc-uments via WAP . . . . . . . . . . . . . . . . . . . . . 11
Apostolos Tatsis, Vanyo Peychev, Krassimira Panovska, AdamZeinov, Directions for the IT Education Improvement inthe Secondary Schools . . . . . . . . . . . . . . . . . . . 12
Zarko Popovic, Stojan Bogdanovic, Miroslav Ciric Minimal Pathsin Semigroups . . . . . . . . . . . . . . . . . . . . . . . . 12
I.G. Rosenberg, Clones and related concepts in Universal Alge-bras and Many-valued Logics . . . . . . . . . . . . . . . 13
Dietmar Scbweigert, On power ordered sets . . . . . . . . . . . 14Bl. Sendov, On the Hausdorff Geometry of Polynomials . . . . 14Slavcho Shtrakov, Hypersubstitutions and Colored Trees . . . . 15Stefan M. Stefanov, Applications of Cutting Plane Methods for
Solving Variational Inequalities . . . . . . . . . . . . . . 15Stanislava Stoilova, On the Uniform Distribution mod1 of Se-
quences . . . . . . . . . . . . . . . . . . . . . . . . . . . 16K. Todorov, On The Minimal Polynomials of the N-Generalized
Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . 16Milen Todorov, Anton A. Penzov, A Colour Quantization Method
with Histogram Partition . . . . . . . . . . . . . . . . . 17Ivan Trenchev, Peter Milanov, The genetic code optimality . . . 18Georgi Tuparov, Object-Oriented Approach for Modeling of Dis-
crete Production Systems . . . . . . . . . . . . . . . . . 19Daniela Tuparova, Katerina Marcheva, The Logical Expressions
and Functions in the Secondary School Courses in Infor-matics and Information Technologies . . . . . . . . . . . 19
Daniela Tuparova, Didactical Aspects in Problem Solving in theSecondary School Courses in Informatics and Informa-tion Technologies . . . . . . . . . . . . . . . . . . . . . . 20
Hristo Shoilev, Vasil Marinov, Representing of Dynamic Knowl-edge by Means of Predicate Sillogistic Logic . . . . . . . 20
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Logic and binary treesIrena Atanasova
South-West University, Blagoevgrad, Bulgaria
The theory of binary trees is a theory with equality. In this theory are definedfollowing concepts: atom(x), tree(x), left(x), right(x), subtree and propersubtree. For every tree x is defined a unary functional symbol listtree(x).The value of listtree is a the tuple whose elements are the leaves of x. In thetheory of trees are defined following functions: size(x) (the number of nodesand leaves in a tree x), leav(x) (the number of leaves in a tree x) and depth(x)(the number of nodes and leaves in the longest path of a tree x). In the tableaumethod for proving the validity are using axioms as assertions and the axiom(x = x).
Hash functions defined by quasigroup enciphering methodsVerica Bakeva∗, Smile Markovski, Danilo Gligoroski
Institute of Informatics, Faculty of the Natural Sciences and Mathematics,Skopie, Macedonia
Hash functions are a special kind of (public) functions, which are used for dif-ferent cryptographic purposes, like signature schemes, message authenticationcode, etc. Using the quasigroups enciphering methods, we propose algorithmsfor construction of hash functions, which are weakly collision-free and stronglycollision free.
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Coregularity In Finite SemigroupsGeorgi Bijev
Techn. Uni. of Sofia, Bulgaria
Some abstract properties of finite semigroups in connection with the coregu-larity are considered in this paper. Some essential invariant characteristics ofsemigroups (with respect to isomorphism) are taken to consideration.
Separable Sets of Variables for the FunctionsKiril Chimev
South-West University, Blagoevgrad, Bulgaria
We will discuss on the essential and strongly essential variables of the functions,subfunctions, structural properties of the functions with respect to their sep-arable sets of arguments, separable and inseparable sets variables, invarianceof the separable sets of variables under the operations replacing of the vari-ables with constants and identifying the variables, functions with two stronglyessential variables, functions possessing variables from the first order etc.
Dynamic routing in IP version 6 environmentSvetoslav Christov∗
South-West University, Blagoevgrad, BulgariaJordan Denev (University of Sofia, Sofia, Bulgaria)
IP version 6 (IPv6) is a new version of the Internet Protocol, designed as a suc-cessor to IP version 4 (IPv4). There are currently two types of routing protocolin the Internet. These are Interior Gateway Protocols (IGP) sometimes called
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Intra-Domain Routing Protocols and Exterior Gateway Protocols (EGP) some-times called Inter-Domain Routing Protocols. The routing protocol describedhere is Routing Information Protocol (RIP) with minimum changes, necessaryfor operation over IPv6.
Essential Inputs and Minimal Tree AutomataI. Damyanov∗, Sl. Strakov
South-West University, Blagoevgrad, Bulgaria
In the paper we continue studying essential inputs of trees and automata ini-tiated in [1]. We distinguish the behaviour of the essential inputs of trees andessential variables for discrete functions. Strongly essential inputs of trees areintroduced too. It is proved that if a tree and an automaton have at leasttwo essential inputs then they have at least one strongly essential input. Aminimization algorithm for trees and automata is proposed. Various examplesfor application in Computer Science are shown.
[1] Sl. Shtrakov, Tree Automata and Essential Input Variables, 2000, Con-tributions to General Algebra 13, Proc. of the Dresden Conf. 2000, VerlagJohannes Heyn, Klagenfurt 2000.
Hypersubstitutions in Computer ScienceK. Denecke
University of Potsdam, Germany
Hypersubstitutions are mappings from a set of operation symbols going intothe corresponding set of terms. The theory of hypersubstitutions and hyperi-dentities is well-developed and can be aplied to
• tree automata (to generalize them to hyper-tree-automata)
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• tree transformations and tree transducers
• the semantical and syntactical hyperunification problem.
Each of these applications will be discussed under the aspect of the advantagesof hypersubstitutions compared with usual substitutions.
Kissing numbers—bounds and constructionsStefan Dodunekov
Bulgarian Academy of Science
The evaluation of the n-th kissing number (the max number of non-intersectingequal spheres in the n-dimensional Euclidean space that touch one sphere ofthe same radius) is considered to be one of the most interesting mathematicalproblems. It has several centuries of fascinating history and connections withmany branches of mathematics as well as with communication theory. Theexact values of kissing numbers are known only for n = 1, 2, 3, 8 and 24.
Fixed Point Clones in Computer ScienceZoltan Esik
Dept. of Computer Science, University of Szeged, Hungary
Previous work has resulted in a complete description of the equational lawssatisfied by the fixed point operation in the computationally significant modelsincluding ordered and metric models, continuous semirings, and 2-categories.The desciption takes the form of the axioms of Iteration Theories, or IterationClones, cf. [1, 2, 6]. In the lecture I will review the axiomation of iterationclones by the Conway identities and the group-identities [3, 4]. This complete-ness result confirms a conjecture of J. H. Conway in a very general setting. I
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will also provide some applications of iteration clones to solve axiomatizationproblems in computer science (see, e.g., [5, 7]).
[1] S. L. Bloom and Z. Esik: Iteration Theories: The Equational Logic ofIterative Processes, Springer, 1993.[2] S. L. Bloom and Z. Esik: The equational logic of fixed points, TheoreticalComputer Science, 179(1997), 1–60.[3] Z. Esik: Group axioms for iteration, Information and Computation, 148(1999),131–180.[4] Z. Esik: The power of the group identities for iteration, Int. J. Algebra andComputation, 10(2000), 349–373.[5] Z. Esik: Axiomatizing the least fixed point operation and binary supremum,in: proc. Computer Science Logic, 2000, LNCS 1862, Springer, 2000, 302–316.[6] S. L. Bloom, Z. Esik, A. Labella and E. Manes: Iteration 2-theories, AppliedCategorical Structures, 9(2001), 173–216.[7] Z. Esik: The equational theory of fixed points with applications to gener-alized language theory,in: proc. Developments in Language Theory, Vienna,July 2001, 25–44.
Approaches to a problem of constructing free algebras andexamples of free groupoids
Lidija Goracinova Ilieva∗
Pedagogical Faculty ”Gotse Delchev”,MacedoniaSmile Markovski (Institute of Informatics, Faculty of the Natural Sciences
and Mathematics, Skopie, Macedonia)
The difficulties of obtaining a description of the free algebras in a variety givenby a set of defining identities depends on the complexity of the interrelation-ships between the identities in the variety. Although there is no general methodto resolve this problem in each particular case, some approaches turn out to beuseful in the attempt to construct free algebras: direct approach by chosing afree algebra - model, description of the congruence generated by the identities,
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term reduction, replacement scheme, construction by a chain of partial alge-bras, and construction with a sequence of algebras. In that way were obtaineddescriptions of free groupoids in many varieties.
On a class of binary matricesKrassimir Iordjev
South-West University, Blagoevgrad, Bulgaria
The set Λkn of all n×n matrices with k identities in every row and every column
is discussed. Fixing an entry, let πkn and νk
n be the numbers of all matrices inΛk
n with respectively 1 and 0 in this entry. The relation between πkn and νk
n isfound. Some applications of this relation are given.
Colored Solid Varieties of SemigroupsJoerg Koppitz
University of Potsdam, Germany
A hypersubstitution of type τ is a mapping from the set of the operation sym-bols in the set of all terms of type τ , which preserves the arity. We want toconsider so-called colored hypersubstitutions. For this we give the set of termsa coloration by marking the operation symbols in all term by natural numbers.A colored hypersubstitution is a mapping from the set of all natural numbersin the set of all hypersubstitutions of type τ . If we apply a colored hypersub-stitution of type τ to a term of type τ we apply of each one of the operationsymbols the hypersubstitution which is the image of its color. So, one can in agiven term apply on the same operation symbol different hypersubstitutions ifit occurs on different places.Similar to solid varieties of type τ we can define colored-solid varieties of typeτ . We will describe colored-solid varieties of semigroups.
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Classes of Automata Defined by Quasi-orders on a Free MonoidJelena Kovacevic∗
University of Nis, Faculty of Sciences and Mathematics, Cirila i Metodija 2,P. O. Box 224, 18000 Nis, Yugoslavia
Miroslav Ciric (University of Nis, Faculty of Sciences and Mathematics, NisYugoslavia), Tatjana Petkovic (University of Nis, Faculty of Sciences andMathematics, Nis Yugoslavia, and Turku Centre for Computer Science,
Turku, Finland), Stojan Bogdanovic (University of Nis, Faculty of Economics,Nis, Yugoslavia)
For a given quasi-order ξ on a free monoid X∗, an X-automaton A is calledξ-directable if there exists a word u ∈ X∗ such that av = bv for every wordv ∈ X∗ such that uξv and every pair of states a, b ∈ A. It is easy to see thatξ-directable automata are just a special type of directable automata, but for anappropriate choice of a quasi-order ξ as special cases of ξ-directable automatawe obtain various important classes of automata such as ordinary directableautomata, the well-known definite automata and the so-called piecewise di-rectable automata, introduced and studied in a recent paper by T. Petkovicand M. Steinby.We study some general properties of ξ-directable automata, as well as of cer-tain related kinds of automata: trap ξ-directable, ξ-trapped and generalizedξ-directable automata. Interesting connections between the mentioned classesof automata, quasi-orders on a free monoid and ideals of a free monoid areestablished.
On the number of k-valued functions with given range of theirsubfunctions
Dimiter Kovachev∗, Iliya GyudzhenovSouth-West University , Blagoevgrad, Bulgaria
Let M and R be sets of variables of a function where M is not a subset of R.
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In this paper we generalize some of the results published in [1]. The numberof n-variable k-valued functions is found which:
• Assume exactly b different values among k possible values and each sub-function of these functions with arguments from M assumes exactly adifferent values, where a is less than or equal to b.
• Assume exactly q different values of k possible values and the spectrumof M with respect to R for these functions is a singleton containing theelement q.
The way of counting can also be used for constructing the functions underconsideration.
[1] Dimiter. St. Kovachev, n the number of some k-valued functions of nvariables, Union of Bulgarian Mathematician, Mathematics and Education inMathematics, Proceedings of Thirtieth Spring Conference of the Union of Bul-garian Mathematicians, Borovets, April 8-11, 2001, pp. 176–181.[2] D. St. Kovachev, On the Number of Discrete Functions with a Given Range,General Algebra and Applications, Proceedings of the 59th Workshop on Gen-eral Algebra, Potsdam 2000, edited by K. Denecke and H.-J. Vogel, pp. 125–134.
On a Characteristic of Subsets of Abelian groupsBoris V. Novikov
University of Kharkov, Ukrain
Below G denotes a finite elementary Abelian 2-group, T is its subset containing1. The following notion can be useful for studying non-linear codes:Definition 1. We say that T has the defect ≤ n (def T ≤ n) if |T \ aT | ≤ nfor every a ∈ T . An element a ∈ T is called (n)-element if |T \ aT | = n.Evidently, def T ≤ |G \ T | for every T ⊂ G.Definition 2. T is standard if | < T > | = T + def T , where < T > is thesubgroup generated by T .Proposition 1. If def T = 1 then T is standard.
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Definition 3. Let def T = n. T is degenerate if for all distinct (n)-elementsa, b ∈ T their product ab does not belong to T .Proposition 2. If def T = 2 then T is either standard or degenerate.This assertion is wrong if def T = 3:Example. Let |G| = 16, G =< a, b, c, d >, T = {1, a, b, c, d, ab, cd}. Thendef T = 3 and T is neither standard nor degenerate.
Computer Code for Fuzzy Matrix OperationsKetty Peeva∗
TU-Sofia, Faculty of Applied Mathematics and InformaticsPeter Manoilov (TU-Sofia, Faculty of Computer Science), Boriana Deliiska
(LTU-Sofia, Faculty of Manegement)
A computer code is developed under Visual C++ for various fuzzy relationcompositions and fuzzy matrix operations. The interface is organized user-friendly with message and hint dialog boxes. This is our next step in creatinga fuzzy relation calculus package.
Artificial neural networks: introduction and some examplesN. Pencheva∗
Bulg. Acad. Sci., Institute of Physiology, SofiaP. Milanov (Bulg. Acad. Sci., Institute of Mathematics)
Many aspects of brain function could be modelled using a direct network ofneurones, which co-ordinate their firing. Any neurone fires if a weighted sumof the inputs to it from other neurones exceeds its threshold. The synapticweight between any two neurones indicates the contribution, which the firingof the first neurone makes to the total input of the second neurone. Artificial
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neural networks, composed of varying numbers of neurones (organized in lay-ers), with different sets of weights and different connection architectures, couldrepresent a wide range of logical functions and possess considerable potentialfor modelling some mental activities. There are two distinct phases of oper-ation of an artificial neural network: the learning or training stage and theoperation stage.To illustrate some of these ideas we analyze and comment neural networks takenfrom literature, which have proven their usefulness as tools for drug design.Various types of artificial neural networks have been used to turn a blind searchfor novel drug-like molecules into an informed search and to select well-definedsubsets of compounds from the accessible chemical space. The following tasksin drug design can be performed by artificial neural networks: (i) classificationof large data sets by self-organizing networks; (ii) further nonlinear modellingof quantitative structure-activity relationships my multi-layered feedforwordnetworks, self-organizing networks or hybrid architectures; and (iii) predictionof molecular properties.Following this general scheme we comment successful examples leading toactivity-enriched compound collections. The present material helps for bet-ter understanding of: (i) the important contribution of the artificial neuralnetworks in drug-design process; and (ii) the dialogue between the disciplinessuch as mathematics, biochemistry, cognitive sciences etc.
A Multimedia System to Test the Knowledge Acquired at theComputer Graphics Course
Anton A. Penzov∗, Galya HristovaSouth-West University, Blagoevgrad, Bulgaria
The course of computer graphics read at South-West University is of an in-terdisciplinary nature. The students study and inquire into the theory andpractice of creating both vector and raster pictures on computer. Because ofthat it is very important to use an effective and impartial way for evaluatingthe obtained knowledge. A multimedia system is realized and applied for test-ing knowledge of the students in the Computer Graphics course. The results
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achieved by the students are classified and analyzed using a multimedia testsystem. The multimedia system is a complete Windows application and couldalso be adjusted for the Internet.
Algebraic invariants of knots and their applications in Chemistry,Biochemistry anf Phisics
Irina Hristova PetrovaTU-Sofia, Sofia, Bulgaria
Different algebraic invariants of knots are considered as well as computer algo-rithms for their computation. Knots are used for synthesis of new molecules,forunderstanding DNA structure and statistical mechanics.
Information system for querying structured documents via WAPVanyo Peychev
FMI, University of Sofia, Sofia, Bulgaria
One of the major problems in the Internet World is to find out an appropriatedocument contents as result from searching procedure.WAP extends the Internet services for document contents management. Locat-ing the essential contents into documents is most important task in the WAPworld.In this article we present one decision for searching into document contents.
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Directions for the IT Education Improvement in the SecondarySchools
Apostolos Tatsis∗, University of Ioannina, GreeceKrassimira Panovska, Adam Zeinov (South-West University, Blagoevgrad)
Vanyo Peychev, (FMI, University of Sofia, Sofia, Bulgaria)
One of the skills, which contemporary people cannot leave without (the societynowadays requires), is the IT culture. The process of this culture buildingup is long and requires the educational process to take into consideration theIT trends as a whole. An important issue in the process of the students ITeducation at the Secondary schools is their acquaintance with these problemsand the education to be done in accordance with the modern IT technologies.Key technologies and their role in building up a better IT culture in the studentsare presented in this article.
Minimal Paths in SemigroupsZarko Popovic∗, Stojan Bogdanovic,
University of Nis, Faculty of Economics, Nis, YugoslaviaMiroslav Ciric (University of Nis, Faculty of Sciences and Mathematics, Nis
Yugoslavia)
To an arbitrary semigroup S we can associate the graphs (S,−→) and (S, )corresponding respectively to the relations −→ and on S defined as follows:
a −→ b ⇔ (∃n ∈ N) bn ∈ S1aS1
andab ⇔ a −→ b −→ a.
The graph (S,−→) one considers as a digraph, while (S, ) is an undirectedgraph.General properties of these graphs were studied by M. S. Putcha (1974). Thewell-known theorems of T. Tamura (1972) and M. S. Putcha (1974) assert that
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two elements a and b belong to the same component in the greatest semillaticedecomposition of S if and only if there exists a path from a to b in (S,−→), orequivalently, if there exists a path between a and b in (S, ). The structure ofsemigroups in the length of minimal paths in the graph (S,−→) are boundedwas described by S. Bogdanovic and M. Ciric (1996), and the structure ofsemigroups in which the length of minimal paths in the graph (S, ) are boundedwas described by S. Bogdanovic, M. Ciric and Z. Popovic (2000).If S is a completely π-regular semigroup, then the minimal paths from-to, orequivalently, between elements of S in the graphs (S,−→) and (S, ), whichcorrespond to S, will be described using the minimal paths from-to, or equiv-alently, between the idempotent elements of S.
Clones and related concepts in Universal Algebras andMany-valued Logics
I.G. RosenbergUniversity of Montreal, Canada
Several mathematical objects (permutation groups, monoids of selfmaps, clones,certain partial clones and clones of uniformly delayed operations) appearing inalgebra, combinatorics, many valued logics and switching theory can be de-scribed by relations. We explain the best understood case of clones on finiteuniverses and we mention the connection to the two well-known problems: themaximal subgroups of the symmetric group and the concrete representationproblem of finite lattices as congruence lattice of finite universal algebras. Weconclude with the extensions to infinite universes, delayed operations, partialoperations and the (open) case of described large clones of hyperoperations byrelations.
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On power ordered setsDietmar Scbweigert
University of Kaiserslautern, Germany
Power ordered sets are a generalization of Boolean algebras and are defined byinjective monotone maps.In spite of the complicated definition this concept has many applications inmulitcriteria optimization, decision theory, game theory and the theory of vot-ing. We like only to mention the application of the problems; spanning trees,shortest paths, traveling salesman tours and on optimal knapsacks. Further-more we can extend this concept for every relation and we have examples ofgraphs and groups. Therefore this theory stands betweeen the sets and rela-tions between power sets and power relations and therefore proposition logicand predicate logic.
On the Hausdorff Geometry of PolynomialsBl. Sendov
Bulgarian Academy of Science, Sofia
A set of n points A = {z1, z2, . . . , zn} on the complex plane define the monicpolynomial p(z) = (z − z1)(z − z2) · · · (z − zn) with derivative p′(z) = n(z −ζ1)(z − ζ2) · · · (z − ζn−1). The Geometry of polynomials studies the relationsbetween the sets A and A′ = {ζ1, ζ2, . . . , ζn−1}. The classical Gauss-LucasTheorem asserts that the convex hull of A contains A′.In 1958 we formulated the following conjecture in the Geometry of Polynomials.Conjecture. If all the zeros of the polynomial p(z) = (z − z1)(z − z2) · · · (z −zn) (n ≥ 2) lie in the unit disk D = D(0, 1) = {z : |z| ≤ 1}, then for every zk
the disk D(zk, 1) = {z : |z − zk| ≤ 1} contains at least one zero of p′(z).Up to now, Conjecture 1 was neither proved nor disproved, although more than80 related papers have been published on this topic. In attempts to prove thisconjecture, many new results have been obtained and many other problems
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and conjectures have been formulated. Our purpose in this lecture is to reviewsome of these results and to formulate some new problems from the point ofview of Hausdorff distance between the set of zeros and the set of critical pointsof a polynomial.
Hypersubstitutions and Colored TreesSlavcho Shtrakov
South-West University, Blagoevgrad, Bulgaria
Hypersubstitutions are homomorphisms between the sets of terms (trees) fullydetermined by their images of the set of operation symbols, considered as terms.Multi hypersubstitutions work over so called n−colored trees in an analogousway. They are introduced and studied in two Ms. Theses of my students in1999.The discussion in the present paper is when a such multi hypersubstitution isan one-to-one mapping on the set of terms of a given type. The concept ofexpansible and contracting hypersubstitution is considered, too.
Applications of Cutting Plane Methods for Solving VariationalInequalities
Stefan M. StefanovSouth-West University, Blagoevgrad, Bulgaria
In this paper, some generalized monotone and pseudomonotone mappings areconsidered and their properties are studied. Cutting plane methods, devel-oped originally for solving integer linear programming problems, are appliedfor solving variational inequality problems with generalized monotone (pseu-domonotone) mappings. A convergent analytic center cut method for monotonevariational inequalities is also presented.
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On the Uniform Distribution mod1 of SequencesStanislava Stoilova
South-West University, Blagoevgrad, Bulgaria
Some new results are proved for uniform distribution of sequences and measuresfor the irregularity of the distribution of the sequences in [0, 1).A measure is the so-called diaphony. The classical definition of the diaphony isbased on using the trigonometric function system and it is defined by Zinterhof.In 1997 Hellekalek and Leeb defined dyadic version of the diaphony, the so-called dyadic diaphony, based on the Walsh functional system.We will consider other two versions of the diaphony, based on the Chrestenson-Levy functional system and Price functional system. It is proved that the twoversions of the diaphony are measures for uniform distribution and the exactorder of the diaphony of one dimentional sequences is found; the so-calledsequences of Faure.A other results are shown in connection with criteria for uniform distribution.The so-called modified integrals of Price and modified integrals of Haar aredefined their connection with uniform distribution mod1 is shown. Using theclassic results, we obtain the analogues of LeVeque and Erdos-Turan inequali-ties in terms of this modified integrals.
On The Minimal Polynomials of the N-Generalized QuaternionsK. Todorov
South-West University, Blagoevgrad, Bulgaria
The quaternion algebra ha2 is a four-dimensional algebra (over the field of realnumber R), with the standard basis {1, i, j, k}. We will denote the elements1, i, j, k of ha2 by e0, e1, e2, e3.Multiplication is determined by the rules of the multiplication of the quaterniongroup lQ2. The norm N(q) = |q| of the quaternion
q = a0e0 + a1e1 + a2e2 + a3e3, a0, a1, a2, a3 ∈ R.
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is defined as N(q) = |q| = a20 + a2
1 + a22 + a2
3 = qq = qq. The minimalpolynomial, satisfied by the quaternion q is h(x) = x2 − 2a0x + N(q).The quaternion group lQ2 can be considered also as a semigroup with the fol-lowing genetic code: lQ2 = 〈a, b : a = bab and b = aba〉.Here we shall show a generalization lQn of the quaternion group lQ2 obtain instudying a semigroup, generated by the elements of the set Mn = {a1, a2, . . . ,an} subject only to the relations ai = ajaiaj for every two elements ai, aj ∈ Mwith j 6= i.The quaternion algebra han (of the quaternion group rtn) over the field ofreal number R is a 2n-dimensional algebra with the standard basis {ei, i =0, 1, . . . , 2n}. Multiplication is determined by the rules of the multiplicationof the quaternion group rtn.
Any quaternion q =7∑
i=0
cifi ∈ rt3 satisfies the minimal polinomial
h(x) = x2 − 2(cof0 + c7f7)x + N(q), where N(q) =3∑
i=0
cif0 +7∑
i=4
cif7.
Any quaternion q =15∑
i=0
cifi ∈ lQ4 satisfies the minimal polinomial
h(x) = x4 + B1x2 + B2x + B3, where Bi = Bi0f0 + Bi7f7, i = 1, 2, 3.
A Colour Quantization Method with Histogram PartitionMilen Todorow∗, Anton A. PenzovSouth-West University, Blagoevgrad
Several colour quantization methods are analysed and described. Each algo-rithm has different computational complexity. The error from quantization in-creases with decreasing the computational price for execution of the algorithm.The effectiveness of each algorithm depends on the different color distributionin the color image for quantization.A new approach for colour image quantization is proposed. The colour his-togram of the image is generated and structured in a binary tree. Subsequently
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the colour histogram is partitioned into groups and for each group a represen-tative colour is selected. Finally the original image is rewritten in accordancewith the generated representative colours. By attempting to preserve the char-acter of colour distribution in both original and rewritten image, the proposedmethod produces smaller error from quantization and runs faster than mostcolour quantization methods.
The genetic code optimalityIvan Trenchev∗
South-West University, Blagoevgrad, BulgariaPeter Milanov (Bulg. Acad. Sci., Institute of Mathematics)
The genetic code could be considered as a system of storage, transmission,execution and regulation of the information encoded in the genes. The notionof resistance of the genetic code against the effect of mutations, which areequivalent of the errors inherent to all information systems is analyzed.It is to be expected that contemporary genetic code is a structure ensures max-imum resistance to mutation effects. So it is worthy to analyze this problem.It is can be shown that different theoretical codes built from a fixed numberof triplets resist to the effect of mutation differently depending on the relativepositions of their codons in the 64 possible divisions.The previous works in this area analyzed the optimality of groups of tripletstranslated into the same amino acid. It hasnt been measured correctly theresistance of the whole genetic code to the creation of non-synonym mutations.In the present paper we define measures of resistance of the genetic code andcorresponding optimization principles.It is observed a good correspondence between the contemporary genetic codeand the theoretical ones.
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Object-Oriented Approach for Modeling of Discrete ProductionSystems
Georgi TuparovSouth-West University, Blagoevgrad, Bulgaria
In the article an approach for constructing an object-oriented model of a dis-crete production system in order to support concepts of virtual manufacturingis presented. The proposed approach is based on object-oriented paradigm ofObject Modeling Technique (OMT) and Fusion Method (FM). The discussedapproach supports three levels in modeling: static model, functional model anddynamic model. This way of modeling provides conditions for easily convertingto the formal modeling tools such as Petri Nets, Process Algebra etc.
The Logical Expressions and Functions in the Secondary SchoolCourses in Informatics and Information Technologies
Daniela Tuparova∗
South-West University, Blagoevgrad, BulgariaKaterina Marcheva (High Language School, Blagoevgrad, Bulgaria)
The logical operations, expressions and functions take a wide range in the sec-ondary school courses in informatics and information technologies (IT). Theyare used in modules like ”Algorithms”, ”Programming languages”, ”Word pro-cessing”, ”Spreadsheets”, ”Databases” and ”Internet”. Therefore the main fo-cus of the paper is an example for problem solving system, that could be usedin teaching logical operations, expressions and functions across above modules.
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Didactical Aspects in Problem Solving in the Secondary SchoolCourses in Informatics and Information Technologies
Daniela TuparovaSouth-West University, Blagoevgrad, Bulgaria
In the Secondary School Courses in Informatics and Information Technologiesthe problem solving is a power tool for involving and learning the main concepts.The main focus of the paper is the problem solving in different modules at thesecondary school courses in informatics and information technologies (IT) andits stages of decision. Also the main principles in constructing of problemsolving system are discussed. One example for problem solving system usefulin the module ”Spreadsheets” is proposed.
Representing of Dynamic Knowledge by Means of PredicateSillogistic LogicHristo Shoilev∗
Department Computer Systems, Technical University SofiaVasil Marinov, (Department of Mathematics and Informatics, Technical
University Sofia)
In this work is described the representing of dynamic knowledge by means ofpredicate sillogistic logic. The language for describing is predicate sillogistic.The thesises with which the elementary records were created are the axioms,and the rules for the building of new thesises are the rules for conclusion.Modus Ponens and Substitution of values.
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