Browsing Mathematical Sciences by Title
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Halaj, Katarina (Beograd , 2017)[more][less]
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Glišović, Nataša (Beograd , 2018)[more][less]
Abstract: In this doctoral dissertation the modelling process has been taken into consideration in the presence of uncertainty. Two types of problems were analyzed: one is the optimization of the benefit/costs tradeoff during the distribution of the projects and the other is the classification of data described by the attributes among which some are missing. The basic problems during the modelling of the decision making in the presence of uncertainty are the choice of the adequate treatment of uncertainty and the choice of the method for making a decision. One of the aims of the work is investigating the benefits of applying the metaheuristic algorithms on the considered optimization problems. The main measure for the evaluation of their performances is the value of objective function (for both problems: optimization of benefit/costs tradeoff during the project scheduling and clustering of incomplete data). Considering the project scheduling problem the level of satisfaction related to the problem constraints could also be taken into account. The other evaluation criteria of the applied metaheuristic methods is the time required for finding the solution. The influence of the parameters which control the algorithms of the metaheuristic methods is examined, as well as their appropriate values leading to the maximum performances of the implementation could be reached on the tested examples of the considered problems. As for the optimization problem of the profit/costs tradeoff, the uncertainty is modelled by applying the triangle fuzzy problems and then the metaheuristic methods, simulated annealing and genetic algorithm were applied for solving the obtained fuzzy optimization problem. The tested problems are formulated by the fuzzification method which was suggested by (Ribeiro et al. 1999). The represented experimental results for the set of fuzzy problems show the efficiency of the applied methods: simulated annealing and genetic algorithm. Genetic algorithm seems to produce slightly better solution than the simulated annealing. However, both methods out performed the existing form the literature for about 20%. The secund part of the work deals with the clustering data problem with the missing values of the attributes and making decisions in such circumstances. The main phases in solving the considered problem are finding the most appropriate distance, which will be used in the cases when the data are missing for some reasons and choosing the method for solving the clustering problem. As the theoretical and practical contribution, the metric, based on the logic principles, was proposed. By applying the probability, the theorem was proved defining the values of the weighting coefficients related to attributes that describe the objects for clustering. The proposed metric was implemented in the variable neighborhood search metaheuristic method as well as in some of its modifications. The implemented methods have been applied on the real life problems from the literature. Classifying the patients who suffer from some auto-immune diseases, stored in the database of Clinical Centre of Serbia, the precision of the clustering of 93.33% was achieved. As another real life example, seven databases of the European Commision (Board), which contain the data for the mail service, have been analyzed. The clustering efficiency of 90% - 96.96% was achieved. In order to compare the efficiency of the approach based on the variable neighborhood search method, nine databases available on the internet were used and the obtained results were compared with the existing ones from the literature. The experiments showed large stability of variable neighborhood search method: in eight out of nine cases the best solution was reached in all hundred repetitions. Besides that, the quality of the obtained solutons have considerably surpassed the results from the literature. URI: http://hdl.handle.net/123456789/4710 Files in this item: 1
Nglis_DoktorskaDisertacija.pdf ( 3.362Mb ) -
Trajkov, Perica (Beograd , 2016)[more][less]
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Zečević, Petar (Beograd , 2023)[more][less]
URI: http://hdl.handle.net/123456789/5630 Files in this item: 1
v1_masterPetarZecevic.pdf ( 918.2Kb ) -
Radin, Ivana (MATEMATIČKI FAKULTET UNIVERZITETA U BEOGRADU , 2010)[more][less]
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Milanković, Milutin (Beograd , 1913)[more][less]
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Hardy, G. H. (CAMBRIDGE UNIVERSITY PRESS , 1954)[more][less]
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Adamović, Dušica (Beograd , 2013)[more][less]
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Kurepa, Djuro (Zagreb , 1953)[more][less]
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Jovović, Ivana (Belgrade , 2013)[more][less]
Abstract: This dissertation deals with an application of some linear algebra techniques for solving problems of reduction of system of linear operator equation of the form A(x1) = b11x1 + b12x2 + : : : + b1nxn + '1 A(x2) = b21x1 + b22x2 + : : : + b2nxn + '2 ... A(xn) = bn1x1 + bn2x2 + : : : + bnnxn + 'n; where B = [bij ]n n is matrix over the eld K, A is linear operator on the vector space V over K and where '1; '2; : : : ; 'n are vectors in V . In particular, we consider reduction of such system under the action of the general linear group GL(n;K) and also reduction by using the characteristic polynomial B( ) of the matrix B and recurrence for the coe cients of the adjugate matrix of the characteristic matrix I B of the matrix B. The idea is to use rational and Jordan canonical forms to reduce the linear system of operator equations to an equivalent partially reduced system, i.e. to decompose the initial system into several uncoupled systems. This represents a new application of doubly companion matrix introduced by J.C. Butcher in [5]. In this work we are also concerned with transformation of the linear system of operator equation into totally reduced system, i. e. completely decoupled system of higher order linear operator equations. This results are related to results given by T. Downs in [13]. The thesis consists of two parts. The rst part deals with properties of rational and Jordan canonical form. We start with Fundamental Theorem of Finitely Generated Modules Over a Principle Ideal Domain. If we consider nite dimensional vector space V over K as module over the ring K[x] of polynomials in x with coe cients in K, the Fundamental Theorem implies that there is a basis for V so that the associated matrix for B is in rational or Jordan form. The rst section is adapted from Abstract Algebra of D. S. Dummit i R. M. Foote [14]. In the second section we look more closely at Hermite, Smith, rational and Jordan form and establish the relation between them. The structure of the similarity transformation matrix is also described. Some of theorem are considered from several aspects. This section provides a detailed exposition of normal forms using [14, 19, 37, 34, 35, 20, 57] and [1, 22, 48, 52, 54, 60]. The second part concerns with author's original contribution and it relies on papers [42, 43]. First we illustrate methods of the partial and the total reduction of systems v in two or three unknowns and then we study reductions of systems in n unknowns. The partial reduction requests changing of basis so that the system matrix is in the rational or Jordan form. We also treat the total reduction of the obtained partially reduced systems in this manner. Subsection 4.4. "Total Reduction for Linear Systems of Operator Equations with System Matrix in Companion Form" is one interesting way to proceed consideration started in previously mentioned works. It is based on papers of L. Brand [3, 4]. The fth section is generalization of the forth. Here we examine systems in n unknowns and with di erent linear operators. We introduce the notion of characteristic polynomial in more than one unknown - generalized characteristic polynomial and a method for total reduction by nding adjugate matrix of the generalized characteristic matrix of the system matrix. The sixth section is a summary of applications and examples of methods for partial and total reduction. There are some examples of the rst and higher order linear systems of di erential equations and di erent approaches for calculating rational and Jordan canonical forms. The last section is devoted to the study of di erential transcendence of the solution of the rst order linear system of di erential equations with complex coe cients, where exactly one of the following meromorphic functions '1; '2; : : : ; 'n is di erentially transcendental, using method of total reduction. We review some of the standard facts on di erential transcendence following books [45, 44, 39, 50, 7, 33, 15, 23, 36]. URI: http://hdl.handle.net/123456789/2585 Files in this item: 1
Jovović_Ivana.pdf ( 2.063Mb ) -
Ćirić, Ninoslav (Belgrade)[more][less]
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Paripović, Slađana (Beograd , 2012)[more][less]
URI: http://hdl.handle.net/123456789/5157 Files in this item: 1
MASTER_RAD_SLADJANA_PARIPOVIC.pdf ( 1.447Mb ) -
Simovljević, Jovan (GODIŠNJAK NAŠEG NEBA ZA 1957, Naučno delo , 1956)[more][less]
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Manojlović, V. Jelena (Belgrade , 1999)[more][less]
URI: http://hdl.handle.net/123456789/2491 Files in this item: 1
Manojlovic_Jelena_DR teza.pdf ( 829.5Kb ) -
Mišković, Vojislav (Akademija Prirodnih Nauka , 1930)[more][less]
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Maksimović, Zoran (Beograd , 2001)[more][less]
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Milanković, Milutin (Belgrade , 1947)[more][less]
URI: http://hdl.handle.net/123456789/524 Files in this item: 1
MilutinMilankovicOsnivaciPrirodnihNauka.pdf ( 4.583Mb ) -
Hilbert, David (Srpska akademija nauka , 1957)[more][less]
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Hilbert, David (Belgrade , 1957)[more][less]
URI: http://hdl.handle.net/123456789/435 Files in this item: 1
DavidHilbertOsnoveGeometrije.pdf ( 1.055Mb ) -
Lukačević, Ilija (Naučna Knjiga Beograd , 1980)[more][less]
URI: http://hdl.handle.net/123456789/5446 Files in this item: 1
Ilija_Lukacevic_Teorija_relativnosti_OCR.pdf ( 9.697Mb )