Mathematics
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Albijanić, Miroljub (Beograd , 2016)[more][less]
Abstract: The main concept of this thesis is the connection of abstract theory and applied mathematical analysis in the universal mathematical system. The Mathematics is being applied practically in great number of cases, for example, in the application of mathematical statistics, numerical analysis, application in electrical engineering, contribution to computer developing etc. At the same time, in scientific perspective, it reached the level of incredible abstractions (topological spaces, vector spaces et al.). Due to these facts, it has become necessary to advance the teaching of mathematical analysis in technological universities, but also the very methods of teaching as well. This thesis contains both a theoretical and an empirical research. The theoretical research sheds light onto the notions of abstraction and application and gives examples from next teaching subjects: Lagrange’s Theorem, Convexity and Consequences. Taylor’s Formula. Hardy’s Approach to Calculating Surfaces of Flat Shapes. Fourier’s Series and their Application. Banach FixedPoint Theorem and its’ Application. The empirical research consists of two parts: a questionnaire and a test. This research shows how the relation between theory and application is seen by students, how they perceive the teaching of mathematics and which learning instruments they use. The research also shows how students solve simple problems and which type of problems they solve more sucessfully. The sample consists of 429 students of electrical engineering, construction and mechanical engineering for the Questionnaire and 450 students of the same universities for the Test. Students that participated in the research are studying at the University of Belgrade, University of Novi Sad and University of Niˇs. The results of the research confirmed that students from technological universities have a favourable attitude towards mathematics and that they see its significance in its application, i.e. in its use value. Students have clearly defined attitude on the idea that a good lecture by a professor is one which can be understood, which is well articulated, and which motivates students to take part in it. They point out the significance of examples that have elements of application. Visual presentations also enhance the success in solving problems. The research shows that students did not acquire the skill to apply their knowledge of mathematical analysis in solving tasks and problems. Theoretical clarification of notions of abstraction and application, followed by a displayfive topics of mathematical analysis confirm that abstract theory and applied mathematical analysis are interconnected and conjoint in the universal mathematical system. On the basis of results recommendations are defined concerning innovational approaches to teaching, such as planning the lectures and meliorating the contents, asking questions, and also an intelligent prospect, lecture improvement and learning instruments application. This way, it is confirmed that a methodically well organized lecture helps a better understanding of the relation between abstraction and application of mathematical analysis. URI: http://hdl.handle.net/123456789/4447 Files in this item: 1
doktorska_disertacija_MAlbijanic.pdf ( 4.920Mb ) 
Mihajlović, Bojana (Beograd , 2016)[more][less]
Abstract: The subject of this dissertation belongs to scientific field of spectral graph theory, a young branch of mathematical combinatorics, i.e. graph theory, which finds important applications in many areas, such as chemistry, physics, computer science, telecommunications, sociology, etc., and various fields of mathematics. Spectral graph theory connects basic properties and the structure of a graph with characteristics of the spectra of its matrices (adjacency matrix, Laplacian matrix, etc.). In this dissertation we only work with the adjacency matrix. The second largest eigenvalue of the adjacency matrix of a graph (or, simply, second largest eigenvalue of a graph), as well as its distance from the largest eigenvalue, are very important especially in applications of spectral graph theory in computer science. The property of a graph that one of its eigenvalues does not exceed some given value is a hereditary one; therefore, many of the investigations of this kind have been directed at finding the maximal allowed graphs, or minimal forbidden graphs for that property. In this dissertation we determine some classes of graphs whose second largest eigenvalue does not exceed some given value, and, for that purpose, we develop some very useful tools. In methodological sense, investigations in this dissertation represent a combined approach consisting of application of the algebraic apparatus and methods of spectral graph theory and combinatorial reasoning, whilst at some stages the expert system newGRAPH has been used. The dissertation consists of eight chapters, each of which is divided into subchapters. In the beginning, some important previous work is shown, and afterwards we present some original elements of the algebraic and combinatorial apparatus that speed up and simplify the further work. We define some mappings between certain families of graphs, some of which preserve the sign of the expression 2 2 , and, using them, we describe and systematize some (already known) results in a new way. Further on we completely determine all maximal reflexive tricyclic cacti which are not RSdecidable and whose cycles do not form a bundle, from the classes 1 R and 3 R , and we give some partial results about the class 2 R , using previously induced mappings (until now only the graphs from the remaining class 4 R have been completely determined [40], [46]). Next, we completely describe all minimal forbidden graphs in the class of bicyclic graphs with a bridge, and all minimal forbidden graphs in the class 3 R  the approach that so far has never been used with reflexive graphs. Then we determine the maximal number of the cycles for RSundecidable reflexive cacti whose cycles do form a bundle, and, therefore, generally for RSundecidable reflexive cacti and we describe three classes of maximal reflexive RSundecidable reflexive cacti that contain a bundle. Further on, some of the previous results are generalized: the generalized RStheorem is stated and proved (socalled GRStheorem) for any r , r 0 ; previously induced mappings are generalized, their properties are proved and various examples of classes of graphs with the property 2 r (for r 0 ) are given. Based on this, we describe all GRSundecidable maximal graphs for the property 2 2 in the class of unicyclic and multicyclic graphs, and also all RSundecidable maximal θgraphs for this property as well as all GRSundecidable maximal trees with the property 2 5 1 2 . Furthermore, we investigate the limit 3 (as in [28]) and we describe all trees with the diameter 3 and the diameter larger than 8, with the property 2 3 , as well as all GRSundecidable multicyclic cacti with the same property. Finally, we introduce and apply socalled σmodifications of Smith trees. We describe seven σmodifications and corresponding extensions, and we notice the appearance in (already known) results in the class of multicyclic reflexive cacti with 4 cycles. Applying some extensions to certain families of tricyclic cacti, we obtained the results in the class of multicyclic reflexive cacti with 4 cycles, using a different approach [48]. Finally, in the conclusion, we suggest some possible directions of further investigations. URI: http://hdl.handle.net/123456789/4445 Files in this item: 1
Mihailovic_Bojana.pdf ( 6.960Mb ) 
Milošević, Nela (Beograd , 2015)[more][less]
Abstract: This dissertation examines simplicial complexes associated with commutative rings with unity. In general, a combinatorial object can be attached to a ring in many di erent ways, and in this dissertation we examine several simplicial complexes attached to rings which give interesting results. Focus of this thesis is determining the homotopy type of geometric realization of these simplicial complexes, when it is possible. For a partially ordered set of nontrivial ideals in a commutative ring with identity, we investigate order complex and determine its homotopy type for the general case. Simplicial complex can also be associated to a ring indirectly, as an independence complex of some graph or hypergraph which is associated to that ring. For the comaximal graph of commutative ring with identity we de ne its independence complex and determine its homotopy type for general commutative rings with identity. This thesis also focuses on the study of zerodivisors, by investigating ideals which are zerodivisors and de ning zerodivisor ideal complex. The homotopy type of geometric realization of this simplicial complex is determined for rings that are nite and for rings that have in nitely many maximal ideals. In this part of the thesis, we use the discrete Morse theory for simplicial complexes. The theorems proven in this dissertation are then applied to certain classes of commutative rings, which gives us some interesting combinatorial results. URI: http://hdl.handle.net/123456789/4421 Files in this item: 1
Nela_Milosevic_Teza.pdf ( 20.62Mb ) 
Kordić, Stevan (Beograd , 2016)[more][less]
Abstract: Constrain satisfaction problems including the optimisation problems are among the most important problems of discrete mathematics with wide area of application in mathematics itself and in the applied mathematics. Dissertation study optimisation problem and presents an original method for finding its exact solution. The name of the method is Sedimentation Algorithm, which is introduced together with two heuristics. It belongs to the class of branchandbound algorithms, which uses backtracking and forward checking techniques. The Sedimentation Algorithm is proven to be totally correct. Ability of the Sedimentation Algorithm to solve different type of problems is demonstrated in dissertation by its application on the Boolean satisfiability problems, the Whitehead Minimisation Problem and the Berth Allocation Problem in container port. The best results are obtained for Berth Allocation Problem, because its modelling for Sedimentation Algorithm includes all available optimisation techniques of the method. The precise complexity estimation of the Sedimentation Algorithm for the Berth Allocation Problem is established. Experimental results verify that the Sedimentation Algorithm is capable to solve the Berth Allocation Problem on the state of art level. URI: http://hdl.handle.net/123456789/4413 Files in this item: 1
StevanKordic.pdf ( 2.477Mb ) 
Zeada, Samira (Beograd , 2015)[more][less]