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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 ) 
Lalović, Ana (Beograd , 2016)[more][less]
Abstract: The goal of this thesis is to reduce multidimensional space of galactic properties to the smallest number of dimensions su cient to describe them. For this purpose, the statistical analysis is applied over the parameters that describe fundamental galactic properties on the morphologically representative sample of 2180 galaxies. The sample of galaxies used in this thesis is based on the Arecibo Legacy Fast ALFA (Alfalfa) blind HI survey. The importance of an HI blind survey lies in the fact that galaxies are chosen on the basis of their gas content (HI) solely, thus free of optical selection e ects. From the initial sample counting 10000 galaxies, 2180 of them were chosen, since for this subsample the optical spectroscopy from the Sloan Digital Sky Survey (SDSS) was available and moreover the photometry in the UV (Galaxy Evolution Explorer, GALEX), and optical (SDSS) to the nearinfrared (Two Micron All Sky Survey, 2MASS). Parameters are selected according to the previously established correlations between fundamental galactic properties, relying on the previous work. They are extensively tested and confronted between each other to be chosen from the larger parametric space. To select parameters, we rst measured stellar kinematics using publicly available code (pPXF), and tested both empirical and synthetic stellar libraries. In particular, we have measured the velocity dispersion and the higher moments of the lineofsight velocity distribution function. This is the largest galaxy sample created so far with detailed stellar kinematics measured including higher moments of the lineofsight velocity distribution function. The sample size allows statistical tests to be applied to the higher moment of the velocity distribution function (h4), with respect to the di erent groups of morphological galaxy types. Various tests agree with the previous indication that elliptical and lenticular galaxies have the same origin. Further, we have measured the line strength indices for several absorption lines (Lick indices), since some of them are good proxies to galaxy ages and metallicity, also the fundamental galactic properties. In the nal statistical analysis, metallicity proves to be of no importance, but the inclusion of galaxy ages in the analysis, the results change signi cantly. The last step in the parameter selection is the modelling of the galaxies' surface brightness pro les with the Sersic pro le, that is performed in this thesis with the Gal t code. The velocity dispersion measured, along with the Sersic index and effective radius of the Sersic pro le takes the role in the dynamical mass calculation, being the fundamental galactic property and hence used in the nal statistical analysis. Finally, we have taken the mass of the gas component and maximal rotational velocity from the radiospectroscopy and Kron magnitudes (i.e. colours) from the ultraviolet/ optical/nearinfrared photometry (GALEX/SDSS/2MASS databases). After extensive testing, we have chosen the colour calculated from ultraviolet and optical magnitudes (NUV r colour), for the nal statistical analysis. It is worth noting that previous analysis of the galactic properties lack velocity dispersion, as well as the colour with the ultraviolet component, although it is a direct proxy to the speci c star formation rate in the galaxy. This particular colour makes correlations among analysed parameters stronger and proves to be more important than optical colours. Finally, when the proper parametric space of galactic properties is formed (velocity dispersion, colour, luminosity, Petrosian radii R50 and R90, dynamical, HI and stellar masses, maximal rotational velocity and the galaxy ages), the correlation analysis is performed to inspect correlations between parameters. This analysis con rms relations that are already known to hold. Then the principal component analysis is done with the purpose of nding and identifying the smallest number of galactic properties responsible for the nal products of galaxy evolution, as we see today, in the local Universe. The results of the corresponding analysis are the following: there are at least three statistically important, independent components. The rst and the most important component cannot be identi ed with either galactic property, but presents the mixture of several properties: dynamical mass, mass of the stellar and gas component, luminosity and Petrosian radii R50 and R90. Relaying on the previous work, this component may be identi ed with the "size" of the galaxies. The second component, mostly in uenced by the galactic colour, may be identi ed with the "aspect" of the galaxies. The colour was not found to be important in previous work. The galaxy ages can be identi ed with the third principal component. There is a hint on the fourth component, dominated by the maximal rotational velocity that can be identi ed with the speci c angular momentum of galaxies. Although not proven to be statistically important, it may become so in the larger sample of galaxies which will provide the information of the true peak of the galaxies' rotational curves, since the singlebeam HI spectra may show the single maximum and this may not be the true maximum. Also, the rotational velocity includes the inclination correction, another questionable parameter in the analysis. To conclude: there are at least three, and possibly four dimensions of the multidimensional galactic space, as we see today. URI: http://hdl.handle.net/123456789/4446 Files in this item: 1
Lalovic_Ana3.pdf ( 11.44Mb ) 
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 ) 
Landau, L. D.; Lifšic, E. M. (Beograd , 1952)[more][less]

Leko, Marko; Plavšić, Milan (Beograd , 1973)[more][less]