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Bosiljčić, Andrijana (Beograd , 2024)[more][less]
URI: http://hdl.handle.net/123456789/5752 Files in this item: 1
v1_masterAndrijanaBosiljcic.pdf ( 3.400Mb ) -
Jovanović, Milica (Beograd , 2024)[more][less]
Abstract: The analysis of Grassmann manifolds, which were first introduced in the 19th century, is one of the classical problems in the algebraic topology. When analyzing topological spaces, it is always useful to determine their cohomology algebra. The cohomology of Grassmann manifolds is already well known, but their covering spaces, so called oriented Grassmann manifolds, are far less examined. The oriented Grassmann manifold ˜Gn,k is defined to be the space of oriented k-dimensional subspaces of Rn. In this dissertation we analyze the cohomology algebra of oriented Grassmann manifolds ˜Gn,k with integer and modulo 2 coe!cients, predominantly the case k = 3. The dissertation comprises three chapters. The first chapter is an introduction where an overview of known results and necessary tools is given. In the second chapter we study the cohomology with the modulo 2 coe!cients. First of all, the known results in the case k = 2 are presented. Next, we move onto the case k = 3 where the partial description of the cohomology algebra is given. This section is based on papers published in the last several years. We give an overview of these results in the thesis, and we also present original results for n close to a power of two. In the last part of this chapter, we investigate the cohomology algebra of the manifold ˜G2t,4, and that is as far as we have come with the examination of modulo 2 cohomology. The third chapter is dedicated to the integral cohomology. This chapter, like the previous one, also splits in several sections, depending on the value of k. When k = 2, the integral cohomology is completely determined, and we present the proof for n odd. When k = 3, only the integral cohomology of ˜Gn,3, n → {6, 8, 10}, has been determined so far, while for k ↭ 4 only some partial results are known. In this segment we also analyze the connection between the integer and the modulo 2 cohomology algebra of these Grassmannians by analyzing the morphism between them induced by the modulo 2 reduction. URI: http://hdl.handle.net/123456789/5751 Files in this item: 1
Milica_Jovanovic_doktorat.pdf ( 1.723Mb ) -
Mrkela, Lazar (Beograd , 2024)[more][less]
Abstract: This dissertation examines two discrete location problems and their bi- objective variants. The first problem under consideration is the maximal covering location problem with user preferences and budget constraints imposed on facility opening. This variant of the maximal covering problem has not been previously studied in the literature. Unlike the classical maximal covering problem, the variant proposed in this dissertation includes user preferences for locations, where users are assigned to the location with opened facility that they prefer the most. Additionally, different locations have different costs for establishing facilities, and the available budget for opening facilities is limited. This problem is solved using the Variable Neighborhood Search (VNS) method, and the results were compared with the ones obtained by an exact solver on modified instances from the literature. Furthermore, an existing variant of the maximal covering problem is also addressed, which imposes the limit on the number of opened facilities instead of limiting the budget for opening facilities. The second problem examined is the regenerator placement in optical networks. In optical networks, signal quality degrades with distance, necessitating the place- ment of costly devices to restore the signal. This dissertation studies an existing model where the set of possible regenerator locations and the set of user nodes are different, defining the problem as generalized. The generalized regenerator place- ment problem in optical networks is also solved using the Variable Neighborhood Search method, with results compared to the best available solutions from the lit- erature. Bi-objective variants of these problems are defined as well. For the maximal covering location problem, user preferences are included as weighted factors in the total covered demand, forming the first objective function. The second objective function represents the number of uncovered users and aims to ensure fairness in the model. In the regenerator placement problem for optical networks, it is assumed that, due to budget constraints, uninterrupted communication between all pairs of user nodes may not be feasible. Each pair is assigned a weight, and the sum of the weights of connected pairs constitutes the first objective function, while the second objective function represents the cost of placing regenerators. These bi-objective variants are solved using an adapted multi-objective version of the Variable Neigh- borhood Search method, and the results are compared with general evolutionary algorithms. URI: http://hdl.handle.net/123456789/5750 Files in this item: 1
lazar_mrkela_doktorska_disertacija.pdf ( 17.56Mb ) -
Lukić, Katarina (Beograd , 2024)[more][less]
Abstract: n this dissertation, we start from the curvature tensor of the pseudo- Riemannian manifold or the algebraic curvature tensor on a vector space with a (possibly indefinite) scalar product. The duality, proportionality and orthogonal- ity principles of Osserman tensors are studied as they are properties of curvature tensors that are characteristic of Riemannian Osserman manifolds. The estab- lished principles are generalized to the pseudo-Riemannian case and are observed in two directions. On the one hand, we are interested whether these principles follow from Osserman’s conditions, and on the other, to what extent Osserman’s conditions are a consequence of established principles. Quasi-Clifford tensors are introduced as a generalization of Clifford tensors, and then some sufficient condi- tions are given under which the totally duality principle holds for quasi-Clifford tensors, and an example of a pseudo-Riemannian Osserman tensor is presented for which the duality principle does not hold. The theorem on the existence of the algebraic curvature tensor for the given Jacobi operators is proved, which is used to prove the results on the principle of proportionality. The principle of orthogonality is devised as a new potential characterization of Riemannian Osser- man tensors. Every Riemannian Jacobi-orthogonal tensor is an Osserman tensor, while Clifford and two-root Riemannian Oserman tensors are Jacobi-orthogonal. Generalizations of the orthogonality principle in the pseudo-Riemannian case are presented, especially in the cases of small dimensions 3 and 4. URI: http://hdl.handle.net/123456789/5749 Files in this item: 1
katarina_lukic_teza.pdf ( 2.186Mb ) -
Jovanović, Miljana (Beograd , 2024)[more][less]
Abstract: One of the main objectives of the Gaia mission of the European Space Agency is to construct a celestial reference frame at the wavelengths of the optical domain, Gaia CRF. This frame needs a link to the International Celestial Reference Frame – ICRF which is fixed with respect to distant objects (quasars). The objects serving for the purpose of linking are required to be visible in both domains (optical and radio). A set of 47 such objects has been proposed and included which in the radio domain have no detected extended emission. The mentioned objects are active galactic nuclei (AGN) the brightness of which varies over the whole electromagnetic spectrum. The brightness change may be due to activity in different AGN regions, but also to external factors. Such variations can lead to changes in the photocentre position and, consequently, to changes of the object coordinates. In order to establish which objects are suitable for linking these two frames we have examined the brightness variation in the optical domain. The objects have been observed from 2013 in the V and R bands. We have analysed the brightness, colour (V − R) and optical spectral index (α). It has been established that for the majority of objects the brightness is variable, or possibly variable. Almost 15% of all objects have significant changes in their brightness (more than 1 mag), only ∼10% are stable with minor brightness changes of ∼0.3 mag. The results concerning the change analysis of the colour and α are also presented. Based on these results 17 objects are chosen as suitable for linking ICRF to Gaia CRF. The results of the analysis, as well as the observed values, are essential for the examination of these objects because of their importance in astrometry, also in astrophysics. These data are relevant to a better understanding of formation and evolution of galaxies. URI: http://hdl.handle.net/123456789/5748 Files in this item: 1
MiljanaJovanovic-teza.pdf ( 6.418Mb )