http://hdl.handle.net/123456789/12 http://hdl.handle.net/123456789/5791 METAHEURISTIČKE METODE VIŠEKRITERIJUMSKE OPTIMIZACIJE I PRIMENE NA DISKRETNE LOKACIJSKE PROBLEME Mrkela, Lazar 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. Mon, 01 Jan 2024 00:00:00 GMT http://hdl.handle.net/123456789/5789 KOHOMOLOŠKA ALGEBRA GRASMANOVIH MNOGOSTRUKOSTI ORIJENTISANIH TRODIMENZIONALNIH RAVNI U EUKLIDSKOM PROSTORU Jovanović, Milica 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. Mon, 01 Jan 2024 00:00:00 GMT http://hdl.handle.net/123456789/5787 Hermitske strukture i geodezijske linije na četvorodimenzionalnim hiperboličkim prostorima Babić, Marijana The only non-compact four-dimensional rank-one symmetric spaces are the complex hyperbolic plane CH2 and the four-dimensional real hyperbolic space RH4. As connected homogeneous manifolds of negative sectional curvature, these spaces admit the structure of a four-dimensional real solvable Lie group equipped with a left-invariant metric. This Lie group appears naturally in the Poincar´e half-space model of real hyperbolic space and in the Siegel paraboloid model of the complex hyperbolic plane. The boundary of the paraboloid model carries the structure of the Heisenberg group. Hermitian structures consist of a left-invariant Riemannian metric together with a compatible complex structure. In this thesis, all such structures are classified and their geometric properties are studied. It is shown that every Riemannian metric on real hyperbolic space admits a two-dimensional sphere of Hermitian complex structures. In the case of the complex hyperbolic plane, some metrics admit exactly four distinct Hermitian complex structures, while others admit a two-dimensional sphere of such structures. Their curvature properties, holonomy groups, and self-duality are investigated. It is shown that the standard metric on the complex hyperbolic plane is the unique K¨ahler metric within the obtained classification, whereas all Riemannian metrics on real hyperbolic space are Einstein. Geodesics on the solvable Lie groups of the spaces CH2 and RH4, with respect to all possible left-invariant Riemannian metrics, are studied in this thesis using the Euler–Arnold equations. These equations effectively reduce a system of secondorder differential equations on a Lie group to a system of first-order equations on the corresponding Lie algebra. Numerical solutions of these equations enable the visualization of geodesics and geodesic spheres. Fri, 01 May 2026 00:00:00 GMT http://hdl.handle.net/123456789/5783 INTEGRALNE SREDINE KOMPOZICIONOG OPERATORA NA PROSTORIMA HOLOMORFNIH FUNKCIJA Dmitrović, Dušica The study of integral means of the composition of functions defined on the unit disk D in the complex plane dates back to the 1920s, with one of the earliest results in this area being Littlewood’s subordination principle. When investigating the norm of composition operators on certain spaces of holomorphic functions, a natural need arises to study the relationship between the integral means of the composition f ◦ φ and those of the function f itself. Littlewood’s principle is one of the main tools used to establish this connection. However, it is not the only one. In this dissertation, additional methods for studying the relationship between these integral means are presented. By applying these methods, two-sided estimates for the norm of the composition operator Cφ on spaces of mixed norm Hp,q,α are obtained in the form K1 ≤ ∥ Cφ ∥Hp,q,α→Hp,q,α ≤ K2, where the constants K1 and K2 depend on the parameters p, q, α and |φ(0)|. Furthermore, the monotonicity of the integral mean of a holomorphic function f on the unit disk D, denoted by Mp,q,α[f ](ρ, R, s) , is investigated, where 0 < p, q, α < ∞, 0 ≤ ρ < R ≤ 1 and 0 ≤ s ≤ 1. One consequence of this result is the monotonicity of the norm ∥f ∥p,q,α in mixed norm spaces with respect to the parameters p, q, α. One of the operators that can be represented as an integral of weighted composition operators Tt is the Hilbert matrix operator H acting on the weighted Bergman spaces Ap γ . Moreover, it is known that the operator H is bounded if and only if 1 < γ + 2 < p, and in this case, the following lower bound for the norm of the operator holds: ∥H∥Ap γ →Ap γ ≥ π/ sin (γ+2)π p . When γ > 0 and p ≥ 2(γ + 2), it is known that the norm is equal to this constant. In studying the norm of the operator H, after applying Minkowski’s theorem, the application of Minkowski’s inequality reduces the problem to estimating the norm of the operator Tt. As a result of this analysis, in the case where γ < 0 a new upper bound for the norm of the operator H is obtained, while in the case where γ > 0, the interval on which the norm equals the constant π/ sin (γ+2)π p is extended. Finally, the dissertation presents a refinement of Littlewood’s subordination principle under an additional injectivity assumption, together with applications of the new inequality to the Rogosinski theorem and to norm estimates for compositions of functions on weighted Bergman spaces. Thu, 19 Feb 2026 00:00:00 GMT