http://hdl.handle.net/123456789/4 http://hdl.handle.net/123456789/5828 POLINOMIJALNA ENTROPIJA INDUKAOVANIH DINAMIČKIH SISTEMA NA HIPERPROSTORIMA Đorić, Maša In this thesis, we applied a method for computing the polynomial entropy of dynamical systems with a finite non-wandering set to certain hyperspaces of the interval and the circle. We subsequently generalized these results to the case where the non-wandering set is not finite. We proved that the polynomial entropy of the induced map C(f ) is equal to 2, that of Fn(f ) is equal to n, and that of 2f is infinite, when f is a homeomorphism of the interval or the circle. Furthermore, we generalized these results for the n-fold symmetric product of the interval and the circle to an arbitrary compact metric space X and extended them to the suspension of the space Fn(X). In addition to the results on hyperspaces, we presented results concerning polynomial entropy on certain one-dimensional continua, such as local dendrites and regular curves. We also constructed a family of pointwise periodic homeomorphisms on a continuum whose polynomial entropy attains every value in [0, +∞]. http://hdl.handle.net/123456789/5816 Novel goodness-of-fit tests based on independence-type characterizations Halaj Mileusnić, Katarina The main objective of this dissertation is to develop novel goodness-of- fit tests based on independence characterizations and to investigate their proper- ties. The starting point is the fact that certain distributions can be uniquely char- acterized by the independence of suitable functions of random variables. This observation makes it possible to reduce goodness-of-fit testing for a hypothe- sized model to the problem of detecting departures from the corresponding in- dependence condition. Accordingly, the proposed test statistics are constructed by comparing empirical versions of joint functionals with the product of the cor- responding marginal functionals, where the resulting discrepancy serves as the basis for new goodness-of-fit tests for different classes of distributions. One of the contributions of the dissertation is the development of goodness- of-fit tests for the geometric distribution based on Ferguson’s independence char- acterization. In this setting, the general framework of comparing joint and marginal empirical functionals is implemented through V-empirical probability generating functions. Furthermore, a novel approach to goodness-of-fit testing for absolutely continuous distributions is proposed, relying on the discrepancy between the joint U-empirical distribution function of an appropriately defined random vector and the product of its marginal U-empirical distribution func- tions. Another part of the research focuses on circular data. A goodness-of-fit test for the wrapped normal distribution is proposed, where the general methodology is adapted to the periodic structure of observations on the circle. The theoretical part of the dissertation is devoted to the derivation of the asymptotic properties of the proposed test statistics, drawing on the theory of U- and V- statistics as well as on the theory of U-empirical processes. An addi- tional contribution is the establishment of new asymptotic results for finite linear combinations of weakly degenerate V-statistics, both in settings with known pa- rameters and in settings involving parameter estimation. These findings provide the theoretical foundation for a class of combined tests in which appropriately chosen weights regulate the contribution of individual components and can be used to enhance the overall efficiency of the testing procedure. The practical performance of the proposed methods will be assessed through extensive simulation studies and the analysis of several real data sets. The sim- ulation study will investigate the finite-sample properties of the tests under the null hypothesis and a range of relevant alternatives, while the real-data applica- tions will demonstrate their usefulness in practical statistical problems. In this way, the empirical part of the dissertation will complement the theoretical de- velopments and provide evidence of the practical applicability of the proposed methodologies. http://hdl.handle.net/123456789/5804 DINAMIKA AUTONOMNIH KOMPETITIVNIH LOTKA-VOLTERINIH DINAMIČKIH SISTEMA Branković, Danijela In this dissertation is performed an analysis of the stability of equilibriums of the two-dimensional autonomous competitive Lotka–Volterra dynamical system. Necessary and suf- ficient conditions are determined for equilibriums (without the origin) to be asymptotically sta- ble or unstable in [0, +∞)2, as well as necessary and sufficient conditions so that the observed dynamical system has no equilibriums in (0, +∞)2. The appropriate ecological interpretati- on is also given. It is also obtained that four transcritical bifurcations occur in the observed dynamical system if it is analyzed on R2. It is also investigated an analysis of the stability of a three-dimensional autonomous com- petitive Lotka–Volterra dynamical system. It is also performed an analysis of the stability of equilibriums lying on the axes or in the interior of the planes of 𝑛-dimensional autonomous competitive Lotka–Volterra dynamical system. The appropriate ecological interpretation is al- so given. Conditions for equilibriums lying on the axes or in the interior of the planes to be asymptotically stable or unstable in [0, +∞)𝑛 are established. The system of the Friedmann equations, that describes dynamics of the ΛCDM model of the universe, firstly represented and analyzed as a three-dimensional, and afterwards as an 𝑛-dimensional autonomous dynamical system of a class of Lotka–Volterra dynamical systems, with density parameters of the universe’s constituents as dependent variables, is investigated. The appropriate physical interpretation, including observing the evolution of the universe in the frame of the linear stability theory, is presented. Analytical solutions of these systems represent new parametrizations of density parameters of the universe’s constituents regarding to the scale expansion factor of the universe. http://hdl.handle.net/123456789/5803 MATEMATIČKI MODELI I METODE OPTIMIZACIJE ZA REŠAVANJE VARIJANTI PROBLEMA r-REZERVNOG CENTRA Tasić, Jelena This dissertation examines the p-next center problem (PNCP) and three of its variants that have not previously been addressed in the literature. All the problems consi- dered involve determining locations for establishing service centers, focusing on the user in the most unfavorable position. In practice, this may be the user who is farthest from their assigned health clinic, or the customer farthest from the local market. The objective is to ensure that these users travel the shortest possible distance to a service center. Since these are NP-hard problems, standard solvers such as CPLEX are unable to provide optimal, or often even feasible, solutions for larger instances. The p-next center problem reflects the realistic possibility that one or more centers may suddenly fail. In such cases, users assigned to a closed center are redirected to the (nearest) backup center, and the goal is to determine the locations for centers so as to minimize the maximum of all total distances traveled by users. In this dissertation, a skewed variable neighborhood search method (SVNS) is proposed for solving the p-next center problem, which incorporates a fast interchange heuristic within the local search phase. The method is tested on the well-known OR-LIB set of instances containing up to 900 nodes, and the results are compared with the best results from the literature. As an extension of the previous problem, the concept of facilitated communication be- tween centers is considered, and the p-next center problem with a discount factor (ωPNCP) is defined to incorporate this idea. A mathematical formulation of the problem is provided and solved using the CPLEX solver. A basic variable neighborhood search (BVNS) method is proposed for solving the problem, and the potential benefits achievable through enhanced communication between centers are analyzed on a set of instances from the literature that include up to 900 nodes. To further adapt the p-next center problem to practical needs, the conditional p-next center problem (CPNCP) is defined. This problem is applicable to the expansion of existing business networks while retaining existing centers where there is a possibility of sudden center failures. A mathematical model is proposed, instances with up to 900 nodes are generated, and the problem instances are solved using the CPLEX solver. A variable neighborhood search method is proposed for solving this problem. Two approaches to business network expansion are analyzed, along with potential long-term savings that can be achieved by their application. The maximal covering p-next center problem (MCPNCP) with binary and partial cove- rage is defined. The objective is to maximize the total demand of users that are covered, that is, users whose distance to their backup center does not exceed a given radius. Two mathematical models are proposed. Instance with up to 400 nodes are generated and the proposed models are compared using the CPLEX solver. A skewed variable neighborhood search method is proposed for solving the problem, and the results are compared with those obtained by the CPLEX solver.