http://hdl.handle.net/123456789/4 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. http://hdl.handle.net/123456789/5782 Metode za efikasno rešavanje dominacijskih problema na velikim grafovima Kapunac, Stefan This dissertation addresses methods for efficiently solving several important variants of domination problems on graphs, with a particular focus on large-scale instances that frequ- ently appear in real-world systems. Domination problems have numerous applications in the analysis and management of complex networks, including social, telecommunication, transport, and biological networks. The study covers four problems: minimum weight total domination, minimum weight independent domination, k-strong Roman domination, and the canonical mi- nimum domination problem on large graphs. For the minimum weight total domination problem, a variable neighborhood search approach is proposed, with carefully designed mechanisms for shaking, local search, and fitness function evaluation. The results show that the proposed algorithm achieves optimal solutions on small and medium instances and outperforms competing approaches on large graphs. Additionally, an application of this problem for accelerating information spreading in social networks is proposed. For the minimum weight independent domination problem, two new integer linear pro- gramming models are developed. Solving these models finds optimal solutions for all smaller instances while demonstrating superior performance compared to competing exact approaches on larger graphs. In addition, a greedy heuristic is proposed that outperforms competing greedy methods on most instances. In the case of k-strong Roman domination, a greedy heuristic based on node coverage information is developed, along with a metaheuristic approach based on variable neighborhood search that uses the greedy algorithm for initialization. This problem is particularly challenging due to the exponential complexity of solution feasibility verification, leading to the introduction of the concept of quasi-feasibility that enables efficient feasibility assessment during the search. Experimental results show that the proposed algorithm consistently outperforms the greedy approach and existing competing methods, especially on larger graphs. The practical value of the algorithm is illustrated through a case study involving the optimal positioning of fire stations and vehicles in urban municipalities to ensure the entire city is safe in the event of k simultaneous fires. For the minimum domination problem, a new hybrid approach called IRIS is proposed. IRIS is designed as a general-purpose framework that bridges the gap between exact integer linear programming solvers and heuristic search by iteratively fixing selected variables to reduce the search space. Тhe novelty lies in its flexible subproblem construction mechanism, which can be tailored using various selection strategies. In this study, we implement and evaluate a specific configuration of IRIS that utilizes historical statistical data and a node-coverage-based heuristic to intelligently identify variables for fixing. This targeted approach allows the ILP solver to find high-quality solutions for large-scale instances that are computationally prohibitive for exact methods. Experimental results demonstrate that IRIS achieves competitive performance com- pared to the best existing methods, establishing it as a valid alternative for solving domination and potentially other NP-hard problems. http://hdl.handle.net/123456789/5781 U- AND V-STATISTICS FOR INCOMPLETE DATA AND THEIR APPLICATION TO MODEL SPECIFICATION TESTING Aleksić, Danijel This dissertation addresses the problem of model specification testing in situa- tions where data are incomplete, utilizing the existing theory of non-degenerate and weakly degenerate U- and V-statistics. The first two chapters lay the theoretical groundwork by pre- senting essential concepts related to U- and V-statistics and the general mathematical frame- work of missing data analysis, which serve as the foundation for the new results developed in subsequent chapters. In Chapter 3, a novel test for assessing the missing completely at random (MCAR) assump- tion is introduced. This test demonstrates improved control of the type I error rate and supe- rior power performance compared to the main competitor across the majority of the simulated scenarios examined. Chapter 4 explores the application of Kendall’s test for independence in the presence of MCAR data. It provides both theoretical insights and simulation-based comparisons of the complete-case analysis and median imputation, pointing out their individual advantages and drawbacks. Chapter 5 focuses on testing for multivariate normality when data are incomplete. It rig- orously establishes the validity of the complete-case approach under MCAR and proposes a bootstrap method to approximate p -values when imputation is employed. Additionally, vari- ous imputation techniques are evaluated with respect to their impact on the type I error and the power of the test. Finally, Chapter 6 adapts the energy-based two-sample test to handle missing data by intro- ducing a weighted framework that makes full use of all available observations. Alongside some theoretical developments, the chapter presents two distinct bootstrap algorithms for p -value estimation under this approach. Additionally, the performance of several imputation methods is examined in this context, and appropriate bootstrap algorithm is proposed for that setting. http://hdl.handle.net/123456789/5780 ZAJEDNIČKI SPEKTRALNI RADIJUS ŠUR - ADAMAROVOG PROIZVODA SKUPA MATRICA I ŠUR - ADAMAROVI MNOŽIOCI SA PRIMENAMA NA DERIVACIONE NORMA NEJEDNAKOSTI ZA OPERATORE Bogdanović, Katarina In the rst and the second chapter of dissertation we prove some new inequalities for the spectral radius, essential spectral radius, oper- ator norm, measure of non-compactness and numerical radius of Hadamard (Schur) weighted geometric means of positive kernel operators on Banach function and sequence spaces. The list of extensions and re nings of known inequalities has been expanded. Some new inequalities and equalities for the generalized and the joint spectral radius and their essential versions of Hadamard (Schur) geometric means of bounded sets of positive kernel op- erators on Banach function spaces have been proved. There are additional results in case of non-negative matrices that de ne operators on Banach sequence spaces. In the third part we present some inequalities for opera- tor monotone functions and (co)hyponormal operators and give relations of Schur multipliers to derivation like inequalities for operators. In particular, let Ψ, Φ be s.n. functions, p ⩾ 2 and φ be an operator monotone function on [0, ∞) such that φ(0) = 0. If A, B, X ∈ B(H) and A and B are strictly ac- cretive such that AX−XB ∈ CΨ(H), then also AXφ(B)−φ(A)XB ∈ CΨ(H) and ||AXφ(B) − φ(A)XB||Ψ ⩽ r φ A+A∗ 2 − A+A∗ 2 φ′ A+A∗ 2 A+A∗ 2 −1 A(AX − XB)B B+B∗ 2 −1 r φ B+B∗ 2 − B+B∗ 2 φ′ B+B∗ 2 Ψ . under any of the following conditions: (a) Both A and B are normal, (b) A is cohyponormal, B is hyponormal and at least one of them is normal, and Ψ := Φ(p)∗ , (c) A is cohyponormal, B is hyponormal and ||.||Ψ is the trace norm ||.||1. Alternative inequalities for ||.||Ψ(p) norms are also obtained.