http://hdl.handle.net/123456789/12 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. Thu, 01 Jan 2026 00:00:00 GMT http://hdl.handle.net/123456789/5800 POLINOMIJALNA ENTROPIJA ZA MORSOVE GRADIJENTNE SISTEME I LOGISTIČKO PRESLIKAVANJE Perić, Milan This thesis presents a method for calculating the polynomial entropy of the topolog- ical dynamic system with finitely many non-wandering points. A special coding is adapted for such systems. Thanks to this coding the polynomial entropy can be bounded by the number of specific mutually singular points in the closures of stable manifolds of non-wandering points. This method was applied to Morse gradient systems. It is shown that the polynomial entropy of the Morse gradient system is bounded by n(F ) − 1, where n(F ) is the number of different Morse indices of critical points of the Morse function F. If Morse gradient systems on mani- folds of dimension n has only critical points of indices 0 and n, it is proved that the polynomial entropy is equal to 1, and if the system has critical points of indices 0, n/2 and n, it is proved that polynomial entropy is equal to 2. The polynomial entropy for different parameter values in logistic map has also been calculated, and it has been shown that the polynomial entropy distinguishes the systems of low complexity with drastically different behaviours, which cannot be distinguished by the topological entropy. The example of the homeomorphism of the con- nected compact metric space that is not equicontinuous and with vanishing polynomial entropy is also given. Fri, 01 Jan 2021 00:00:00 GMT http://hdl.handle.net/123456789/5799 UTAPANJA PROSTORA HARMONIJSKIH FUNKCIJA SA MEŠOVITOM NORMOM U OGRANIČENIM OBLASTIMA U Rn Jovanović Spasojević, Tanja In this thesis, subjects of consideration are the embeddings theorems of weighted Bergman spaces in Lp-spaces, as well as embeddings theorems of harmonic mixed norm spaces. The first part of the thesis generalizes the theorems of embeddings Bergman spaces into Lp(μ)-spaces, where μ is a Borel measure on a given domain. They have been earlier studied on domains such as unit ball and upper half-space. Generalization refers to bounded domains Ω ⊂ Rn with C1 boundary. This embedding will be valid to any p > 0, whenever the measure of the spaces Lp satisfies the Carledon condition. Reverse the direction will be valid only in case if p > 1 + α+2 n−2 . The second part of the dissertation also generalizes the embeddings theorems of mixed norm spaces of harmonic functions on a unit ball, where the generalization is applied to the domain Ω ⊂ Rn with C1 boundary. However, in addition we are obtaining another important result relating to the limitation of the maximum operators in the mixed norm on the general domain for the class of QNS functions. Sat, 01 Jan 2022 00:00:00 GMT http://hdl.handle.net/123456789/5798 EKSTREMALNI PROBLEMI BRAUNOVOG KRETANJA I DRUGIH SLUČAJNIH PROCESA Jovalekić, Milica Let M be a maximum and let N be a minimum of the non-negative martingale X1, X2, . . . , Xn. It is well known, that if X1 = 1, then γ(‖M ‖1) ≤ E (Xn log Xn) and γ(‖N ‖1) ≤ E (Xn log Xn) , where γ(x) = x − 1 − log x, for all x > 0. In this thesis, we prove the analogue of this result in the case when 1 < p < ∞, by proving that δp (‖M ‖p p ) ≤ ‖Xn‖p and δp (‖N ‖p p ) ≤ ‖Xn‖p, where δp(x) = ( 1 − 1 p ) x 1 p + 1 p x 1 p −1, for all x > 0. We also obtain a probabilistic proof of the fact min ρ∈D(Qn) ∫ Qn dx1 . . . dxn ρ (x1, . . . , xn)p−1 ∏n j=1 xαj +1 j = n∏ j=1 ( p p − αj − 1 )p , where p > 1, αj < p − 1 for j = 1, . . . , n and D (Qn) is family of all densities on the n-dimensional unit cube Qn = (0, 1)n in Rn. This provides the proof of the multidimensional weighted Hardy inequality. Namely, if f : Rn + → (0, ∞) is a measurable function, p > 1 and αj < p − 1 for j = 1, . . . , n, then ∫ Rn + n∏ j=1 xαj j Hnf (x)p dx ≤ n∏ j=1 ( p p − αj − 1 )p ∫ Rn + n∏ j=1 xαj j f (x)p dx, where Hnf (x) = 1 x1 . . . xn ∫ x1 0 · · · ∫ xn 0 f (t) dt, is a multidimensional Hardy operator, x = (x1, . . . , xn) ∈ Rn +, t = (t1, . . . , tn) and dt = dt1 . . . dtn. Let B(t) be a standard planar Brownian motion and r(θ) be the length of the projection of B[0, 1] on the line generated by the unit vector eθ = (cos θ, sin θ), where 0 ≤ θ ≤ π. We nd the common distribution function F of the random variables r(θ). Namely, we prove that F(x) = 8 ∞∑ n=1 ( 1 x2 + 1 (2n − 1)2π2 ) exp ( − (2n − 1)2π2 2x2 ) , for every x > 0. As immediate consequence, lower bound for the expected diameter of the set B[0, 1], better than known, is obtained. Namely, it is known that Ed ≥ 1.601, where d is the diameter of the set B[0, 1]. In this thesis we show Ed ≥ 1.856. Sat, 01 Jan 2022 00:00:00 GMT