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Todić, Bojana (Beograd , 2024)[more][less]
Abstract: This dissertation deals with the coupon collector problem, which in its simplest (classical) form can be formulated as follows: A collector wants to collect a set of n distinct coupons, by buying a single coupon each day. The random variable of interest is the waiting time until the collection is completed. The goal of the dissertation is to propose and analyze three new generalizations of the classical coupon collector problem obtained by introducing additional coupons with special purposes into the set of n standard coupons. The first two chapters are devoted to the results on the classical coupon collector problem and the known generalizations obtained by introducing additional coupons into the coupon set ([1], [2], [39], [54]). New results are presented in chapters 3,4, and 5. The third chapter of the dissertation is dedicated to the case where, in addition to the standard coupons, the coupon set consists of a null coupon (which can be drawn, but does not belong to any collection), and an additional universal coupon, that can replace any standard coupon. For the case of equal probabilities of standard coupons, the asymptotic behavior (as n → ∞) of the expectated value and variance of the waiting time for a fixed size subcollection of a collection of coupons is obtained when one or both probabilities of additional coupons are fixed, and the remaining coupons have equal small probabilities. These results, published in [27], generalize part of the results in [2]. The same problem is analyzed using a Markov chain approach, which led to the determination of the fundamental matrix and some related features of the collection process (probability that the coupon collection process ends in a certin way). These results are contained in the paper [24]. For the case of unequal probabilities of standard coupons, a class of bounds is derived for the first and second moments of the waiting time until the end of the experiment by using majorization techniques and refining the bounds proposed in [51]. The quality of the proposed bounds is tested in numerical experiments, and the specific bounds from the class with the most desirable properties are given. These results are published in [26]. The fourth chapter of the dissertation deals with the generalization in which the additional coupon (so called, penalty coupon) interferes with the collection of standard coupons in the sense that the collection process ends when the absolute difference between the number of collected standard coupons and the number of collected penalty coupons is equal to n. This generalization can be seen as a special case of the random walk with two absorbing barriers. The distribution and a simple upper bound on the first moment of the corresponding waiting time are determined by combinatorial considerations. The application of the Markov chain approach led to obtaining the fundamental matrix. These results are published in [53]. In the fifth chapter of the dissertation another additional coupon (so called reset coupon) is introduced, which acts as a reset button, in the sense that the set of coupons drawn up to time (day) t becomes empty if the reset coupon is drawn on day t+1. In the case of unequal probabilities of standard coupons, the distribution of the corresponding waiting time is obtained by combinatorial considerations. For the case of equal probabilities of obtaining standard coupons, For the case of equal probabilities, applying the first step analysis for the correspondingly constructed Markov chains led to the expressions for the expected waiting time and its simple form in terms of the beta function. These results are used for analysing the asymptotic behavior (when the size of the collection tends to infinity) of the expected waiting time, taking into account possible values of the probability of obtaining a reset coupon. These results are published in [25]. Setting the probabilities of the additional coupons to zero, all three generalizations of the coupon collector problem defined and analyzed in this dissertation as well as the obtained results reduce to the corresponding results for the classical coupon collector problem. URI: http://hdl.handle.net/123456789/5677 Files in this item: 1
Todic_Bojana_disertacija.pdf ( 1.004Mb ) -
Vicanović, Jelena (Beograd , 2024)[more][less]
Abstract: A convex continuous-time maximization problem is formulated and the nec- essary optimality conditions in the infinite-dimensional case are obtained. As a main tool for obtaining optimal conditions in this dissertation we use the new theorem of the alternative. Since there’s no a differentiability assumption, we perform a linearization of the problem using subdifferentials. It is proved that the multiplier with the objective function won’t be equal to zero. It was also shown that if the linear and non-linear constraints are separated, with additional assumptions it can be guaranteed that the multiplier with non-linear constraints will also be non-zero. In the following, an integral constraint is added to the original convex problem, so that a Lyapunov-type problem, i.e. an isoperimetric problem, is considered. Lin- earization of the problem using subdifferentials proved to be a practical way to ignore the lack of differentiability, so the optimality conditions were derived in a similar way. It is shown that the obtained results will also be valid for the vector case of the isoperimetric problem. Additionally, the optimality conditions for the smooth problem were considered. On the minimization problem, it was shown that the necessary conditions of Karush-Kuhn-Tucker type will be valid with the additional regularity constraint condition. Also, any point that satisfies the mentioned optimality conditions will be a global minimum. URI: http://hdl.handle.net/123456789/5676 Files in this item: 1
J.Vicanovic_doktorska_disertacija.pdf ( 2.198Mb ) -
Kostić, Petar (b , 2023)[more][less]
Abstract: The models of radio synchrotron emission of supernova remnants (SNRs) imply uniform density ahead of shock wave, so the evolution of luminosity is usu- ally studied in such an environment, most often through the surface-brightness-to- diameter dependence, the Σ–D relation. This field aims to better understand the SNR evolution, the emission models, but also the methods for determining their distance. It is not an easy task because of a very large scatter in the Σ–D Milky Way sample. The dissertation puts a different perspective at the Σ–D relation (usually treated as power-law function), assuming that non-uniform environment around the stars considerably affects its shape and slope, that may vary during the SNR expansion. It makes the ambient density structure an important factor whose impact must be investigated. The numerical code for hydrodynamic (HD) simulations and the emission model were developed. The 3D HD simulations were performed in different non-uniform environments, including low-density bubbles and a variety of clumpy models. Based on the simulation results, a semi-analytical 3D spherically-symmetric model of HD and Σ–D evolution of SNRs in clumpy medium was developed, which is used to generate large Σ–D samples. The results show that after entering the clumpy medium the SNR brightness enhances, but afterward the Σ–D slope steepens, shortening the brightness evolu- tion lifetime. Despite the evident increase in slope in clumpy medium, the Galactic sample average slope flattens at ≈ 13–50 pc. After analyzing the generated SNR samples in clumpy medium it is concluded that the significant flattening and scatter in Galactic sample originates in sporadic emission jumps of individual SNRs in a limited diameter interval. The additional analyses of selection effects are needed to investigate these issues. URI: http://hdl.handle.net/123456789/5606 Files in this item: 1
Kostic_Petar_disertacija.pdf ( 1.947Mb ) -
Protić, Danijela (Beograd , 2023)[more][less]
Abstract: Anomaly detection is the recognition of suspicious computer network behavior by comparing unknown network traffic to a statistical model of normal network behavior. Binary classifiers based on supervised machine learning are good candidates for normality detection. This thesis presents five standard binary classifiers: the k-nearest neighbors, weighted k-nearest neighbors, decision trees, support vector machines and feedforward neural network. The main problem with supervised learning is that it takes a lot of data to train high-precision classifiers. To reduce the training time with minimal degradation of the accuracy of the models, a two-phase pre-processing step is performed. In the first phase, numeric attributes are selected to reduce the dataset. The second phase is a novel normalization method based on hyperbolic the tangent function and the damping strategy of the Levenberg-Marquardt algorithm. The Kyoto 2006+ dataset, the only publicly available data set of real-world network traffic intended solely for anomaly detection research in computer networks, was used to demonstrate the positive impact of such pre-processing on classifier training time and accuracy. Of all the selected classifiers, the feedforward neural network has the highest processing speed, while the weighted k-nearest neighbor model proved to be the most accurate. The assumption is that when the classifiers work concurrently, they should detect either an anomaly or normal network traffic, which occasionally is not the case, resulting in different decision about the anomaly, i.e. a conflict arises. The conflicting decision detector performs a logical exclusive OR (XOR) operation on the outputs of the classifiers. If both classifiers simultaneously detected an anomaly or recognized traffic as normal, their decision was no conflict had occurred. Otherwise a conflict is detected. The number of conflicts detected provides an opportunity for additional detection of changes in computer network behavior. URI: http://hdl.handle.net/123456789/5599 Files in this item: 1
Danijela Protic - Doktorska Disertacija.pdf ( 3.143Mb ) -
Kovjanić, Milorad (Beograd , 2023)[more][less]
URI: http://hdl.handle.net/123456789/5598 Files in this item: 1
Milorad Kovjanić - Disertacija.pdf ( 2.190Mb )