Mathematics
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Stefanović, Seđan (Beograd , 2025)[more][less]
Abstract: The subject of the dissertation is the investigation of the relation of strong BJ orthogonality in C∗-algebras. For two elements a and b of C∗-algebra A, we say that a is strong BJ orthogonal to b, if for all c ∈ A holds ‖a + bc‖ ⩾ ‖a‖ and we write a ⊥S b. If it is also true that b ⊥S a, then we say that a and b are mutual strong BJ orthogonal and write a ⊥⊥S b. To this relation, we associate an undirected graph Γ(A) (which we call an orthograf), where the vertices are the nonzero elements of the C∗-algebra A, with the identification of an element and its scalar multiple; while there is an edge between two vertices a and b if a ⊥⊥S b. We will show that for any C∗-algebra A, different from three simple C∗-algebras, and for any two non-isolated vertices a and b in the orthograph, we can find vertices c1, c2, c3 ∈ Γ(A) such that a ⊥⊥S c1 ⊥⊥S c2 ⊥⊥S c3 ⊥⊥S b. We will also describe the isolated vertices of the graph Γ(A) for any C∗-algebra A. Finally, in the case of finite-dimensional C -algebras, we will determine the diameter of Γ(A), i.e., the minimum number of elements required to connect any two vertices. URI: http://hdl.handle.net/123456789/5754 Files in this item: 1
Stefanovic_Srdjan_doktorska_disertacija.pdf ( 990.4Kb ) -
Jovanović, Milica (Beograd , 2024)[more][less]
Abstract: 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. URI: http://hdl.handle.net/123456789/5751 Files in this item: 1
Milica_Jovanovic_doktorat.pdf ( 1.723Mb ) -
Lukić, Katarina (Beograd , 2024)[more][less]
Abstract: n this dissertation, we start from the curvature tensor of the pseudo- Riemannian manifold or the algebraic curvature tensor on a vector space with a (possibly indefinite) scalar product. The duality, proportionality and orthogonal- ity principles of Osserman tensors are studied as they are properties of curvature tensors that are characteristic of Riemannian Osserman manifolds. The estab- lished principles are generalized to the pseudo-Riemannian case and are observed in two directions. On the one hand, we are interested whether these principles follow from Osserman’s conditions, and on the other, to what extent Osserman’s conditions are a consequence of established principles. Quasi-Clifford tensors are introduced as a generalization of Clifford tensors, and then some sufficient condi- tions are given under which the totally duality principle holds for quasi-Clifford tensors, and an example of a pseudo-Riemannian Osserman tensor is presented for which the duality principle does not hold. The theorem on the existence of the algebraic curvature tensor for the given Jacobi operators is proved, which is used to prove the results on the principle of proportionality. The principle of orthogonality is devised as a new potential characterization of Riemannian Osser- man tensors. Every Riemannian Jacobi-orthogonal tensor is an Osserman tensor, while Clifford and two-root Riemannian Oserman tensors are Jacobi-orthogonal. Generalizations of the orthogonality principle in the pseudo-Riemannian case are presented, especially in the cases of small dimensions 3 and 4. URI: http://hdl.handle.net/123456789/5749 Files in this item: 1
katarina_lukic_teza.pdf ( 2.186Mb ) -
Lukić, Žikica (Beograd , 2024)[more][less]
Abstract: The main goal of this dissertation is twofold. In the first part, two novel two- sample tests for matrix data are presented. The theoretical properties of these novel tests are investigated in the context of testing orthogonal invariance in distribution, while the empirical values are presented in other cases. The tests are not distribution-free under H0. Therefore, their quality is investigated through a power study by implementing the warp-speed bootstrap algorithm. The novel tests are applied to multiple cases of real data, primarily originating in the field of finance. These tests are the first of their kind for two-sample tests of positive definite symmetric matrix distributions and are based on Laplace and Hankel transforms. The second part of this dissertation addresses problems related to data segmentation (or change point detection). Two novel classes of univariate tests for offline data segmentation are outlined, and their theoretical properties are studied. The powers are estimated using the permutation bootstrap algorithm, and the novel tests are shown to have higher test powers than the well-known tests based on the characteristic function. The location of the change point is estimated, and the novel tests are empirically demonstrated to possess greater precision. These tests are applied to two distinct datasets from meteorology and macroeconomics, further emphasizing their applicability in real-case scenarios. Moreover, the two-sample test based on the Hankel transform is modified to address change point problems. The asymptotic properties of this novel test are derived. A power study is presented, demonstrating the quality of the novel test in small-sample scenarios. The novel test is applied to financial data, emphasizing the practical applicability of this approach. This represents the first test for change point inference based on integral transforms for matrix data. URI: http://hdl.handle.net/123456789/5747 Files in this item: 1
lukic_Zikica_Disertacija_com2.pdf ( 1.909Mb ) -
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 )