Mathematics
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Babanić, Mirko (Beograd , 2022)[more][less]
Abstract: The preparatory part of the dissertation, which leads to the basic one, is based on returnvariance parameters that represent two key random variables of the model devised by Markowitz. The research used historical data that in themselves reflect all available information absorbed by the financial market, and therefore, we can consider them not only homogeneous but also absolute (for reasons of realization). Therefore, an analytical procedure of approximation by a sixthdegree polynomial was performed on such data, which represent combinations of values of average returns and variances of portfolio returns, thus establishing a relation that is explicitly expressed by an algebraic sixthdegree polynomial equation. After that, further analytical procedure determined the conditions for the existence of both the minimum and the tangent portfolio and redefined the terms: efficient portfolio set, preference toward risk, risk aversion, and indifference line. The central topic of the dissertation, the revision of Tobin 's separation theorem, is formulated and proved through three theorems, one basic and two auxiliary. URI: http://hdl.handle.net/123456789/5593 Files in this item: 1
Mirko Babanic  Disertacija.pdf ( 1.504Mb ) 
Mutavdžić, Nikola (Beograd , 2023)[more][less]
Abstract: In this PhD thesis we investigate bounds of the gradient of harmonic and harmonic quasiconformal mappings. We also discuss such bounds for functions that are harmonic with respect to the hyperbolic metric or certain other metrics. This research has been motivated by some recent results about Lipschitzcontinuity of quasiconformal mappings that satisfy the Laplace gradient inequality. More precisely, the mappings we consider are solutions of the Dirichlet problem for the Poisson equation and can be considered as a generalization of harmonic mappings. Besides the ball, we also work with general domains on which solutions of the Dirichlet problem are defined, as well as general codomains. Finally, we announce new results that have been formulated for regions of C1,αsmoothness, both as the domain and the codomain. Besides presenting the main results, we give an overview of general notions from differential geometry and recall some of the properties of hyperbolic metric in an ndimensional ball. We also state properties of harmonic and subharmonic functions with respect to the hyperbolic metric, which are analogous to some classical results from the theory if harmonic functions and Hardy’s theory. It turns out that the gradients of hyperbolic harmonic functions behave differently from those of euclidean harmonic functions. A similar conclusion is obtained for the family of Tαharmonic functions. Namely, unlike the space of harmonic functions, the solution of the Dirichlet problem in the space of Tαharmonic functions is shown to be Lipschitzcontinuous when so is the boundary function. In addition, we investigate Höldercontinuity of the solution of the Dirichlet problem for the Poisson equation in the euclidean and hyperbolic metric. We will present versions of the Schwarz lemma on the boundary for pluriharmonic mappings in Hilbert and Banach spaces. These results will follow from the version of the Schwarz lemma for harmonic mappings from the unit disc to the interval (1, 1) without the assumption that the point z = 0 maps to itself. Furthermore, we show a version of the boundary Schwarz lemma for harmonic mappings from a ball to a ball, not necessarily of the same dimension. The proof uses a version of the Schwarz lemma for multivariable functions, first considered by Burget. This result is obtained by integrating the Poisson kernel over socalled polar caps. The assumption that point z = 0 maps to itself is again not needed, thus yielding a generalization of a recent result by D. Kalaj. At the end of this section, it is demonstrated that the analogous result is false in the case of hyperbolic harmonic functions. In a certain sense, this means that the Hopf lemma is not valid for hyperbolic harmonic functions. Amongst various versions of the Schwarz lemma, we have been investigating bounds of the modulus for classes of holomorphic functions f on the unit disc whose index If fulfils certain geometric conditions. These classes are a generalization of the star and αstar functions, previously investigated by B. N. Örnek. Our method is based on using Jack’s lemma and can be applied in certain more general cases. As an illustration, we derive the sharp bounds for the modulus of a holomorphic function f with index If whose codomain is a vertical strip, as well as bounds for the modulus of the derivative of f at point z = 0. Moreover, we give a bound for the rate of growth of the modulus of holomorphic functions on disk U that map point z = 0 to itself and whose codomain is a vertical strip. URI: http://hdl.handle.net/123456789/5582 Files in this item: 1
Doktorska_Disertacija_Nikola_Mutavdzic.pdf ( 939.2Kb ) 
Merkle, Ana (Beograd , 2023)[more][less]
Abstract: Many new developments in the filed of probability and statistics focus on finding causal connections between observed processes. This leads to considering dependence relations and investigating how the past influence the present and the future. The well known concept of Granger (1969) causality is closely related to the idea of local dependence introduced by Schweder (1970). Granger studied time series, while Schweder considered Markov chains. The concept was later extended to more general stochastic processes by Mykland (1986). All this concepts incorporate the time into consideration dependence. The dissertation consist of four chapters. New results are presented in the fourth chap ter. The main aim of this doctoral dissertation is to determine di↵erent concepts of stochastic predictability using the well known tool of conditional independence. Follow Granger’s idea, relationships between family of sigma  algebras (filtrations) and between processes in continuous ti me were considered since continuous time models dependence represent the first step in various applications, such as in finance, econometric practice, neuroscience, epidemiology, climatology, demographic, etc. In this dissertation the concept of dependence between stochastic processes and filtration is introduced. This concept is named causal predictability since it is focused on prediction. Some major characteristics of the given concept are shown and connections with known concept of dependence are explained. Finally, the concept of causal predictability is applied to the processes of di↵usion type, more precisely, to the uniqueness of weak solutions of Ito stochastic di↵erential equations and stochastic di↵erential equations with driving semi martingales. Also, the representation theorem in terms of causal predictability is established and numerous examples of applications of the given concept are presented such as application in financial mathematics in the view of modeling default risk, in Bayesian statistics. The idea for the future might be to deal with the case of progressive stochastic predictability, i.e. the generalization of stochastic predictability from fixed time to stopping time. URI: http://hdl.handle.net/123456789/5572 Files in this item: 1
DOKTORAT_finalnaVerzija.pdf ( 1.785Mb ) 
Vrećica, Ilija (Beograd , 2022)[more][less]
Abstract: First part of dissertation examines sumsets hA = {a1 + · · · + ah ∈ Z d : a1, . . . , ah ∈ A}, where A is a finite set in Z d . It is known that there exists a constant h0 ∈ N and a polynomial pA(X) such that pA(h) = hA for h ⩾ h0. However, little is known of polynomial pA and constant h0. Cone CA over the set A contains information about hA, for all h ∈ N. When A has d + 2 elements, polynomial pA and constant h0 can be explicitly described. When A has d + 3 elements, an upper bound is found for the number of elements of hA. Second part of dissertation examines Selmer groups of elliptic curves in the con gruent number family. A squarefree natural number is congruent if and only if there exists a right triangle with area n whose sides all have integer lengths. It is known that n is a congruent number if and only if elliptic curve En : y 2 = x 3 − n 2x has nonzero rank as an algebraic group. Selmer groups of isogenies on En are interesting, because their rank is not smaller than the rank of En, so when the Selmer groups have rank zero, then the elliptic curve En also has rank zero. Elements of these Selmer groups can be represented as partitions of a particular graph, from which one may find the distribution of ranks of Selmer groups. URI: http://hdl.handle.net/123456789/5533 Files in this item: 1
Teza_Ilija_Vrecica.pdf ( 1.817Mb ) 
Lelas, Nikola (Beograd , 2022)[more][less]
Abstract: The subject of the dissertation is the problem of simultaneous nonvanishing of quadratic twists of elliptic curve 𝐿functions and quadratic Dirichlet 𝐿functions (and their derivatives) at the centar point 𝑠 = 1 2 , in the situation when these objects are defined over the rational function field over finite field F𝑞. For each monic polynomial 𝐷 ∈ F𝑞 [𝑡], there is a corresponding quadratic Dirichlet character 𝜒𝐷. Associated to such character, there is the corresponding Dirichlet 𝐿 function 𝐿(𝑠, 𝜒𝐷), as well as the quadratic twist 𝐸 ⊗ 𝜒𝐷 of a fixed elliptic curve 𝐸/F𝑞 (𝑡), whose 𝐿function is denoted by 𝐿(𝑠, 𝐸 ⊗ 𝜒𝐷). A lower bound for the number of polynomials 𝐷 ∈ H∗ 2𝑔+1 such that 𝐿 URI: http://hdl.handle.net/123456789/5532 Files in this item: 1
Disertacija_nikola_lelas.pdf ( 2.523Mb )