Browsing Mathematics by Title
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Perović, Aleksandar (Belgrade)[more][less]
Abstract: The interpretation method is a characteristic common for all results from this thesis. The thesis consists of five chapters and two appendices. A brief overview of the contents of the thesis and the obtained results are presented in Chapter 1. Logical background and the well-known notions the basic notions, definitions and properties from forcing are given in the appendices of the thesis. An elementary proof of equivalence between Cohen forcing and forcing with propositional Lindenbaum algebras is presented in Chapter 2. Dense embedding and the interpretation method are used in that proof. A complete axiomatization of the notion of qualitative probability is presented in Chapter 3. Probabilistic logic LPP_2 LPP_2^FR(n) and LPP^S are extended with the qualitative probability operator π. Several formal techniques as infinite rules, elimination of quantifiers and interpretation method (implicitly), are used to prove the extended completeness theorem and decidability for these logics. In Chapter 4 of the thesis a complete axiomatization of the logic with polynomial weight formulas is presented and the extended completeness theorem is proved. Applications of the interpretation method are given. By using that method the compactness theorem for the non-archimedean valued probabilistic logics is proved in Chapter 5. URI: http://hdl.handle.net/123456789/100 Files in this item: 1
phdAleksandarPerovic.pdf ( 700.0Kb ) -
Laban, Miloš (Belgrade , 1980)[more][less]
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Janković, Svetlana (Belgrade)[more][less]
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Karadžić, Lazar (Belgrade)[more][less]
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Petrović, Vojislav (Novi Sad)[more][less]
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Ušćumlić, Momčilo (Belgrade , 1965)[more][less]
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Radenović, Stojan (Belgrade)[more][less]
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Krstić, Mihailo (Beograd , 2025)[more][less]
Abstract: This doctoral dissertation addresses the integration of functions taking values in spaces of bounded operators and in spaces of complex measures on a given σ-algebra. The mentioned integrability is considered in a more general sense than that required in the theory of weak integration of vector-valued functions. The first part of the dissertation deals with the integrability of families of operators. If (Ω, M, μ) is a space with a positive measure μ and (At)t∈Ω is a family of operators from B(X, Y ), where X and Y are Banach spaces, then μ-integrability of the function Ω ∋ t 7 → ⟨Atx, y∗⟩ ∈ C is required for every x ∈ X and y∗ ∈ Y ∗. In this case, we prove that the quantity sup∥x∥=∥y∗∥=1 R Ω ⟨Atx, y∗⟩ dμ(t) is finite. This expres- sion allows us to define a norm on the corresponding vector space of families of operators. Furthermore, for every E ∈ M, one obtains an operator R E At dμ(t) in B(X, Y ∗∗), whose defining property is ⟨y∗, R E At dμ(t) x⟩ = R E ⟨Atx, y∗⟩ dμ(t) for every x ∈ X and y∗ ∈ Y ∗. The second part of the dissertation deals with the integrability of families of measures. If (λx)x∈X is a family of complex measures on (Y, A), where (X, B, μ) is a space with a positive measure μ, and if for every A ∈ A the function X ∋ x 7 → λx(A) ∈ C is μ-integrable, then the quantity supA∈A R X |λx(A)| dμ(x) is finite. This allows us to define a norm on the corresponding vector space of families of measures. In this case, for every B ∈ B there exists a complex measure R B λx dμ(x) on A such that R B λx dμ(x) (A) = R B λx(A) dμ(x) for every A ∈ A. The dis- sertation is organized as follows. The first part (Chapters 2–4) deals with the integration of functions taking values in B(X, Y ). Chapter 2 provides a survey of the known results on the integration of functions in B(H), where H is a separable Hilbert space, and presents original results extending the existing theory. In Chapter 3, the developed theory is applied to the Laplace transform of B(H)-valued functions, which has been previously considered in the literature. Chapter 4 is significant because it generalizes the integrability of functions taking values in B(X, Y ). This type of integration was first defined in [8]. The second part of the dissertation (Chapter 5) deals with the integration of functions taking values in spaces of complex measures on a given σ-algebra. The introduced type of integration is more general than Pettis concept and has been considered in [6, 7]. These works represent a natural ex- tension and application of the experiences gained from working with functions taking values in operator spaces, including original results of the candidate with coauthors. Numerous concrete examples are included, making this abstract material much more illustrative. URI: http://hdl.handle.net/123456789/5779 Files in this item: 1
Disertacija_M_Krstic.pdf ( 3.184Mb ) -
Janeva, Biljana (Skopje)[more][less]
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Stojanović, Miroslava (Belgrade)[more][less]
URI: http://hdl.handle.net/123456789/350 Files in this item: 1
phdMiroslavaStojanovic.pdf ( 2.145Mb ) -
Ćetković, Simon (Belgrade)[more][less]
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Kostić, Aleksandra (Beograd , 2021)[more][less]
Abstract: This dissertation examines simplicial complexes associated with cyclotomic po-lynomials and irreducible characters of finite solvable groups. In the process of analysis ofthe associated objects special attention is paid to the noncommutativity of the examinedstructures.A collection of simplicial complexes can be associated to an algebraic object such as acyclotomic polynomial. In most cases, the homotopy type of associated simplicial complexesgives us complete information about the coefficients of the cyclotomic polynomial. The onlyexceptions are cyclotomic polynomials whose degree is a product of three different primenumbers and this case is the focus of research in this doctoral dissertation. When it ispossible, the homotopy type of a simplicial complex associated with the polynomialΦpqr(x),wherep,qandrare different prime numbers, is determined by using the discrete Morsetheory. However, in special cases, the simplicial complexes associated with the polynomialΦpqr(x)have a noncommutative fundamental group, thus providing a new noncommutativeinvariant of this type of polynomial. Complex presentations that appear as presentations ofthe fundamental groups of associated simplicial complexes are analyzed using Fox’s calculus.This thesis also focus on the study of simplicial complexes associated to a set of irreduciblecharacters of a finite solvable group. Two types of simplicial complexes are attached to aset of irreducible characters of a finite solvable group — character degree complex and primedivisor complex. The examination of the fundamental group of these types of simplicial com-plexes provides better understanding of the structure of the irreducible characters of finitesolvable groups. URI: http://hdl.handle.net/123456789/5096 Files in this item: 1
Teza-Aleksandra_Kostic.pdf ( 1.009Mb ) -
Peruničić, Predrag (Belgrade , 1984)[more][less]
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Bjelica, Momčilo (Beograd , 1990)[more][less]
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Božović, Nataša (Belgrade)[more][less]
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Surla, Katarina (Novi Sad , 1980)[more][less]
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Jevtić, Miroljub (Belgrade)[more][less]
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Ivanović, Marija (Beograd , 2022)[more][less]
Abstract: This dissertation focuses on the Roman domination problem and its two modifications. Improvements and relaxations of two integer linear programming for- mulations for the Roman domination problem from the literature are introduced, proved to be equivalent to the existing ones despite of the variables relaxation and usage of fewer number of constraints and compared by standard optimization solvers, CPLEX and Gurobi. Improved formulations can be equally used as original ones, but, as it can be seen from numerical results, for some instances, they can be more useful. Given the fact that old and new formulations can not be used for some large size instances, and that algorithms for solving Roman domination problem are mostly defined for some particular graph classes, the aim of this research was to find a new algorithm that can be used for solving Roman domination problem on all graph classes and all graph sizes. Although the Roman domination problem belongs to the class NP, presented algorithm is able to find solution value equal to optimal solution value on very large number of instances in less then a second. For the first modification of the Roman domination problem, named Restrained Roman domination problem, a new mixed integer linear programming formulation is intro- duced and, to the best of the author’s knowledge, this formulation is the first in the literature. For the second modification of the Roman domination problem, the Weak Roman domination problem, an improved integer linear programming formu- lation is presented. Improved formulation is also proved to be correct, equivalent to the existing formulation from the literature and compared using standard op- timization solvers, CPLEX and Gurobi. Numerical results showed the advantage of the improved formulation on almost all tested instances. Additionally, an im- proved linear-time algorithm for solving the Weak Roman domination problem on block graphs is introduced and, similarly to the Roman domination problem, a new algorithm, based on the variable neighborhood search method is presented. With the new variable neighborhood search based algorithm we aimed to find solution of the Weak Roman domination problem equal to the optimal on very large number of tested instances. For instances for which some solution value is found but not proved to be an optimal, presented algorithm provided the new lower-bounds. Even more, for some instances, where optimization solvers were not able to prove optimality or to find any solution, new solutions are found. URI: http://hdl.handle.net/123456789/5431 Files in this item: 1
MarijaIvanovic_ ... a_saPotpisanimIzjavama.pdf ( 1.958Mb ) -
Pejović, Tadija (Belgrade)[more][less]
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Gardašević Filipvić, Milanka (Beograd , 2011)[more][less]