Browsing Mathematics by Title
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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]
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Jovanikić, Branko (Belgrade)[more][less]
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Torgašev, Aleksandar (Belgrade)[more][less]
URI: http://hdl.handle.net/123456789/101 Files in this item: 1
phdAleksandarTorgasev.pdf ( 83.67Mb ) -
Kašanin, Radivoj (Belgrade)[more][less]
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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 Lipschitz-continuity 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,α-smoothness, 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 n-dimensional ball. We also state properties of harmonic and sub-harmonic 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 Tα-harmonic functions. Namely, unlike the space of harmonic functions, the solution of the Dirichlet problem in the space of Tα-harmonic functions is shown to be Lipschitz-continuous 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 so-called 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 α-star 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 ) -
Tošić, Dušan (Belgrade , 1984)[more][less]
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Malešević, Jovan (Belgrade)[more][less]
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Svetlik, Marek (Beograd , 2020)[more][less]
Abstract: In this dissertation we consider various versions of the Schwarz lemma and theSchwarz-Pick lemma for holomorphic, harmonic and harmonic quasiregular mappings. Inaddition, in order to present new results, an overview of the results that can be considered asclassical is given. As one of the most important consequence of the Schwarz-Pick lemma forholomorphic mappings, an introduction of the hyperbolic distancedΩon the simply connecteddomainsΩ C(such thatΩ6=C) is given in details, as well as the connection of that distanceand holomorphic mappings. All versions of the Schwarz lemma and the Schwarz-Pick lemma for harmonic mappingsare shown as assertions analogous to the corresponding claims for holomorphic mappings.In the proofs of these assertions, the properties of the hyperbolic distance and Euclideanproperties of hyperbolic disks are used. Firstly, we considered some versions of the Schwarzlemma for harmonic mappings from the unit disk to the interval (-1,1) and then for harmonicmappings of the unit disk into itself, without the assumption thatz= 0is mapped to itself bythe corresponding map. Thereby, the corresponding inequalities were shown to be sharp andextremal mappings were found. By using the strip and half plane method, simple proofs ofthe Schwarz-Pick lemma for real-valued harmonic mappings are given, as well as the simpleproofs of their corollaries that are formulated in terms of corresponding hyperbolic distances.For both holomorphic and harmonic mappings a version of the Schwarz lemma have beenformulated and proved in the case where the values of these mappings and values of thenorms of their differentials, at the pointz= 0, are given. Also, in that case we showed thatthe corresponding inequalities are sharp and extremal mappings were found. It has also beenshown that the same methods can be used to obtain Harnack’s inequalities for harmonicmappings, as well as for their generalizations.Furthermore, we give simple proofs of a version of the Schwarz-Pick lemma for harmonicquasiregular mappings whose codomain is a half plane or a strip. One version of the Schwarzlemma for harmonic quasiregular mappings from the unit disk into a strip is obtained thanksto the appropriate (which seems unexpected) inequality satisfied by the Euclidean and hyper-bolic distances on the strip. By using the properties of the Gaussian curvature we also showthat harmonic quasiconformal mappings of the hyperbolic domain into convex hyperbolicdomain are quasi-isometries of the corresponding metric spaces.The introduction of the hyperbolic distance is shown in two ways. The first way is classi-cal one. Starting from the hyperbolic metric on the unit disk, first we define the hyperboliclength of theC1curve and then the hyperbolic distance between two given points. Thesecond one is based on the axiomatic foundation of the absolute plane geometry. Startingfrom the theorem related to the existence and the uniqueness (up to the unit for length) ofthe distance in the absolute plane (which is in accordance with the basic geometry relations- between and congruence), we simultaneously derive the formula for that distance in twomodels of that plane. One of these models is the set of complex numbersC, observed as amodel of the Euclidean plane and the second one is the unit disk that is considered as thePoincaré disk model of hyperbolic plane. URI: http://hdl.handle.net/123456789/5091 Files in this item: 1
svetlik_marek-phd.pdf ( 1.279Mb ) -
Ilić, D. Ivana (Belgrade , 2013)[more][less]
Abstract: For the sequence of heavy-tailed and possibly dependent random variables with the missing observations the estimation of the tail-index is considered. Under minimal but verifiable assumption of ''extremal dependence'' we proved the consistency of geometric-type estimator (Brito and Freitas, 2003). We extended results from Mladenovic and Piterbarg (2008) and proved the consistency and the asymptotic normality of the Hill estimator. Illustrative examples are provided. URI: http://hdl.handle.net/123456789/2485 Files in this item: 1
Elektronska verzija.pdf ( 3.250Mb )