Browsing Mathematics by Title

Kašanin, Radivoj (Belgrade)[more][less]

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 ) 
Tošić, Dušan (Belgrade , 1984)[more][less]

Malešević, Jovan (Belgrade)[more][less]

Svetlik, Marek (Beograd , 2020)[more][less]
Abstract: In this dissertation we consider various versions of the Schwarz lemma and theSchwarzPick 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 SchwarzPick 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 SchwarzPick 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 SchwarzPick lemma for realvalued 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 SchwarzPick 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 hyperbolic 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 quasiisometries of the corresponding metric spaces.The introduction of the hyperbolic distance is shown in two ways. The first way is classical 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_marekphd.pdf ( 1.279Mb ) 
Ilić, D. Ivana (Belgrade , 2013)[more][less]
Abstract: For the sequence of heavytailed and possibly dependent random variables with the missing observations the estimation of the tailindex is considered. Under minimal but verifiable assumption of ''extremal dependence'' we proved the consistency of geometrictype 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 ) 
Jovanović, Milan (Beograd , 2015)[more][less]
Abstract: Early papers dealing with socalled stressstrength problems were published in the middle of the 20th century. This topic, which belongs to the reliability theory, is still very active nowadays, which can be seen through the number of published papers dealing with it  around ten each year. In this dissertation, some methods for estimation of the reliability parameter for a system with independent stress and strength are presented. Also, two new models are introduced and some estimators of the reliability parameter for each of them are derived. The dissertation is divided into four chapters. In the rst chapter, some basic terms are introduced and some examples from real life, illustrating big possibilities for application of the results from this scienti c eld, are described. Sorted based on the stress and strength distributions, a chronological overview of all research activities dealing with these topics, to the author's best knowledge, is presented. Some special func tions, which are later used for calculations, along with their main properties are shown. The expressions for the reliability parameter for some stress and strength distributions are either derived or listed. The second chapter is devoted to different methods used for point esti mation, as well as for interval estimation of the reliability parameter of a system. For each methods estimators of the reliability parameter for some stress and strength distributions are either derived or listed. In the third chapter, a new model is introduced. In this model, the stress has geometric, while the strength has Poisson distribution. This is one of the rst, if not the rst, appearances in the literature, where the stress and strength distributions do not belong to the same family of distributions. For this model, the reliability parameter is estimated using different methods and decision on optimal estimators for usage in practice is based on the simulations. In the fourth chapter, another model is introduced, with the stress and strength distributions which are not only from different families of distribu tions, but also do not belong to the same type of distributions. The stress has geometric, while the strength has exponential distribution. The reliabil ity parameter for this model is also estimated using different methods, and the decision on optimal estimators for usage in practice is once again based on the simulations. URI: http://hdl.handle.net/123456789/4352 Files in this item: 1
Jovanovic_Milan_teza.pdf ( 4.636Mb ) 
Mladenović, Pavle (Belgrade , 1985)[more][less]

Petrić, Jovan (Belgrade , 1960)[more][less]

SaklŠnajder, Zagorka (None)[more][less]
URI: http://hdl.handle.net/123456789/175 Files in this item: 1
phdZagorkaSaklSnajder.pdf ( 2.498Mb ) 
Kilibarda, Goran (Belgrade)[more][less]

Ilić Stepić, Angelina V. (Beograd , 2012)[more][less]

Ušan, Janez (Belgrade)[more][less]

Autor, Nepoznat (None)[more][less]

Karamata, Jovan (Belgrade)[more][less]

Lažetić, Nebojša (Belgrade , 1980)[more][less]

Ušćumlić, Šćepan (Belgrade)[more][less]

Čukić, Ljubomir (Belgrade , 1981)[more][less]

Čukić, Ljubomir (Beograd , 1981)[more][less]
URI: http://hdl.handle.net/123456789/4093 Files in this item: 1
Linearna_preslikavanja.PDF ( 17.30Mb ) 
Šćepanović, Ranko (Belgrade , 1985)[more][less]