Mathematics
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Milovanović - Aranđelović, Marina (Beograd , 2016)[more][less]
Abstract: The aim of this dissertation is to present new properties of some classes of contractive mappings, and applications of this results to nonlinear analysis. It contain six chapters. The first chapter gives basic properties of semi-metric spaces. New result is extension of Borel - Lebesque theorem to semi metric spaces. The second chapter contain generalization of Niemytzki’s fixed point theorem. In third chapter the notion of measure of non-compactness are extended to class of semimetric spaces. New fixed point theorem for condensing mappings defined on semi-metric spaces is presented. Further three chapters contains new common fixed points theorems of Mair - Keeler type, new results for accretive mappings and new common fixed point results for mappings defined on probabilistic metric spaces. URI: http://hdl.handle.net/123456789/4482 Files in this item: 1
DisertacijaMilovanovic-Arandjelovic_Marina.pdf ( 1.599Mb ) -
Milošević, Bojana (Beograd , 2016)[more][less]
Abstract: Goodness of t and symmetry tests occupy a signi cant part of nonparametric statistic. Most of classical tests are based on the distance between the assumed distribution function and its consistent estimate, empirical distribution function. The symmetry tests are analogously constructed. A new approach that is especially attractive in recent years is making tests based on characterizations of di erent types. Those tests use U-empirical distribution functions (generalized empirical ones), U-empirical transforms (eg. Laplace transform, characteristic functions etc.) and U-empirical moments of distributions. The main advantage of these tests is that they are often free of some distribution parameters. Therefore they are suitable for testing composite hypothesis. For purpose of comparison of tests the Bahadur e ciency has become very popular. One of the reasons is that it does not require the asymptotic normality of test statistics. In addition, Bahadur and Pitman e ciencies very often locally coincide. It turns out that for determining Bahadur e ciency it is necessary to nd large deviations function under null hypothesis. If that is not possible, usually the approximate Bahadur e ciency is used. It requires the existence of asymptotic distribution and the asymptotic behavior of its tail under null hypothesis, and the limit in probability under alternative distribution. The goals of the thesis are the construction of new goodness of t and symmetry tests based on U-statistics and V -statistics, deriving the asymptotic distribution of proposed statistics, their large deviation functions and Bahadur e ciencies or their approximations. The thesis is divided into two parts. The rst part consists of three chapters. In the rst chapter the theory of U-statistics and V -statistics as well as U-empirical and V -empirical distribution function and some other empirical transforms is presented. The second chapter is devoted to the U-statistics and V -statistics with estimated parameters. The third chapter deals with asymptotic e ciency of nonparametric tests. Most of the chapter is devoted to Bahadur e ciency. In the same chapter the large deviation function for a new class of tests statistics is derived. This result is presented in [69]. The second part, which starts with the fourth chapter, is dedicated to new tests based on U-statistics and V -statistics. In the fourth chapter some type of characterizations which are used within the next chapters for construction of tests are presented. They include characterizations based on equidistribution of some statistics among which the characterizations of symmetric distributions stand out, then those based on functional equations that the distribution function satis es, those based on the independence of statistics and those based on moments. Two new characterizations of symmetric distributions are also presented. The fth chapter deals with new goodness of t tests. There are four new exponentiality tests, two new goodness of t tests for a family of Pareto distribution, as well as two new goodness of t tests for logistic distribution (see [66], [69]). Also one new class of uniformity tests which can be used as goodness of t test for any predetermined continuous distribution is proposed (see [67]). The sixth chapter is devoted to new symmetry tests. Five new symmetry tests are proposed. In the seventh chapter there are new exponentiality tests based on U-statistics and V -statistics with estimated parameters. A special attention is given to new tests based on U- empirical Laplace transforms. Beside those, some tests based on U-empirical moments are also presented. For new presented tests local Bahadur and/or Pitman e ciency is calculated. In the nal chapter, a brief review of some applications in time series analysis is shown. URI: http://hdl.handle.net/123456789/4470 Files in this item: 1
BojanaMilosevic_doktorat.pdf ( 1.897Mb ) -
Jovanović, Mirko (Beograd , 2016)[more][less]
Abstract: This dissertation is the contribution to the Metric fixed point theory, the area that has recently been rapidly developing. It contains five chapters. The first chapter gives the proof of one already known lemma. This lemma is used in the proof of Banach’s theorem for orbital complete metric spaces. The second chapter contains the proofs of eight theorems, which generalize some known results from the theory of fixed points in metric spaces (Boyd-Wong’s, ´ Ciri´c ’ s, Pant’ s, and other). Some of these theorems are modifications of the known ones, while three are completely new. Three theorems are proven in the third chapter. They generalize the result of fixed point of mapping defined in compact metric space given by Nemytzki, as well as one generalization of Edelstein’s theorem. The proofs and stated corollaries of some theorems are original. Chapter four discusses b-metric spaces as a generalization of metric spaces. The generalization of Zamfirescu’s theorem of b-metric spaces is presented as well as some of its applications. A new result concerning weakly almost contractive mappings is also determined. Chapter five contains some new results in cone metric spaces. Two theorems are presented as the analogue of the same theorems in the setting of standard metric spaces. A completely new theorem is established which results in the Banach’s theorem in cone metric spaces whereby the cone does not need to be normal. A generalization of Fisher’s theorem in cone metric spaces over a regular cone is also proven. Almost all results in this dissertation are confirmed by corresponding examples, which explain how these results differ from the already known results. URI: http://hdl.handle.net/123456789/4458 Files in this item: 1
Mirko_Jovanovic_dr.pdf ( 1.375Mb ) -
Jandrlić, Davorka (Beograd , 2016)[more][less]
Abstract: Application of association rule and support vector machine technique for T cell epitope prediction Abstract: Data mining is an interdisciplinary sub eld of computer science, including various scienti c disciplines such as: database systems, statistics, machine learning, arti cial intelligence and the others. The main task of data mining is automatic and semi-automatic analysis of large quantities of data to extract previously unknown, nontrivial and interesting patterns. Rapid development in the elds of immunology, genomics, proteomics, molecular biology and other related areas has caused a large increase in biological data. Drawing conclusions from these data requires sophisticated computational analyses. Without automatic methods to extract data it is almost impossible to investigate and analyze this data. Currently, one of the most active problems in immunoinformatics is T cell epitope identi cation. Identi cation of T - cell epitopes, especially dominant T - cell epitopes widely represented in population, is of the immense relevance in vaccine development and detecting immunological patterns characteristic for autoimmune diseases. Epitope-based vaccines are of great importance in combating infectious and chronic diseases and various types of cancer. Experimental methods for identi cation of T - cell epitopes are expensive, time consuming, and are not applicable for large scale research (especially not for the choice of the optimal group of epitopes for vaccine development which will cover the whole population or personalized vaccines). Computational and mathematical models for T - cell epitope prediction, based on MHC-peptide binding, are crucial to enable the systematic investigation and identi cation of T - cell epitopes on a large dataset and to complement expensive and time consuming experimentation [16]. T - cells (T - lymphocytes) recognize protein antigen(s) only when degradated to peptide fragments and complexed with Major Histocompatibility Complex (MHC) molecules on the surface of antigen-presenting cells [1]. The binding of these peptides (potential epitopes) to MHC molecules and presentation to T - cells is a crucial (and the most selective) step in both cellular and humoral adoptive immunity. Currently exist numerous of methodologies that provide identi cation of these epitopes. In this PhD thesis, discussed methods are exclusively based on peptide sequence binding to MHC molecules. It describes existing methodologies for T - cell epitope prediction, the shortcomings of existing methods and some of the available databases of experimentally determined linear T - cell epitopes. The new models for T - cell epitope prediction using data mining techniques are developed and extensive analyses concerning to whether disorder and hydropathy prediction methods could help understanding epitope processing and presentation is done. Accurate computational prediction of T cell epitope, which is the aim of this thesis, can greatly expedite epitope screening by reducing costs and experimental e ort. These theses deals with predictive data mining tasks: classi cation and regression, and descriptive data mining tasks: clustering, association rules and sequence analysis. The new-developed models, which are main contribution of the dissertation are comparable in performance with the best currently existing methods, and even better in some cases. Developed models are based on the support vector machine technique for classi cation and regression problems. À new approach of extracting the most important physicochemical properties that in uence the classi cation of MHC-binding ligands is also presented. For that purpose are developed new clustering-based classi cation models. The models are based on k-means clustering technique. The second part of the thesis concerns the establishment of rules and associations of T - cell epitopes that belong to di erent protein structures. The task of this part of research was to nd out whether disorder and hydropathy prediction methods could help in understanding epitope processing and presentation. The results of the application of an association rule technique and thorough analysis over large protein dataset where T cell epitopes, protein structure and hydropathy has been determined computationally, using publicly available tools, are presented. During the research on this theses new extendable open source software system that support bioinformatic research and have wide applications in prediction of various proteins characteristics is developed. A part of this thesis is described in the works [71][82][45][42][43][44][72][73] that are published or submitted for publications in several journals. The dissertation is organized as follows: In section1 is illustrated introduction to the problem of identifying T - cell epitopes, the importance of mathematical and computational methods in this area, vii as well as the importance of T - cell epitopes to the immune system and basis for functioning of the immune system. In section 2 are described in details data mining techniques that are used in the thesis for development of new models. Section 3 provides an overview of existing methods for predicting the T - cell epitopes and explains the work methodologies of existing models and methods. It pointed out the shortcomings of existing methods which have been the motivation for the development of new models for the T - cell epitope prediction. Some of the publicly available databases with the experimentally determined MHC binding peptides and T - cell epitope are described. In section 4 are presented new developed models for epitopes prediction. The developed models include three new encoding schemes for peptide sequences representation in the form of a vector which is more suitable as input to models based on the data mining techniques. Section 5 reports results of presented new classi cation and regression models. The new models are compared with each other as well as with currently existing methods for T cell epitope prediction. Section 6 presents the research results of the T - cell epitopes relationship with ordered and disordered regions in proteins. In the context of this chapter summary results are presented which are shown in more detail in the published works [71][82][45][44]. Section 7 concludes the dissertation with some discussion of the potential signi cance of obtained results and some directions for future work. URI: http://hdl.handle.net/123456789/4457 Files in this item: 1
doktorskaTezaDavorkaJandrlic.pdf ( 7.938Mb ) -
Radičić, Biljana (Beograd , 2016)[more][less]
Abstract: In thisdissertation, k-circulantmatricesareconsidered,where k is an arbitrary complexnumber.Themethodforobtainingtheinverseofanon- singular k-circulantmatrix,foranarbitrary k ̸= 0, ispresented,andusing that method,theinverseofanonsingular k-circulantmatrixwithgeometric sequence (witharithmeticsequence)isobtained,foranarbitrary k ̸= 0 (for k = 1). Usingthefullrankfactorizationofàmatrix,theMoore-Penrosein- verseofasingular k-circulantmatrixwithgeometricsequence(witharithme- tic sequence)isdetermined,foranarbitrary k (for k = 1). Foranarbitrary k, the eigenvalues,thedeterminantandtheEuclideannormofa k-circulantma- trix withgeometricsequencei.e.witharithmeticsequencearederived,and boundsforthespectralnormofa k-circulantmatrixwithgeometricsequence are determined.Also, k-circulantmatriceswiththe rstrow (F1; F2; :::;Fn) i.e. (L1;L2; :::;Ln), where Fn i.e. Ln is the nth Fibonaccinumberi.e.Lucas number,areinvestigatedandtheeigenvaluesandtheEuclideannormsof suchmatricesareobtained.BoundsforthespectralnormsoftheHadamard inversesoftheabovematrices,foranarbitrary k ̸= 0, arealsodetermined. Attheendofthisdissertation,theeigenvalues,thedeterminantandbounds for thespectralnormofa k-circulantmatrixwithbinomialcoe cientsare derived,andboundsforthespectralnormoftheHadamardinverseofsuch matrix, foranarbitrary k ̸= 0, aredetermined. URI: http://hdl.handle.net/123456789/4456 Files in this item: 1
Disertacija_Biljana_Radicic.pdf ( 1.609Mb )