Mathematical Sciences
Subcommunities within this community
Collections in this community
Recent Submissions

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 (BoydWong’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 bmetric spaces as a generalization of metric spaces. The generalization of Zamfirescu’s theorem of bmetric 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 semiautomatic 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. Epitopebased 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 MHCpeptide 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 antigenpresenting 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 newdeveloped 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 MHCbinding ligands is also presented. For that purpose are developed new clusteringbased classi cation models. The models are based on kmeans 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, kcirculantmatricesareconsidered,where k is an arbitrary complexnumber.Themethodforobtainingtheinverseofanon singular kcirculantmatrix,foranarbitrary k ̸= 0, ispresented,andusing that method,theinverseofanonsingular kcirculantmatrixwithgeometric sequence (witharithmeticsequence)isobtained,foranarbitrary k ̸= 0 (for k = 1). Usingthefullrankfactorizationofàmatrix,theMoorePenrosein verseofasingular kcirculantmatrixwithgeometricsequence(witharithme tic sequence)isdetermined,foranarbitrary k (for k = 1). Foranarbitrary k, the eigenvalues,thedeterminantandtheEuclideannormofa kcirculantma trix withgeometricsequencei.e.witharithmeticsequencearederived,and boundsforthespectralnormofa kcirculantmatrixwithgeometricsequence are determined.Also, kcirculantmatriceswiththe 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 kcirculantmatrixwithbinomialcoe 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 ) 
Hodžić, Sandra (Beograd , 2016)[more][less]
Abstract: In recent years there has been increasing interest in modeling the physical and chemical processes with equations involving fractional derivatives and integrals. One of such equations is the subdi usion equation which is obtained from the di usion equation by replacing the classical rst order time derivative by a fractional derivative of order with 0 < < 1: The subject of this dissertation is the initialboundary value problem for the subdi usion equation and its approximation by nite di erences. At the beginning, the onedimensional equation is observed. The existence and the uniqueness of weak solution is proved. The stability and the convergence rate estimates for implicite and the weighted scheme are obtained. The main focus is on twodimensional subdi usion problem with Laplace operator as well as problem with general secondorder partial di erential operator. It is assumed that the coe cients of the di erential operator satisfy standard ellipticity conditions that guarantees existence of solution in appropriate spaces of Sobolev type. In that case, apart from above mensoned, we constructed the additive and the factorized di erence schemes. We investigated their stability and convergence rate depending on the smoothness of the input data and of generalized solution. URI: http://hdl.handle.net/123456789/4455 Files in this item: 1
Disertacija_Sandra_Hodzic.pdf ( 913.2Kb ) 
Glavaš, Lenka (Beograd , 2015)[more][less]
Abstract: The subject of this doctoral dissertation is related to the problems of extreme values in strictly stationary random sequences. It belongs to the topical area of probability and statistics, broadly applicable to real life situations and in many scienti c elds. It relies on large number of seminal articles and monographs. The main aim of the dissertation is to determine the asymptotic behavior of maxima of some incomplete samples from the rstorder autoregressive processes with uniform marginal distributions. The dissertation consists of three chapters. New results (the theoretical ones and the results of computer simulations) are presented in the third chapter. Two types of the uniform ARp1q process pXnqnPN are considered: positively correlated and negatively correlated process, with the lag one correlation p1q : CorrpXn 1;Xnq equal to 1 r and 1 r , respectively, where r ¥ 2 is the parameter of the underlying process. Let pcnqnPN be a nonrandom 0 1 sequence, such that lim nÑ8 1 n n ¸j 1 cj p P r0; 1s. This sequence of degenerate random variables is introduced with the purpose to correspond to the sequence pXnq in the following sense: r.v. Xj is observed if cj 1, otherwise r.v. Xj is not observed (missing observation). Let us use the notation: the r.v. Mn : max 1¤j¤n Xj is maximum of the complete (size n) sample from the random sequence pXnq, and the r.v. Mn is what is called partial maximum, i.e. the maximal element of incomplete sample tXj : cj 1; 1 ¤ j ¤ nu. Based on di erent, speci c deterministic sequences pcnq it is proved that the limiting distribution, as n Ñ 8, of the twodimensional random vector Mn;Mn , is not uniquely determined by the limit value p. This appears as a consequence of the fact that for the uniform ARp1q process one of the weak dependence conditions does not apply. Namely, the uniform ARp1q process does not satisfy the local condition under which clustering of extremes is restricted. As a consequence of this property, some interesting conclusions about asymptotic joint distributions of random variables Mn and Mn are reached. In the cases when the partial maximum Mn is determined by an arbitrary point process there are presented results obtained by simulations. The rst two chapters are rather informative. Having in mind interest in studying the asymptotic behavior of linearly standardized twodimensional componentwise maxima the role of the rst chapter is to anticipate the concept of multivariate extreme values. In the second chapter the basic terms in the time series analysis are formulated, with the accent on the linear stationary models, especially on rstorder autoregressive models. The special attention is dedicated to the uniform ARp1q processes, their properties and existing results concerning their extremal behavior. Still open questions are mentioned in the conclusion, in the very end of the third chapter. URI: http://hdl.handle.net/123456789/4454 Files in this item: 1
Doktorska_disertacija_Lenka_Glavas.pdf ( 1.195Mb )