Browsing Mathematics by Title
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Bulatović, Jelena (Belgrade)[more][less]
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Borovićanin, Bojana (Kragujevac, Serbia , 2008)[more][less]
Abstract: Different spectral characterizations of certain classes of graphs are considered in this dissertation. The large number of papers concerning this topic, indicates that problems of this kind are very interesting in spectral graph theory. This dissertation, beside Preface and References with 46 items, consists of two chapters: 1. Harmonic graphs, 2. Graphs with maximal index. Harmonic graphs are introduced and studied in details in Chapter 1. This chapter consists of four sections. In section 1.1 the definition of harmonic graphs, as well as their basic properties, are given. Harmonic trees are discussed in section 1.2. In section 1.3 we characterize harmonic graphs with small number of cycles; in particular, all unicyclic, bicyclic, tricyclic and tetracyclic graphs are determined. Finally, in section 1.4, we determine all connected 3-harmonic graphs with integral spectrum. The solution of maximal index problem in certain classes of graphs is given in Chapter 2. This chapter consists of four sections. In sections 2.1 and 2.2 we review some results related to the index of a graph. The emphasis is on graphs with given number of both vertices and edges; in particular we discuss graphs having the fixed number of pendant edges, too. In section 2.3 we give the solution of maximal index problem in the class of connected tricyclic graphs with n vertices and k pendant edges. Finally, in section 2.4, we determine graphs with maximal index among all connected cactuses with n vertices. URI: http://hdl.handle.net/123456789/1834 Files in this item: 1
disertacija_Bojana Borovicanin.pdf ( 1.939Mb ) -
Jovanović, Irena (Beograd , 2014)[more][less]
Abstract: Spectral graph theory is a mathematical theory where graphs are considered by means of the eigenvalues and the corresponding eigenvectors of the matrices that are assigned to them. The spectral recognition problems are of particular interest. Between them we can distinguish: characterizations of graphs with a given spectrum, exact or approximate constructions of graphs with a given spectrum, similarity of graphs and perturbations of graphs. In this dissertation we are primarily interested for the similarity of graphs, where graphs with the same number of vertices and graphs of different orders are considered. Similarity of graphs of equal orders can be established by computation of the spectral distances between them, while for graphs with different number of vertices the measures of similarity are introduced. In that case, graphs under study are usually very large and they are denoted as networks, while the measures of similarity can be spectraly based. Some mathematical results on the Manhattan spectral distance of graphs based on the adjacency matrix, Laplacian and signless Laplacian matrix are given in this dissertation. A new measure of similarity for the vertices of the networks under study is proposed. It is based on the difference of the generating functions for the numbers of closed walks in the vertices of networks. These closed walks are calculated according to the new spectral formula which enumerates the numbers of spanning closed walks in the graphlets of the corresponding graphs. The problem of the characterization of a digraph with respect to the spectrum of AAT , apropos ATA, where A is the adjacency matrix of a digraph, is introduced. The notion of cospectrality is generalized, and so the cospectrality between some particular digraphs with respect to the matrix AAT and multigraphs with respect to the signless Laplacian matrix is considered. URI: http://hdl.handle.net/123456789/4233 Files in this item: 1
Jovanović_Irena.pdf ( 1.138Mb ) -
Adamović, Dušan (Belgrade , 1965)[more][less]
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Algali, Khola (Beograd , 2019)[more][less]
Abstract: In this thesis we give some new asymptotic formulas for mean values of multiplicative functions of several variables depending on GCD and LCM of arguments. We obtain an asymptotic formula with a power saving error term for the summation function of a family of generalized least common multiple and greatest common divisor functions of several integer variables. Xn 1,...,nk+`≤x [n1,...,nk]a (n1,...,nk)c , [nk+1,...,nk+`]b (nk+1,...,nk+`)d = Ck,a,c;`,b,d (a + 1)k(b + 1)` xk(a+1)+`(b+1) + O xk(a+1)+`(b+1)−1 2+ and Xn 1,...,nk+`≤x [n1,...,nk]a (n1,...,nk)c , [nk+1,...,nk+`]b (nk+1,...,nk+`)d (n1 ...nk)a(nk+1 ...nk+`)b = Ck,a,c;`,b,d xk+` + O xk+`−1 2+ . Also we obtain an asymptotic formula with a power saving error term for the summation function of Euler phi-function evaluated at iterated and generalized least common multiples of four integer variables. Xn 1,n2,n3,n4≤x ϕ [n1,n2]a (n1,n2)c , [n3,n4]b (n3,n4)d = Ca,c;b,d (a + 1)2(b + 1)2 x2a+2b+4 + O x2a+2b+7 2+ . URI: http://hdl.handle.net/123456789/4820 Files in this item: 1
khola_phd_new_ver.pdf ( 665.4Kb ) -
Ranković, Dragana (Beograd , 2011)[more][less]
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Alidema, Rašit (Belgrade , 1980)[more][less]
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Nikolić, Nebojša (Beograd , 2015)[more][less]
Abstract: A Steiner system S(t; k; v) is a set which contains v elements (v-set) and a family of k-subsets (blocks), such that each t-subset appears in exactly one block (v > k > t > 1; v; k; t 2 N). In the case of a (v; k; t)¡covering, each t-subset appears in at least one block of a given family. A Steiner system S(t; k; v) exists if and only if C(v; k; t) = ¡v t ¢±¡k t ¢ , where C(v; k; t) is the cardinality of minimal (v; k; t)¡covering. As the existence of Steiner system S(t; k; v) and the determination of the minimal (v; k; t)¡covering are still open problems, their solutions are known only in some special cases. Besides the review of the previous results related to the problem of the existence of Steiner systems and the problem of determining the minimal (v; k; t)¡covering, several new constructions of (v; k; t)¡covering are given in this paper. Since the number of blocks in (v; k; t)¡covering represents the upper bound on C(v; k; t), a large number of upper bounds are also obtained by using these constructions. In many cases, the obtained upper bounds are better than the best known upper bounds on C(v; k; t). This dissertation gives a new combinatorial construction of minimal (v; 3; 2)¡ coverings, which represents a generalization of Bose and Skolem constructions of the Steiner triple systems STS(6n + 3) and STS(6n + 1). In each of the 6 cases (v = 6n; : : : ; 6n+5), (v; 3; 2)¡covering is obtained by applying certain permutation p to the initial set of blocks. The obtained construction also represents a new proof of the statement that the values of C(v; 3; 2) are equal to SchÄonheim lower bound L(v; 3; 2). Other constructions of (v; k; t)¡coverings, given in this paper, are heuristic. First, we give improved implementation of the well known greedy algorithm. Then, a new greedy algorithm, as well as the theorem which provides a su±cient con- dition for equality of greedy lex and greedy colex coverings are given. Finally, by v using so called LR procedure, three other heuristics are developed and implemented: Large neighbourhood search, Variable neighborhood descent and General variable neighborhood search. Large neighbourhood search is the procedure of alternately destroying and re- pairing a solution in order to improve the incumbent solution. In the proposed algorithm, this is the procedure for systematic removing and adding blocks to the covering, based on LR procedure. By using LR procedure, the blocks which exclu- sively cover the minimal number of t-subsets are removed from (v; k; t)¡covering, and then the uncovered t-subsets are covered with as few blocks as possible. The greedy algorithm is used for the covering of the uncovered t-subsets. Variable neighborhood search is based on the idea of systematic change of neigh- borhood within a local search algorithm in order to avoid the convergence to a local minimum. A variant of the method where changes of neighborhood is performed in a deterministic way is called Variable neighborhood descent. A variant where the local search procedure is replaced by the Variable neighborhood descent method is General variable neighborhood search. The basis of the local search procedure applied in these two heuristics is also LR procedure. Regarding quality of the obtained results and the performance of the methods, Large neighbourhood search, Variable neighborhood descent, and General variable neighborhood search overcome two heuristics for solving the problem of minimal (v; k; t)¡covering known from the literature: Simulated annealing and Tabu search. Unlike the existing heuristics, the proposed heuristics are applicable to arbitrarily (v; k; t)¡covering. By applying aforementioned heuristics, 23 new best known upper bounds on C(v; k; t) are established. URI: http://hdl.handle.net/123456789/4285 Files in this item: 1
phdNikolic_Nebojsa.pdf ( 851.1Kb ) -
Đorić, Dragan (Beograd , 2002)[more][less]
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Jevremović, Vesna (Belgrade)[more][less]
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Valjarević, Dragana (Kragujevac , 2013)[more][less]
Abstract: Jedan od važnih i osnovnih ciljeva nauke je da se među događajima i pojavama utvrde uzročno-posledične veze. Šta se podrazumeva pod pojmom uzročnosti i kako se ona može meriti bila je tema mnogih rasprava. U ekonomiji, Grendžerova uzročnost (C. nj. Granger, 1969) je veoma dobro poznat koncept i jedna od najprimenjivanijih metoda u istraživanjima. Definicija uzročnosti u smislu Grendžera zasniva se na ideji da sadašnjost ili budućnost ne mogu uzrokovati prošlost. Udrugim naučnim oblastima se, takođe, dugo raspravljalo o pojmu uzročnosti. Međutim, do bitnijeg napretka dolazi tek poslednjih dekada. Danas, pojam uzročnosti ima široku primenu u fizici, biološkim i sociološkim naukama, istoriji, medicini posebno u epidemiologiji, ekonomiji i dr. Predmetistraživanjaove doktorske disertacije je statistička teorija uzročnostii njena primena na slaba rešenja stohastičkih diferencijalnih jednačina i martingalnu reprezentaciju. Pokazuje se da je ovaj koncept ekvivalentan sa slabom jedinstvenošću slabih rešenja stohastičkih diferencijalnih jednačina i ekstremnim rešenjima martingalnog problema. Na ovaj koncept se mogu primeniti i vremena zaustavljanja, pa se u skladu s tim dokazuje i ekvivalencija sa ekstremnim rešenjem martingalnog problema za zaustavljene procese, kao i sa lokalno jedinstvenim slabim lokalnim rešenjima. Takođe, koncept uzročnosti se može primeniti i u Teoriji martingala. Naime, ovaj koncept se može dovesti u vezu sa očuvanjem svojstva martingalnosti, ortogonalnim martingalima, stabilnim potprostorima, kao i martingalnim reprezentacijama, koje imaju primenu, naročito u finansijskoj matematici. U glavi 1 dati su osnovni pojmovi iz teorije verovatnoća, definicija slučajnog procesa i pregled njihovih osnovnih osobina. Takođe, ovde su date definicije martingala i semimartingala kao i stohastička integracija u odnosu na semimartingale. U ovoj glavi su date i stohastičke diferencijalne jednačine sa semimartingalima kao i njihova stroga rešenja, dok će o slabim rešenjima više biti reči u glavi 3. Daje se i definicija uzročnosti, koja se zasniva na Grendžerovoj uzročnosti, koju uvodi Mikland (P. A. Mykland š32]), a kasnije njeno uopštenje daje lj. Petrović (š34, 36, 37, 38, 39, 40, 42]). Takođe, data definicija uzročnosti je sa fiksnog vremena proširena na vremena zaustavljanja. Na kraju su dati neki osnovni rezultati koji se odnose na navedene pojmove uzročnosti. U glavi 2 je prikazano kako se koncept statističke uzročnosti može primeniti u Teoriji martingala. Za svojstvo martingalnosti, koje je vezano za filtracije, dokazano je da je očuvanje tog svojstva, kada σ- algebra informacija raste, direktno vezano za koncept uzročnosti. Takođe, može se uspostaviti ekvivalencija između ortogonalnosti martingala i koncepta uzročnosti (64). Isto se može dokazati i za lokalne martingale i za zaustavljene lokalne martingale. Koncept uzročnosti se može povezati i sa stabilnim potprostorima koji sadrže zdesna neprekidne modifikacije martingala oblikaMt = P(A j Ft) (videti 44). U glavi 3 data su definisana slaba rešenja različitih tipova diferencijalnih jednačina koje su generisane semimartingalima. Naime, dokazano je da je koncept uzročnosti ekvivalentan sa slabom jedinstvenošću slabih rešenja (39). Takođe, na slaba rešenja su primenjena i vremena zaustavljanja, pa je proučavana i veza između lokalne jedinstvenosti slabih lokalnih rešenja i koncepta uzročnosti sa vremenima zaustavljanja. Drugačiji pristup rešavanju stohastičkih diferencijalnih jednačina, tj. martingalniproblemjetakođerazmatranidokazanajeekvivalencijaizmeđu ekstremnog rešenja martingalnog problema i uzročnosti (41). Uglavi 4 je razmatrana veza između koncepta uzročnosti i martingalne reprezentacije u odnosu na različite filtracije sa posebnim osvrtom na već poznate rezultate. Pokazano je da se dati rezultati mogu primeniti na slaba rešenja stohastičkih diferencijalnih jednačina generisanih semimartingalima, kao i na martingalni problem koji je pridružen datoj jednačini. Delovi disertacije 2.2, 2.3, 2.4, 3.1, 3.2, 3.4, 3.6 sadrže nove rezultate koji su objavljeni u radovima 39, 41, 44, 64. URI: http://hdl.handle.net/123456789/2788 Files in this item: 1
Dragana_Valjarevic_PhD.pdf ( 553.7Kb ) -
Parezanović, Nedeljko (Belgrade)[more][less]
URI: http://hdl.handle.net/123456789/161 Files in this item: 1
phdNedeljkoSParezanovic.pdf ( 3.500Mb ) -
Stanojević, Jelena (Beograd , 2015)[more][less]
Abstract: The main goal of the thesis is the development of a new suggested trans- form con¯dence intervals for the ratio of the variances of the two samples. Since now, the methods based on the F statistic have been suggested in the literature. However, the defect of that intervals is the huge sensitivity in re- lation with assumption of parameters distribution. Suggested statistic could be modi¯ed. Edgeworth expansion of the t-statistic has found the place in the thesis and based on that intervals have been compared. Also, on the base of the simulation it was point out that Johnsons transformation give better result in the sense of probability covering in regard to F interval and interval based on Halls transformation. Moreover, the con¯dence intervals for the mean and variances for the one and two sample problems have been considered in the dissertation. Especially, the problem of the di®erence of the proportions for the two samples, with the numerical results and data from the insurance. In addition, the existing methods for the estimation of the extreme value index and the high quantiles have been reviewed. Particularly, the direct simulation estimation of the quantile and probability covering of its deviation from the rights value, for Pareto and Gamma distributions, and also for general Pareto distribution have been discussed. The results were obtained by large deviation theory and their generalization on the topological spaces is stated. In this research, beside the probability theory and elemen- tary principles of the classical analysis, methods of the statistical theory and statistical conclusions have been applied. URI: http://hdl.handle.net/123456789/4261 Files in this item: 1
phdJelenaStanojevic.pdf ( 1.063Mb ) -
Vrećica, Ilija (Beograd , 2022)[more][less]
Abstract: First part of dissertation examines sumsets hA = {a1 + · · · + ah ∈ Z d : a1, . . . , ah ∈ A}, where A is a finite set in Z d . It is known that there exists a constant h0 ∈ N and a polynomial pA(X) such that pA(h) = |hA| for h ⩾ h0. However, little is known of polynomial pA and constant h0. Cone CA over the set A contains information about hA, for all h ∈ N. When A has d + 2 elements, polynomial pA and constant h0 can be explicitly described. When A has d + 3 elements, an upper bound is found for the number of elements of hA. Second part of dissertation examines Selmer groups of elliptic curves in the con gruent number family. A squarefree natural number is congruent if and only if there exists a right triangle with area n whose sides all have integer lengths. It is known that n is a congruent number if and only if elliptic curve En : y 2 = x 3 − n 2x has nonzero rank as an algebraic group. Selmer groups of isogenies on En are interesting, because their rank is not smaller than the rank of En, so when the Selmer groups have rank zero, then the elliptic curve En also has rank zero. Elements of these Selmer groups can be represented as partitions of a particular graph, from which one may find the distribution of ranks of Selmer groups. URI: http://hdl.handle.net/123456789/5533 Files in this item: 1
Teza_Ilija_Vrecica.pdf ( 1.817Mb ) -
Aranđelović, Ivan (Beograd , 1999)[more][less]
URI: http://hdl.handle.net/123456789/4137 Files in this item: 1
Stavovi_o_presecanju.PDF ( 2.592Mb ) -
Merkle, Ana (Beograd , 2023)[more][less]
Abstract: Many new developments in the filed of probability and statistics focus on finding causal connections between observed processes. This leads to considering dependence relations and investigating how the past influence the present and the future. The well known concept of Granger (1969) causality is closely related to the idea of local dependence introduced by Schweder (1970). Granger studied time series, while Schweder considered Markov chains. The concept was later extended to more general stochastic processes by Mykland (1986). All this concepts incorporate the time into consideration dependence. The dissertation consist of four chapters. New results are presented in the fourth chap- ter. The main aim of this doctoral dissertation is to determine di↵erent concepts of stochastic predictability using the well known tool of conditional independence. Follow Granger’s idea, relationships between family of sigma - algebras (filtrations) and between processes in continuous ti- me were considered since continuous time models dependence represent the first step in various applications, such as in finance, econometric practice, neuroscience, epidemiology, climatology, demographic, etc. In this dissertation the concept of dependence between stochastic processes and filtration is introduced. This concept is named causal predictability since it is focused on prediction. Some major characteristics of the given concept are shown and connections with known concept of dependence are explained. Finally, the concept of causal predictability is applied to the processes of di↵usion type, more precisely, to the uniqueness of weak solutions of Ito stochastic di↵erential equations and stochastic di↵erential equations with driving semi- martingales. Also, the representation theorem in terms of causal predictability is established and numerous examples of applications of the given concept are presented such as application in financial mathematics in the view of modeling default risk, in Bayesian statistics. The idea for the future might be to deal with the case of progressive stochastic predictability, i.e. the generalization of stochastic predictability from fixed time to stopping time. URI: http://hdl.handle.net/123456789/5572 Files in this item: 1
DOKTORAT_finalnaVerzija.pdf ( 1.785Mb ) -
Jocković, M. Jelena (Belgrade , 2012)[more][less]
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Jocković, Jelena (Beograd , 2012)[more][less]
Abstract: Statistical methodology for dealing with extremes depend on how extreme values are defined. One way to extract extremes from a given sample x1, x2, ..., xN is to consider maxima (minima). The other way is to consider values y1 = x1 − u, y2 = x2 − u, . . . , yn = xn − u, where y1, y2, . . . , yn are sample members above (below) a given predetermined threshold u. These two methods lead to two different approaches in extreme value theory. This doctoral dissertation has two main goals. One of them is to apply the techniques from extreme value framework to certain type of combinatorial problems. The other goal is to contribute to the field of statistical modeling of extremes. The dissertation consists of three chapters. In the first chapter, we introduce generalized extreme value distributions and generalized Pareto distributions (GPD). These two families play key roles in the two approaches to modeling extremes. We set out the theoretical background for both approaches. In the second chapter, we apply the extremal techniques to combinatorial waiting time problems. Precisely, we consider Coupon collector’s problem, defined as follows: elements are sampled with replacement from the set Nn = {1, 2, . . . , n} under assumption that each element has probability 1/n of being drawn. The subject of interest is the waiting time Mn until all elements of Nn or some other pattern are sampled. We focus our attention to the following two cases: 1. Mn is the waiting time until all elements of Nn are sampled at least r times, where r is a positive integer; 2. Mn is the waiting time until all pairs of elements jj, j ∈ Nn are sampled. We present new results related to the asymptotic behavior of the waiting time Mn, if it is known that a large number of trials was performed and the experiment is not over. For both cases, we determine the limiting distribution of exceedances of Mn over high thresholds, and answer some related questions: how to choose a suitable high threshold (depending on n) in order to obtain a limiting distribution; under what conditions the limit does not depend on the threshold; are the generalized Pareto distributions the only possible limits. We also estimate the speed of convergence in both cases. The third chapter of the dissertation is devoted to estimation of parameters and quantiles of the generalized Pareto distributions. We restrict the attention to the two-parameter version of GPD, defined as: Wγ,σ(x) = 1 − e−x , x ≥ 0, γ = 0 1 − 1 + γ σx −1 , x ≥ 0, γ > 0 1 − 1 + γ σx −1 , x ∈ h 0,−σ γ i , γ < 0. Well known problem with this model is inconsistency with the sample data, which is that one or more sample observations exceed the estimated upper bound in case when γ < 0. We propose a new, general technique to overcome the inconsistency problem and improve performance of the existing GPD estimation methods. We apply the proposed technique to methodof- moments and method-of-probability-weighted-moments estimates, investigate its performance through computer simulation and provide some real data examples. Finally, we address the problem of estimating high GPD quantiles. We evaluate the robustness of some estimation methods through simulation study and present a case study from finance (value-at-risk estimation), with special emphasis to certain difficulties related to this field of application. URI: http://hdl.handle.net/123456789/4271 Files in this item: 1
phdJockovic_Jelena.pdf ( 1.687Mb ) -
Petruševski, Ljiljana (Belgrade , 1986)[more][less]
URI: http://hdl.handle.net/123456789/51 Files in this item: 1
phdLjiljanaPetrusevski.pdf ( 1.651Mb ) -
Blagojević, Dragan (Belgrade)[more][less]