Browsing Mathematics by Title

Prešić, Marica (Belgrade)[more][less]

Krstev, Cvetana (Beograd , 1997)[more][less]

Marinković, Silvana (Kragujevac, Serbia , 2011)[more][less]
Abstract: In this dissertation functions and equations in some classes of lattices such as Post algebras, Stone algebras and multiplevalued logics, are studied. The dissertation, beside Preface and References with 46 items, consists of five chapters. In Introduction some basic notations which will be used in next chapters are given. Main results on Boolean functions and equations are exposed in Chapter 2. In Chapter 3, assuming that a general solution is known, the class of reproductive general solutions of the equation in Stone algebra is described. All general solutions of equations in one variable in multiplevalued logic are described in Chapter 4. A necessary and sufficient conditions that given sequence of recurrent inequalities represents solution of some consistent Post equation are given in Chapter 5. Also, it is proved that every Post transformation is the parametric solution of some consistent Post equation. URI: http://hdl.handle.net/123456789/1842 Files in this item: 1
SilvanaMarinkovicDoktorat.pdf ( 360.3Kb ) 
Stipanić, Ernest (Belgrade)[more][less]

Dacić, Rade (Belgrade , 1965)[more][less]

Mitrović, Slobodanka (Belgrade)[more][less]

Lipkovski, Aleksandar (Belgrade , 1985)[more][less]
URI: http://hdl.handle.net/123456789/25 Files in this item: 1
phdAleksandarLipkovski.pdf ( 2.471Mb ) 
Miličić, Miloš (Belgrade , 1982)[more][less]

Malinović, Todor (Novi Sad , 1986)[more][less]

Obradović, Marko (Beograd , 2015)[more][less]
Abstract: First characterizations of probability distributions date to the thirties of last century. This area, which lies on the borderline of probability theory and mathematical statistics, attracts large number of researchers, and in recent times the number of papers on the subject is increasing. Goodnessof t tests are among the most important nonparametric tests. Many of them are based on empirical distribution function. The application of characterization theorems for construction of goodnessof t tests dates to the middle of last century, and recently has become one of the main directions in this eld. The advantage of such tests is that they are often free of distribution parametres and hence enable testing of composite hypotheses. The goals of this dissertation are the formulation of new characterizations of exponential and Pareto distribution, as well as the application of the theory of Ustatistics, large deviations and Bahadur e ciency to construction and examination of asymptotics of goodnessof t tests for aforementioned distributions. The dissertation consists of six chapters. In the rst chapter a review of di erent types of characterizations is presented, pointing out their abundance and variety. The special emphasis is given to the characterizations based on equidistribution of functions of the sample. Besides, two new characterizations of Pareto distribution are presented. The second chapter is devoted to some new characterizations of the exponential distributions presented in papers [65] and [53]. Six characterizations based on order statistics are presented. A special case of one of them (theorem 2.4.3) represents the solution of open problem stated by Arnold and Villasenor [9]. In the third chapter there are basic concepts on Ustatistics, the class of statistics important in the theory of unbiased estimation. Some of their asymptotic properties are given. Uempirical distribution functions, a generalization of standard empirical distribution functions, are also de ned. The fourth chapter is dedicated to the asymptotic e ciency of statistical tests, primarily to Bahadur asymptotic e ciency, i.e. asymptotic e ciency of the test when the level of signi cance approaches zero. Some theoretical results from the monograph by Nikitin [57], and papers [61], [59], etc. are shown. In the fth chapter new results in the eld of goodnessof t tests for Pareto distribution are presented. Based on three characterizations of Pareto distribution given in section 1.1.2. six goodnessof t tests, three of integral, and three of Kolmogorov type, are proposed. In each case the composite null hypothesis is tested since the test statistics are free of the parameter of Pareto distribution. For each test the asymptotic distribution under null hypothesis, as well as asymptotic behaviour of the tail (large deviations) under close alternatives is derived. For some standard alternatives, the local Bahadur asymptotic e ciency is calculated and the domains of local asymptotic optimality are obtained. The results from this chapter are published in [66] and [64]. The sixth chapter brings new goodnessof t tests for exponential distribution. Based on the solved hypothesis of Arnold and Villasenor two classes of tests, integral and Kolmogorov type, are proposed, depending on the number of summands in the characterization. The study of asymptotic properties, analogous to the ones in the fth chapter is done in case of two and three summands, for which the tests have practical importance. The results of this chapter are presented in [39]. URI: http://hdl.handle.net/123456789/4288 Files in this item: 1
phdObradovicMarko.pdf ( 789.3Kb ) 
Aranđelović, Dragoljub (Belgrade)[more][less]

Alagić, Mara (Belgrade , 1985)[more][less]

Jokanović, Dušan (Podgorica)[more][less]

Ivanović, Jelena (Beograd , 2020)[more][less]
Abstract: U teoriji kategorija, koherencija koja je vezana za odre eni tip kategorija, u najgrubljem smislu znaqi komutiranje dijagrama sastavljenih od kanonskihstrelica tih kategorija. Ovo komutiranje mo e biti bezuslovno ili uslovljenozadatim pretpostavkama. U savremenom smislu, koherencija, taqnije teoremekoherencije, podrazumevaju postojanje vernog funktora iz slobodno generisane kategorije datog tipa u kategoriju koja omogu ava proveru jednakosti strelica. Takve su najqex e kategorije qije su strelice relacije ili dijagrami(mnogostrukosti, kobordizmi).Rezultati koherencije su od velikog znaqaja za opxtu teoriju dokaza. Naime,oni obezbe uju formiranje zadovoljavaju eg kriterijuma za jednakost izvo enjau odre enim deduktivnim sistemima i na taj naqin pru aju mogu nost definisanja osnovnog pojma kojim se teorija dokaza bavi.Predmet ove doktorske disertacije je prouqavanje topoloxkih dokaza koherencije i formiranje novih klasa politopa koje u takvim dokazima koherencije mogu poslu iti. Sadr aj disertacije je, dakle, u najve oj meri posve enspomenutim dokazima koherencije, odnosno raznovrsnim geometrijskim realizacijama specifiqnih apstraktnih politopa koji su zadati kombinatorno.Naime, ranih devedesetih godina, Mihail Kapranov je uveo familiju elijskih kompleksa pod nazivom permutoasociedri koja predstavlja ,,hibrid” dveznaqajne familije prostih politopa–familije asociedara i familije permutoedara. Ovaj hibrid je predstavljao prvu geometrijsku interpretaciju udru ivanja komutativnosti i asocijativnosti. Kapranov je pokazao da je datom elijskom strukturom proizveoCWloptu qime je dobio direktan topoloxki dokazkoherencije u simetriqnim monoidalnim kategorijama. Ubrzo nakon toga, Viktor Rajner i Ginter Cigler su ove elijske komplekse realizovali kao familiju konveksnih politopa. Me utim, dobijena familija nije familija prostihpolitopa. S druge strane, i svi asociedri i svi permutoedri jesu prosti. Onipripadaju nestoedarima–xiroko izuqavanoj familiji prostih politopa, kako sakombinatorne strane, tako i sa strane primena u torusnoj topologiji.Polazixte ove teze je da je prirodno prona i prost hibrid ove dve familije, tj. prost permutoasociedar. Koriste i kombinatornu proceduru sliqnuonoj koja je proizvela i same asociedre i permutoedre, u ovoj disertaciji seuvodi apstraktni politop koji odgovara problemu koherencije, a koji se pritommo e realizovati kao prost politop. Preciznije, formirane su klasendimenzionalnih prostih permutoasociedara koje daju topoloxki dokaz simetriqnemonoidalne koherencije, a zatim su te klase i geometrijski realizovane eksplicitnim zadavanjem nejednaqina poluprostora uRn+1koji definixu politopexiii xivu klasama. Ovandimenzionalna realizacija je oznaqena saPAn. Pored toga,u tezi je ponu ena i alternativna realizacija iste familije uz pomo sumaMinkovskog. Naime, uvedena je familijandimenzionalnih politopa, oznaqena saPAn,c, koja je dobijena sumiranjem odre enih politopa. PolitopPAn,cjenormalno ekvivalentan politopuPAnza svakoc∈(0,1]. Req je o specifiqnojrealizaciji, po ugledu na realizaciju nestoedara Aleksandra Postnjikov, kojapodrazumeva da svaki sabirak, grubo reqeno, doprinosi nastajanju taqno jednepljosni rezultuju e sume Minkovskog. Drugim reqima, svaki sabirak dovodi dozarubljivanja teku e parcijalne sume odsecanjem jedne njene strane. Postnjikovje za sabirke koristio simplekse, dok ova disertacija pokazuje da je, u analognoj realizaciji prostog permutoasociedra, za odre ene sabirke neophodno uzetipolitope koji ne samo da nisu simpleksi, ve nisu nu no ni prosti politopi. Usluqaju formiranja familijePAn,1, sabirci su definisani kao konveksni omotaqi skupova taqaka uRn+1xto je znaqajna prednost sa stanovixta programiranja.Osim xto predstavlja direktan topoloxki dokaz teoreme koherencije u simetriqnim monoidalnim kategorijama, poseban znaqaj ove alternativne realizacije je u uspostavljanju jasne veze izme u operacije sumiranja Minkovskogi operacije odsecanja strana permutoedra, tj. njegovog zarubljivanja. Iz oveveze implicitno sledi procedura za analognu realizaciju xire klase prostihpolitopa–familije prostih permutonestoedara.Na kraju, u tezi su date ocene hromatskih brojeva prostog permutoasociedrai nekih znaqajnih nestoedara, sa ciljem prouqavanja mogu e veze ovih klasapolitopa sa torusnim i kvazitorusnim mnogostrukostima.Tokom svih spomenutih istra ivanja, za potrebe ove disertacije je razvijenonekoliko softverskih rexenja (programa i aplikacija) uz pomo raznovrsnihprogramskih jezika i paketa (Java,polymake/Perl,Rhinoceros/ Grasshopper). Najznaqajnija me u njima opisana su u dodatku teze programskim kodom i odgovaraju im ilustrativnim primerima. URI: http://hdl.handle.net/123456789/5104 Files in this item: 1
Ivanovic_Jelena_disertacija.pdf ( 2.983Mb ) 
Pešović, Marko (Beograd , 2021)[more][less]
Abstract: The combinatorial objects can be joined in a natural way with the correspondingcombinatorial Hopf algebras. Many classical enumerative invariants of combinatorial objectsare obtained as a result of universal morphism from the corresponding combinatorial Hopfalgebras to the combinatorial Hopf algebra of quasisymmetric functions.On the other hand, to combinatorial objects we can assign some geometric objects such ashyperplane arrangement or convex polytope. For example, simple graph corresponds to graphicalzonotope and matroid corresponds to matroid base polytope. These classes of polytopes belongto the class of polytopes known as generalized permutohedra. For a generalized permutohedronthere is a weighted quasisymmetric enumerator which for different classes of generalizedpermutohedra represents generalizations of classical enumerative invariants such as Stanley’schromatic symmetric function for graph and Billera−Jia−Rainer quasisymmetric function formatroid.A weighted quasisymmetric enumerator associated with a generalized permutohedron is aquasisymmetric function. For certain classes of generalized permutohedra this enumeratorcoincides with the result of the universal morphism from corresponding combinatorial Hopfalgebra. URI: http://hdl.handle.net/123456789/5207 Files in this item: 1
Pesovic_Marko.pdf ( 1.804Mb ) 
Jelić Milutinović, Marija (Beograd , 2021)[more][less]
Abstract: In this dissertation we examine several important objects and concepts in combinatorialtopology, using both combinatorial and topological methods.The matching complexM(G) of a graphGis the complex whose vertex set is the setof all edges ofG, and whose faces are given by sets of pairwise disjoint edges. These complexes appear in many areas of mathematics. Our first approach to these complexes is newand structural  we give complete classification of all pairs (G,M(G)) for whichM(G) is ahomology manifold, with or without boundary. Our second approach focuses on determiningthe homotopy type or connectivity of matching complexes of several classes of graphs. Weuse a tool from discrete Morse theory called the Matching Tree Algorithm and inductiveconstructions of homotopy type.Two other complexes of interest are unavoidable complexes and threshold complexes.Simplicial complexK⊆2[n]is calledrunavoidable if for each partitionA1t···tAr= [n] atleast one of the setsAiis inK. Inspired by the role of unavoidable complexes in the Tverbergtype theorems and GromovBlagojevi ́cFrickZiegler reduction, we begin a systematic studyof their combinatorial properties. We investigate relations between unavoidable and threshold complexes. The main goal is to find unavoidable complexes which are unavoidable fordeeper reasons than containment of an unavoidable threshold complex. Our main examplesare constructed as joins of selfdual minimal triangulations ofRP2,CP2,HP2, and joins ofRamsey complex.The dissertation contains as well an application of the important “configuration space test map” method. First, we prove a cohomological generalization of Dold’s theorem fromequivariant topology. Then we apply it to Yang’s case of Knaster’s problem, and obtain anew simpler proof. Also, we slightly improve few other cases of Knaster’s problem. URI: http://hdl.handle.net/123456789/5184 Files in this item: 1
Marija_Jelic_Milutinovic.pdf ( 5.212Mb ) 
Stojadinović, Tanja (Beograd , 2013)[more][less]
Abstract: Multiplication and comultiplication, which de ne the structure of a Hopf algebra, can naturally be introduced over many classes of combinatorial objects. Among such Hopf algebras are wellknown examples of Hopf algebras of posets, permutations, trees, graphs. Many classical combinatorial invariants, such as M obius function of poset, the chromatic polynomial of graphs, the generalized DehnSommerville relations and other, are derived from the corresponding Hopf algebra. Theory of combinatorial Hopf algebras is developed by Aguiar, Bergerone and Sottille in the paper from 2003. The terminal objects in the category of combinatorial Hopf algebras are algebras of quasisymmetric and symmetric functions. These functions appear as generating functions in combinatorics. The subject of study in this thesis is the combinatorial Hopf algebra of hypergraphs and its subalgebras of building sets and clutters. These algebras appear in di erent combinatorial problems, such as colorings of hypergraphs, partitions of simplicial complexes and combinatorics of simple polytopes. The structural connections among these algebras and among their odd subalgebras are derived. By applying the character theory, a method for obtaining interesting numerical identities is presented. The generalized DehnSommerville relations for ag fvectors of eulerian posets are proven by Bayer and Billera. These relations are de ned in an arbitrary combinatorial Hopf algebra and they determine its odd subalgebra. In this thesis, the generalized DehnSommerville relations for the combinatorial Hopf algebra of hypergraphs are solved. By analogy with Rota's Hopf algebra of posets, the eulerian subalgebra of the Hopf algebra of hypergraphs is de ned. The combinatorial characterization of eulerian hypergraphs, which depends on the nerve of the underlying clutter, is obtained. In this way we obtain a class of solutions of the generalized DehnSommerviller relations for hypergraphs. These results are applied on the Hopf algebra of simplicial complexes. URI: http://hdl.handle.net/123456789/4306 Files in this item: 1
phdTanjaStojadinovic.pdf ( 13.95Mb ) 
Stojadinović, Tanja (Univerzitet u Beogradu , 2014)[more][less]
Abstract: Multiplication and comultiplication, which de ne the structure of a Hopf algebra, can naturally be introduced over many classes of combinatorial objects. Among such Hopf algebras are wellknown examples of Hopf algebras of posets, permutations, trees, graphs. Many classical combinatorial invariants, such as M obius function of poset, the chromatic polynomial of graphs, the generalized DehnSommerville relations and other, are derived from the corresponding Hopf algebra. Theory of combinatorial Hopf algebras is developed by Aguiar, Bergerone and Sottille in the paper from 2003. The terminal objects in the category of combinatorial Hopf al gebras are algebras of quasisymmetric and symmetric functions. These functions appear as generating functions in combinatorics. The subject of study in this thesis is the combinatorial Hopf algebra of hyper graphs and its subalgebras of building sets and clutters. These algebras appear in di erent combinatorial problems, such as colorings of hypergraphs, partitions of sim plicial complexes and combinatorics of simple polytopes. The structural connections among these algebras and among their odd subalgebras are derived. By applying the character theory, a method for obtaining interesting numerical identities is pre sented. The generalized DehnSommerville relations for ag fvectors of eulerian posets are proven by Bayer and Billera. These relations are de ned in an arbitrary com binatorial Hopf algebra and they determine its odd subalgebra. In this thesis, the generalized DehnSommerville relations for the combinatorial Hopf algebra of hy pergraphs are solved. By analogy with Rota's Hopf algebra of posets, the eulerian subalgebra of the Hopf algebra of hypergraphs is de ned. The combinatorial char acterization of eulerian hypergraphs, which depends on the nerve of the underlying clutter, is obtained. In this way we obtain a class of solutions of the generalized DehnSommerviller relations for hypergraphs. These results are applied on the Hopf algebra of simplicial complexes. URI: http://hdl.handle.net/123456789/3745 Files in this item: 1
phdTanjaStojadinovic.pdf ( 13.95Mb ) 
Todić, Bojana (Beograd , 2024)[more][less]
Abstract: This dissertation deals with the coupon collector problem, which in its simplest (classical) form can be formulated as follows: A collector wants to collect a set of n distinct coupons, by buying a single coupon each day. The random variable of interest is the waiting time until the collection is completed. The goal of the dissertation is to propose and analyze three new generalizations of the classical coupon collector problem obtained by introducing additional coupons with special purposes into the set of n standard coupons. The first two chapters are devoted to the results on the classical coupon collector problem and the known generalizations obtained by introducing additional coupons into the coupon set ([1], [2], [39], [54]). New results are presented in chapters 3,4, and 5. The third chapter of the dissertation is dedicated to the case where, in addition to the standard coupons, the coupon set consists of a null coupon (which can be drawn, but does not belong to any collection), and an additional universal coupon, that can replace any standard coupon. For the case of equal probabilities of standard coupons, the asymptotic behavior (as n → ∞) of the expectated value and variance of the waiting time for a fixed size subcollection of a collection of coupons is obtained when one or both probabilities of additional coupons are fixed, and the remaining coupons have equal small probabilities. These results, published in [27], generalize part of the results in [2]. The same problem is analyzed using a Markov chain approach, which led to the determination of the fundamental matrix and some related features of the collection process (probability that the coupon collection process ends in a certin way). These results are contained in the paper [24]. For the case of unequal probabilities of standard coupons, a class of bounds is derived for the first and second moments of the waiting time until the end of the experiment by using majorization techniques and refining the bounds proposed in [51]. The quality of the proposed bounds is tested in numerical experiments, and the specific bounds from the class with the most desirable properties are given. These results are published in [26]. The fourth chapter of the dissertation deals with the generalization in which the additional coupon (so called, penalty coupon) interferes with the collection of standard coupons in the sense that the collection process ends when the absolute difference between the number of collected standard coupons and the number of collected penalty coupons is equal to n. This generalization can be seen as a special case of the random walk with two absorbing barriers. The distribution and a simple upper bound on the first moment of the corresponding waiting time are determined by combinatorial considerations. The application of the Markov chain approach led to obtaining the fundamental matrix. These results are published in [53]. In the fifth chapter of the dissertation another additional coupon (so called reset coupon) is introduced, which acts as a reset button, in the sense that the set of coupons drawn up to time (day) t becomes empty if the reset coupon is drawn on day t+1. In the case of unequal probabilities of standard coupons, the distribution of the corresponding waiting time is obtained by combinatorial considerations. For the case of equal probabilities of obtaining standard coupons, For the case of equal probabilities, applying the first step analysis for the correspondingly constructed Markov chains led to the expressions for the expected waiting time and its simple form in terms of the beta function. These results are used for analysing the asymptotic behavior (when the size of the collection tends to infinity) of the expected waiting time, taking into account possible values of the probability of obtaining a reset coupon. These results are published in [25]. Setting the probabilities of the additional coupons to zero, all three generalizations of the coupon collector problem defined and analyzed in this dissertation as well as the obtained results reduce to the corresponding results for the classical coupon collector problem. URI: http://hdl.handle.net/123456789/5677 Files in this item: 1
Todic_Bojana_disertacija.pdf ( 1.004Mb ) 
Shafah, Osama (Beograd , 2013)[more][less]