Browsing by Title
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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 ) -
Majstorović, Ivana (Beograd , 2020)[more][less]
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Cvetković, Dragoš (Belgrade , 1987)[more][less]
URI: http://hdl.handle.net/123456789/460 Files in this item: 1
book001DragosCvetkovic.pdf ( 15.14Mb ) -
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 com-plexes 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 calledr-unavoidable if for each partitionA1t···tAr= [n] atleast one of the setsAiis inK. Inspired by the role of unavoidable complexes in the Tverbergtype theorems and Gromov-Blagojevi ́c-Frick-Ziegler reduction, we begin a systematic studyof their combinatorial properties. We investigate relations between unavoidable and thre-shold 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 self-dual 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 well-known 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 Dehn-Sommerville 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 Dehn-Sommerville relations for ag f-vectors 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 Dehn-Sommerville 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 Dehn-Sommerviller 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 well-known 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 Dehn-Sommerville 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 Dehn-Sommerville relations for ag f-vectors 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 Dehn-Sommerville 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 Dehn-Sommerviller 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 ) -
Telebak, Vladimir (MATEMATIČKI FAKULTET UNIVERZITETA U BEOGRADU , 2011)[more][less]
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Muminović, Muhamed (Savez astronomskih društava BiH, ASTRONOMSKA OPSERVATORIJA, Sarajevo , 1985)[more][less]
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Kršljanin, V.; Dimitrijević, M.; Vince, I.; Vujnović, V.; Jankov, S.; Teleki, Dj.; Čabrić, N.; Protić - Benišek, V.; Janković, N.; Protić, M.; Arsenijević, J.; Prosen, M.; Djokić, M. (ASTRONOMSKA OPSERVATORIJA U BEOGRADU i ASTRONOMSKO DRUŠTVO 'Rudjer Bošković' , 1986)[more][less]
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Bačević, Stefan (Beograd , 2017)[more][less]
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Đurđević, Dragan (Beograd , 2013)[more][less]
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Stanković, Danijela (Beograd , 2018)[more][less]
URI: http://hdl.handle.net/123456789/4693 Files in this item: 1
masterStankovicDanijela.pdf ( 2.355Mb ) -
Savović, Vjera (Beograd , 2023)[more][less]
URI: http://hdl.handle.net/123456789/5612 Files in this item: 1
v1_masterVjeraSavovic.pdf ( 655.3Kb ) -
Mandić, Slobodan (Beograd , 2008)[more][less]
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Karamata, Jovan (Belgrade)[more][less]
URI: http://hdl.handle.net/123456789/443 Files in this item: 1
JovanKaramataKompleksanBroj.pdf ( 45.35Mb ) -
Urošević, Marina (Beograd , 2022)[more][less]
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Lazović, Milena (Beograd , 2017)[more][less]
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Babić, Sladjana (MATEMATIČKI FAKULTET UNIVERZITETA U BEOGRADU , 2009)[more][less]
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Avalić, Dijana (Beograd , 2015)[more][less]
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Đurica, Marina (Beograd , 2017)[more][less]