Browsing Doctoral Dissertations by Title
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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 ) -
Berković, Mladen (Beograd , 1977)[more][less]
URI: http://hdl.handle.net/123456789/4100 Files in this item: 1
Konacni_elementi_membrana.PDF ( 3.010Mb ) -
Berković, Mladen (Belgrade , 1977)[more][less]
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Shafah, Osama (Beograd , 2013)[more][less]
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Protić, Petar (Novi Sad , 1986)[more][less]
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Vulanović, Relja (Novi Sad , 1986)[more][less]
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Šarac, Marica (Belgrade)[more][less]
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Romano, Daniel (Belgrade , 1985)[more][less]
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Veljković, Kristina (Beograd , 2016)[more][less]
Abstract: Design of control chart for monitoring central tendency of nongaussian random variables with symmetric or positively skewed distributions is considered. In the case of nongaussian symmetric distributions, modified X bar control chart is proposed in this dissertation. For chosen Student, Laplace, logistic and uniform distributions, theoretical distribution of the standardized sample mean is calculated and approximated with Pearson type II or Pearson type VII distributions. Width of control limits and power of X bar control chart are established, for a given probability of type I error. The results imply that the corresponding Pearson distribution represents very good approximation of the distribution of the standardized sample mean. For implementation of X bar control chart in practice, measures of sample kurtosis are compared and the usage of proposed chart is illustrated on given data. In the case of positively skewed distributions, one sided median control chart for monitoring central tendency of quality characteristics is proposed in this dissertation. For chosen exponential, gamma and Weibull distributions, theoretical distribution of sample median is calculated and approximated with Pearson type I or Pearson type VI distributions. Calculated values of upper control limits and power of median control chart for theoretical distribution of sample median and corresponding Pearson distribution are very close. For implementation of median control chart in practice, measures of sample skewness and sample kurtosis are compared and then proposed median chart is constructed for given data. Besides the statistical design of control charts for monitoring central tendency of nongaussian random variables, their optimal economic statistical design is also considered. Use of genetic algorithms for constrained minimization of expected loss function is proposed in this dissertation. Same symmetric distributions as in the case of statistical design of the X bar control chart and positively skewed distributions as in the case of statistical design of median control chart are chosen. For all chosen distributions of quality characteristic, a corresponding Pearson distribution gives results very close to results based on the theoretical distribution of the standardized sample mean (sample median). URI: http://hdl.handle.net/123456789/4449 Files in this item: 1
Kristinateza.pdf ( 2.651Mb ) -
Zeković, Ana (Beograd , 2015)[more][less]
Abstract: A main focus of the paper is construction of new methods for defining diverse knot distance types - the distance of knots made by crossing changes (Gordian distance) and the distance among knots made by crossing smoothing (smoothing distance). Different ways of knots presentation are introduced, with objective to a mirror curve model. It is presented a purpose of the model, coding of knots, by using the model preferences, as well as introduction of a method to determinate a knots presented by the model and derived all the knots that could be placed to a nets dimensions p×q (p ≤ 4, q ≤ 4). Diverse knot notations are described into details, with a focus to Conway’s notation and its topological characteristics. As it is known, a present algorithms are based on an algebra of chain fractions, that are in close relation with a presentation of rational knots, which results in an absence of a huge number of non-rational knots, in an existing Gordian’s distance tables. The subject of the paper is an implementation of methods with bases on determination of new distances equal 1. The methods are based on a non-minimal presentation of rational and non-rational knots, generation of algorithms established on geometrical characteristics of Conway’s notation and a weighted graph search. The results are organized into Gordian’s distance knots tables up to 9 crossings, and have been enclosed with the paper. In order to append the table with knots having a bigger number of crossings, it has been suggested a method for extension of results for knot families. Using facts of relation among Gordian’s numbers and smoothing numbers, a new method for smoothing number determination is presented, and results in a form of lists for knots not having more then 11 crossings. In conjunction with Conway’s notation concept and the method, algorithms for a smoothing distance are generated. New results are organized in knot tables, up to 9 crossings, combined with previous results, and enclosed with the paper. A changes and smoothing to a knot crossing could be applied for modeling topoisomerase and recombinase actions of DNA chains. It is presented the method for studying changes introduced by the enzymes. A main contribution to the paper is the concept of Conways notation, used for all relevant results and methods, which led to introduction of a method for derivation a new knots in Conways notation by extending C-links. In a lack of an adequat pattern for an existing knot tables in DT-notation, there is usage of a structure based on topological knot concepts. It is proposed a method for knot classification based on Conways notation, tables of all knots with 13 crossings and alternated knots with 14 crossings has been generated and enclosed. The subject of the paper takes into consideration Bernhard-Jablan’s hypothesis for a determination of unknotting number using minimal knot diagrams. The determination is crucial in computation of diverse knot distances. The paper covers one of main problems in knot theory and contains a new method of knot minimization. The method is based on relevance of local and global minimization. 5 There are defined new terms such as a maximum and a mixed unknotting number. The knots that do not change a minimum crossing number, after only one crossing change are taken into consideration for the analyzes. Three classes of the knots are recognized, and called by authors . Kauffman’s knots, Zekovic knots and Taniyama’s knots. The most interesting conclusion correlated with Zekovic knots is that all derived Perko’s knots (for n ≤ 13 crossings) are actually Zekovic knots. Defining this class of knots provides opportunity to emphasize new definitions of specifis featured for well-known Perko’s knots. URI: http://hdl.handle.net/123456789/4255 Files in this item: 1
phdZekovicAna.pdf ( 5.246Mb ) -
Dražić, Milan (Beograd , 1995)[more][less]
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Dražić, Milan (Beograd , 1995)[more][less]
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Takači, Arpad (Novi Sad , 1981)[more][less]
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Mitić, Ljubiša (Belgrade)[more][less]
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Nikić, Mioljub (Belgrade)[more][less]
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Demčenko, Vasilije (Belgrade , 1924)[more][less]
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Stojanović, Rastko (None , 1956)[more][less]
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Vujičić, Veljko (Belgrade , 1961)[more][less]
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Radosavljević, Jovan (Beograd , 2023)[more][less]
Abstract: Graph G = (V,E) is an ordered pair of set of nodes V and branches E. Order graph G is the number of nodes |V |, and its size is the number of branches |E|. Knots u, v ∈ V are adjacent if there is a branch uv ∈ E between them. Distance dist(u, v) nodes u and v G is the length of the shortest path from u to v. The diameter of the graph G is the largest distance dist(u, v) let two nodes in, v. They are discussed in the dissertation graphs of diameter 2. Intuitively, the notion that graphs are dia- meters 2 simple structures; however, they are known to be asymptotically close all graphs of diameter 2. That is why a narrower class is interesting — class D2C of critical graphs of diameter 2, i.e. graphs where the removal of any branches leads to an increase in diameter. In addition, a narrower class of pri- mitive D2C (PD2C) graphs, i.e. D2C graphs that do not have two nodes with the same set of neighbors. In the introductory chapter 2, the basic concepts, algorithms and dings used in the dissertation. They are presented in the following chapters original results regarding diameter graphs 2. Chapter 3 describes the procedure for obtaining a list of D2C graphs of order up to 13. With built-in parallelization, the creation of a list of D2C graphs of order up to 13 it lasted a month. This was a step forward, because previously there was a spi- around all graphs of diameter 2 lines up to 10. The obtained results were used for testing several known hypotheses about graphs of diameter 2. In chapter 4 it is shown that for every m ⩾ 3 a D2C graph containing cli- a ku of size m must have at least 2m nodes. At the same time, with accuracy up to isomorphism, there is exactly one graph of size 2m that contains a clique of characters m. Chapter 5 discusses PD2C graphs with the smallest number of branches. From list of all PD2C graphs of order n ⩽ 13 are selected PD2C graphs of size at most 2n − 4. Only three of the isolated graphs are of size 2n − 5, which is in accordance with the statement of the Erdes-Renji theorem about the lower bound for the size graphs of diameter 2 that do not contain a node adjacent to all other nodes (that limit is 2n − 5). PD2C graphs of size 2n − 4 rows up to 13 sorted are in three groups: • The first group belongs to the Z family, defined in the dissertation, which for each n ⩾ 6 contains exactly one PD2C graph of order n of size 2n − 4. • The second group consists of seven Hamiltonian PD2C graphs of order at most 9 of size 2n−4. In the dissertation it was proved that there is no such Hamil- tone graph of order greater than 11, i.e. that the seven graphs found are the only ones Hamiltonian PD2C graphs of size 2n − 4. • The third group consists of a unique graph that does not belong to any of the first two groups. Based on these results, the hypothesis was formulated that all PD2C graphs re- that n ⩾ 10 and sizes 2n − 4 belong to the family Z. Keywords: graphs, critical graphs of diameter 2, primitive graph- You Scientific field: Computing and informatics Narrower scientific field: Graph theory UDC number: 004.415.5(519.1 URI: http://hdl.handle.net/123456789/5594 Files in this item: 1
disertacijaJovanRadosavljevic.pdf ( 746.0Kb )