Browsing Mathematics by Title
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Tasković, Milan (Belgrade)[more][less]
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Pavlović-Lažetić, Gordana (Belgrade)[more][less]
URI: http://hdl.handle.net/123456789/45 Files in this item: 1
phdGordanaPavlovicLazetic.pdf ( 4.791Mb ) -
Ilić-Dajović, Milica (Belgrade , 1965)[more][less]
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Zekić, Mladen (Beograd , 2021)[more][less]
Abstract: Central place in this thesis occupy the coherence results for certain types of closed categories. Coherence results in category theory usually serve to provide a simple decision procedure for equality of arrows in some category. The approach to coherence that we follow here implies the existence of a faithfull functor from a freely generated category A of certain type to the category B in which an equality of arrows can be easily checked. Category B, which is of the same type as A, usually represents formalisation of some graphical language. Besides coherence, the second most important notion we consider in this thesis is the biproduct. The notion of biproduct in a category incorporates notions of coproduct and product. The main results in this thesis are coherence theorems for three types of closed categories with biproducts – symmetric monoidal closed categories with biproducts, com- pact closed categories with biproducts and dagger compact closed categories with dagger biproducts. Further, we present a new proof of the well-known Kelly-Mac Lane coherence theorem for symmetric monoidal closed categories. The methods we use in that proof are completely proof-theoretical, and one of the key elements in it is the cut-elimination theorem. In all the above coherence results, the graphical language is based on the category of one-dimensional cobordisms. Finaly, we give certain criteria for existence of biproducts in monoidal categories. In this regard, we rely on recent research that characterizes certain type of monoidal categories with finite biproducts by using the existence of right duals of some distinguished objects. Our criteria are a generalization of this result. URI: http://hdl.handle.net/123456789/5307 Files in this item: 1
mladen.zekic.disertacija.pdf ( 1.018Mb ) -
Atanasijević, Ksenija (Belgrade)[more][less]
URI: http://hdl.handle.net/123456789/183 Files in this item: 1
phdKsenijaAtanasijevic.pdf ( 1.987Mb ) -
Arsenović, Mihajlo (Belgrade)[more][less]
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Pevac, Lazar (Belgrade)[more][less]
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Zeada, Samira (Belgrade , 2015)[more][less]
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Zeada, Samira (Beograd , 2015)[more][less]
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Ziada, Samira (Beograd , 2015)[more][less]
Abstract: This thesis has been written under the supervision of my mentor Prof. Aleksandar T. Lipkovski at the University of Belgrade in the academic year 2014- 2015. The topic of this thesis is Classi cation of Monomial Orders In Polynomial Rings and Gr obner Basis. The thesis is divided into four chapters let us review the content and the main contributions to this doctoral thesis. In the rst chapter (page. 1-6), the de nitions, features and examples of the notions basic for this thesis are given. Also, relevant properties of the multivariate polynomial ring and the division with reminder algorithm are recalled. In Chapter 2 (page. 7-22), the classi cation of monomial orderings for multivariate polynomial rings is displayed (given) in detail, with special emphasis on the case of two variables. The connection of this classi cation and the well known classi cation of Robiano is exposed in detail. It is an unusual and a little-known fact that the set of di erent monomial orderings with the natural topology is a Cantor set. Chapter 3 (page. 23-40) and Chapter 4 (page. 41-51) contain the main contributions of the thesis. In Chapter 3, the new approach to the analysis of the division with reminder algorithm is presented, based on set theoretic partial orderings (Section 3.3). Thus, the new evidence for (proof of ) Buchbergers result on niteness of the division procedure, regardless of a choice of the leading terms for the next dividing, is obtained. Likewise, in order to investigate Hilbert's original contribution and links with later considerations, we present his proof of the famous Hilbert basis theorem, on the basis of his original papers (section 3.5). In Chapter 4 (page. 41-51), a result from Chapter 3 is used for another new characterization of Gr obner basis, apparently weaker than well-known ones, to be obtained. Namely, it is shown that the bases G = ff1; : : : ; fkg of an ideal I is Gr obner if and only if, for any f from I, the support SuppG(f) is not empty. This condition is (only outwardly) weaker than a typical condition that the leading term of f belongs to SuppG(f). And describe the Gr obner fan of an ideal I and give an algorithm In a special case of two variables. URI: http://hdl.handle.net/123456789/4304 Files in this item: 1
phdSamiraZiada.pdf ( 465.6Kb ) -
Danić, Dimitrije (, 1885)[more][less]
Abstract: Tema Danićeve disertacije su konformna preslikavanja eliptičkog paraboloida na ravan, prateći definicije i formalizam koje je uveo Gaus (F. Gauss) za tu vrstu preslikavanja. U tom razmatranju izveo je određene parcijalne diferencijalne jednačine koje takođe analizira i rešava. Njegov doprinos bile su metode u rešavanju kompleksnih eliptičkih integrala, uvođenju eliptičkih transformacija i primeni eliptičkih funkcija u rešavanju ovih parcijalnih jednačina. URI: http://hdl.handle.net/123456789/4796 Files in this item: 3
DDanic_thesis_documentation.pdf ( 239.6Kb )DDanic_thesis_transl_SRB.pdf ( 1.764Mb )DDanic_thesis.pdf ( 1.420Mb ) -
Mijajlović, Žarko (Belgrade , 1977)[more][less]
Abstract: Part1. Basic notions of model theory are given. Part2. Dual notions in categories of Boolean algebras and Stone spaces are studied in respect to natural contra-variant functor. The cellularity number of a Boolean algebra B, celB is studied, certain cardinal properties are proved, e.g. it is consistent with ZFC that celB is attained for every Boolean algebra B. Part3. Lindenbaum algebras of first-order theories are studied in details. It is proved that every Boolean algebra is isomorphic to the Lindenbaum algebra B1 of Σ1 formulas of certain first-order complete theory. Stability number ST(k) of a first-order theory T is studied, and it is shown that ST(k) = Ku(k), where Ku(k) is the Kurepa number (Kurepa introduced it in 1935) and T is the theory of dense linear ordering without end-points, while the cardinality of the Stone space of B1(A), A is a model of T, is equal to ded(A), the Dedekind number of the ordering A. Ku(k)= sup{ded(A): A is a model of T, |A|=k}. Part4. Σn Πn ramifications of various notions in model theory are defined and studied, e.g. elementary embeddings, completeness, chains, direct limits, diagram properties, etc. Preservation theorems for these types of formulas are proved. Examples for including ordered structures and algebraic fields are given. Part5. Model completions and elimination of quantifiers are studied. As an application, it is proved that by means of model theory that the classes of Boolean algebras and distributive lattices with the least and the greatest elements are Jonsson’s classes. Algebraic description of saturated models of submodel-complete theories are given, unifying results of Haussdorff (dense linear ordering), Erdös, Gillman (ordered fields) and Boolean algebras (Negrepontis) for homogeneous-universal models. Part6. Here is studied what model-theoretic properties are absolute in ZF in the sense introduced by Levy, i.e. in which cases strong hypothesis (AC, GCH, V=L) can be eliminated from the proof of these properties. It is shown that the following properties of first-order theories are absolute: the consistency, completeness, model-completeness and elimination of quantifiers. These gives new light on model-theoretic proofs of these properties. URI: http://hdl.handle.net/123456789/196 Files in this item: 1
phdZarkoMijajlovic.pdf ( 27.39Mb ) -
Božić, Milan (Belgrade , 1983)[more][less]
Abstract: The thesis consists of five chapters. In the introductory chapter some relational-operational structures building of sets of formulas in calculi RA^+ and R^+ are presented. These structures are used in other chapters in making of the canonical frames of Kripke’ type for semantic of positive fragments of relevant propositional calculi. In Chapter 1, the semantic of these positive fragments, which is a mixture of known Routley & Meyer’s and Maksimova’s semantics, is presented. A new way for the semantic of negation in relevant logics is introduced in Chapter 2. It this way semantic completeness theorems for a large class of expansions of the logic R_min are proved. It is shown that Routley & Meyer’s semantic for the logic R is a special case of that semantic. Relevant modal logics are studied in Chapter 3. A semantic of Kripke’s type by which a completeness of a large class of modal logics whose basics are different relevant calculi with or without negation is introduced. A characterization of a large class modal i.e. Hintikka, schemas is given too. They contain almost all modal schemas characterized by formulas of the first order. Moreover, it is proved that the only known semantic for the calculus R_⊙ is a special case of the semantic given in this chapter. Semantics of relevant calculi which are not distributive are studied in Chapter 4. It is shown that semantics of non-distributive relevant logics radically change Kripke’s semantic. URI: http://hdl.handle.net/123456789/340 Files in this item: 1
phdMilanBozic.pdf ( 36.24Mb ) -
Tošić, Ratko (Belgrade , 1978)[more][less]
Abstract: The thesis consists of five chapters. In Chapter 1 definitions and well-known results from the theory of Boolean algebras and Boolean functions are given. In Chapter 2 of the thesis some properties of Boolean functions, which preserve constants under finite Boolean algebras, are presented by using the component representation. Their consequences about the number of Boolean’s functions are also given. The theorems which are the generalization of Scognmaiglio’s theorem and Andreoli’s theorem for Boolean functions with one variable, are proved in Chapter 3. The following new notions are introduced for monotone logical functions: the profile, the level, homogeneous, the corresponding matrix, etc. Some properties of these functions are shown and some consequences about the number of homogeneous monotone logical functions are presented. In Chapter 4 the applications of monotone Boolean functions in solving the problems of search theory (a branch of the theory of information) are presented. It is shown that the general problem of a type is, in fact, the problem of identifications of homogeneous monotone Boolean functions of the given profile by checking the value of that function for combinations of values of variables. Optimal or almost optimal solutions for some profiles are shown. It is also shown that monotonic logical functions are natural instrument for the generalization of these problems. Some open problems are presented in Chapter 5. URI: http://hdl.handle.net/123456789/356 Files in this item: 1
phdRatkoTosic.pdf ( 3.843Mb ) -
Vujošević, Slobodan (Belgrade , 1981)[more][less]
Abstract: The thesis consists of three chapters. Heyting algebras are studied as an equality category in Chapter 1. The properties of filters and ideals of Heyting algebras are presented together with corresponding properties in distributive nets and Boolean algebras. Free, injective and projective Heyting algebras are presented and a theorem about the representation in algebras with closing is proved. Some properties of Heyting algebras, which are important for study of formal logics closely to intuitionistic logic, are also presented. Complete Heyting algebras are studied in Chapter 2. The family of complete Heyting algebras is obtained by repeating of the construction of the algebra of J-operators. Some properties of this family when the initial Heyting algebras is linear order, are studied. Moreover, the characterization of complete Heyting algebras which can be approximated by complete Boolean algebras is given. Duality of categories of topological spaces and complete Heyting algebras are studied in Chapter 3. Some adjunctions are defined, and for those adjunctions the actions of monad and comonad are studied. It is shown that the category of complete Heyting algebras is reflective in the categories of sets, distributive bounded nets and complete Heyting algebras. It is shown that complete Heyting algebras correspond to "deposited" spaces, and distributive bounded nets correspond to a restriction of Ston’s spaces. URI: http://hdl.handle.net/123456789/89 Files in this item: 1
phdSlobodanVujosevic.pdf ( 3.553Mb ) -
Boričić, Branislav (Belgrade , 1983)[more][less]
Abstract: The thesis consists of four chapters. Chapter 1 contains a general framework for deductive systems and contains a sequence-conclusion natural deduction system for classical first order logic. A sequence NLC_n of intermediate propositional logics is considered in Chapter 2. It is shown that the sequence NLC_n contains three different systems only. These are the classical calculus NLC_1, Dummett's system NLC_2 and the logic NLC_3, an extension of the Heyting propositional logic by the axiom (A⇒B)∨(B⇒C)∨(C⇒A) . It is also shown that the logic NLC_3 is separable. In the sequel, the completeness of NLC_3 with respect to the corresponding Kripke type models having the property that ∀x∀y∀z(xRy∨yRz∨zRx) is proved, as well as its decidability and the independence of logical connectives. It is shown that some subsystems of NLC_3 are separable and that the limits of the considered systems is the Heyting propositional calculus. The logic of the weak law of excluded middle, an extension of the Heyting logic by ¬A∨¬¬A, is considered in Chapter 3. An embedding of classical logic into this logic is described and it is proved that this logic is the minimal one having this property. A Hilbert-type formulation of implication fragment of the Heyting propositional logic formalizing the deducibility relation, is presented in Chapter 4, enabling to define a decision procedure based on a kind of cut-elimination theorem. URI: http://hdl.handle.net/123456789/257 Files in this item: 1
phdBranislavBoricic.PDF ( 5.899Mb ) -
Antić, Miroslava (Belgrade)[more][less]
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Ćirić, Miroslav (Beograd , 1991)[more][less]
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Živaljević, Rade (Belgrade , 1983)[more][less]
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Vukićević, Petar (Berlin , 1894)[more][less]