http://hdl.handle.net/123456789/12 http://hdl.handle.net/123456789/5834 Verovatnosni računi sekvenata i klasifikacija neklasičnih logika zasnovana na entropiji Boričić, Marija After a brief introductory survey, this work is divided into two parts. The first part deals with presence of probability in logic (v. [16], [17], [18], [19], [22], [23] and [24]), and the second one is devoted to the application of entropy in classification of many–valued logics (v. [14], [15], [20], [21] and [25]). The basic idea, dominant in the first part of the work, is to enrich the Gentzen’s sequent calculus LK for propositional classical logic by a kind of probability operator defined over the sequents ¡ ` ¢ in order to express the fact that ”the truthfulness probability of ¡ ` ¢ belongs to the interval [a; b] µ [0; 1]”. We introduce the following four systems: LKprob, LKprob("), NKprob and LKfuzz. The basic form of sequents in LKprob is ¡ `b a ¢ with the above given intended meaning. The system LKprob(") is focused on the Suppes’ forms ¡ `n ¢ enabling to formalize the sentence ”the truthfulness probability of ¡ ` ¢ belongs to the interval [1 ¡ n"; 1] µ [0; 1]”, for some n 2 N. The system NKprob presents a natural deduction counterpart of the sequent calculus LKprob. The models founded on Carnap– Popper–Leblance probability semantics are defined for each of these calculi and accompanied by the corresponding soundness and completeness results. Finally, the calculus LKfuzz is introduced with a more general form of the sequents ¡ `x ¢, where x is an element of a finite lattice, with the aim to describe a fuzzification of LK. The meaning of ¡ `x ¢ is that ”x is the fuzziness measure of ¡ ` ¢”. Models for LKfuzz are given with soundness and completeness results, and a proof– theoretical treatment of LKfuzz includes the cut–elimination theorem. The second part of the work explores the fact that each logical system associated with the partition induced by the corresponding Lindenbaum–Tarski algebra makes it possible to define its entropy. We define the entropy of a logical system based on geometric distribution of measures over matching partition of set of formulae. This iv definition enables the classification of many–valued propositional logics according to their entropies. Asymptotic entropy approximations for some infinite–valued logics are proposed as well. The considered examples include Lukasiewicz’s, Kleene’s and Priest’s three–valued logics, Belnap’s four–valued logic, G¨odel’s and McKay’s m– valued logics, and Heyting’s and Dummett’s infinite–valued logics. http://hdl.handle.net/123456789/5833 Ostatak određenih tipova Gausovih kvadraturnih formula sa specijalnim klasama težinskih funkcija Mihić, Ljubica Mathematical modeling of many phenomena which occur in the natural, technical sciences, economy requires signi cant knowledge of the theory of numerical integration. In the situations where the integral cannot be determined analytically, it is necessary to construct the for- mula which approximates its value with acceptable error. Besides the traditional formulae, the tendencies in the development of this area refer to increment of algebraic degree of precision of the quadrature formula and its error estimation. The subject of this dissertation is Gaussian quadrature formula with increased number of nodes which may have double multiplicity (sections 2 and 4) or arbitrary multiplicity (section 3). When we consider the addition of one node, one of the integration interval ends, the corresponding quadrature formula is called Gauss-Radau, and in the case of adding two new nodes, the appropriate formula is of Gauss-Lobatto type. Beside those formulae, in this dissertation we consider the Kronrod's extension of Gauss-Tur an quadrature formula, which includes the derivatives of the function at nodes. Here, the main aim is to deter- mine efficient error bounds of these formulae with respect to the different types of Chebyshev weight functions and Gori-Micchelli weight function which, in one special case, represents the generalization of the rst Chebyshev weight function. Introductory theorems and well-known statements that are used in the rest of the thesis are presented in the rst section. We described basic properties of orthogonal polynomials participating in the construction of Gaussian quadrature formula. Furthermore, the different types of remainder terms of quadrature formula are de ned, whereby in the following sections we analyze the integral representation of the remainder term. The recent results relating to the increasing number of the nodes of arbitrary multiplicity are exposed. The remaining three sections contain the chapters with original results. The second section is devoted to the estimation of the error of Gauss- Lobatto quadrature formulae with a double nodes at the endpoints. Af- ter detailed analysis of the kernel of the remainder term (see subsection 2.2.1), we proved the Theorems stating the behavior of the maximum modulus of the kernel. We computed xed values mentioned in the The- orems and established the real errors and the error bounds of quadrature formula. In third section we examine the Kronrod's extension of Gauss- Tur an quadrature formulae with respect to the special kind of Gori- Micchelli weight function. This extension, beside the nodes of Gauss- Tur an quadrature formula, also contains the zeros of Chebyshev polyno- mial of the second kind. In those new nodes, we consider just the values of the function (not the values of its derivatives). Previously mentioned Kronrod's extension is actually Chakalov-Popoviciu quadrature formu- la, so, by applying the methods for those formulae, we computed the coefficients and the error of Kronrod's formula. Gautschi-Li hypotheses are proven in the fourth section. Those hypotheses concern the Gauss-Radau quadrature formulae with respect to the Chebyshev weight functions of the second and the third kind. Subsection 4.4 consists of numerical examples and the tables with efficient error bounds. http://hdl.handle.net/123456789/5828 POLINOMIJALNA ENTROPIJA INDUKAOVANIH DINAMIČKIH SISTEMA NA HIPERPROSTORIMA Đorić, Maša In this thesis, we applied a method for computing the polynomial entropy of dynamical systems with a finite non-wandering set to certain hyperspaces of the interval and the circle. We subsequently generalized these results to the case where the non-wandering set is not finite. We proved that the polynomial entropy of the induced map C(f ) is equal to 2, that of Fn(f ) is equal to n, and that of 2f is infinite, when f is a homeomorphism of the interval or the circle. Furthermore, we generalized these results for the n-fold symmetric product of the interval and the circle to an arbitrary compact metric space X and extended them to the suspension of the space Fn(X). In addition to the results on hyperspaces, we presented results concerning polynomial entropy on certain one-dimensional continua, such as local dendrites and regular curves. We also constructed a family of pointwise periodic homeomorphisms on a continuum whose polynomial entropy attains every value in [0, +∞]. http://hdl.handle.net/123456789/5816 Novel goodness-of-fit tests based on independence-type characterizations Halaj Mileusnić, Katarina The main objective of this dissertation is to develop novel goodness-of- fit tests based on independence characterizations and to investigate their proper- ties. The starting point is the fact that certain distributions can be uniquely char- acterized by the independence of suitable functions of random variables. This observation makes it possible to reduce goodness-of-fit testing for a hypothe- sized model to the problem of detecting departures from the corresponding in- dependence condition. Accordingly, the proposed test statistics are constructed by comparing empirical versions of joint functionals with the product of the cor- responding marginal functionals, where the resulting discrepancy serves as the basis for new goodness-of-fit tests for different classes of distributions. One of the contributions of the dissertation is the development of goodness- of-fit tests for the geometric distribution based on Ferguson’s independence char- acterization. In this setting, the general framework of comparing joint and marginal empirical functionals is implemented through V-empirical probability generating functions. Furthermore, a novel approach to goodness-of-fit testing for absolutely continuous distributions is proposed, relying on the discrepancy between the joint U-empirical distribution function of an appropriately defined random vector and the product of its marginal U-empirical distribution func- tions. Another part of the research focuses on circular data. A goodness-of-fit test for the wrapped normal distribution is proposed, where the general methodology is adapted to the periodic structure of observations on the circle. The theoretical part of the dissertation is devoted to the derivation of the asymptotic properties of the proposed test statistics, drawing on the theory of U- and V- statistics as well as on the theory of U-empirical processes. An addi- tional contribution is the establishment of new asymptotic results for finite linear combinations of weakly degenerate V-statistics, both in settings with known pa- rameters and in settings involving parameter estimation. These findings provide the theoretical foundation for a class of combined tests in which appropriately chosen weights regulate the contribution of individual components and can be used to enhance the overall efficiency of the testing procedure. The practical performance of the proposed methods will be assessed through extensive simulation studies and the analysis of several real data sets. The sim- ulation study will investigate the finite-sample properties of the tests under the null hypothesis and a range of relevant alternatives, while the real-data applica- tions will demonstrate their usefulness in practical statistical problems. In this way, the empirical part of the dissertation will complement the theoretical de- velopments and provide evidence of the practical applicability of the proposed methodologies.