http://hdl.handle.net/123456789/4 http://hdl.handle.net/123456789/5840 OCJENE NORME INTEGRALNIH OPERATORA NA PROSTORIMA BESOVA I BLOHA Vujadinović, Đorđije In this work we investigate the various norms of known integral operators such as Bergman projection and certain integral operators induced with Green function. We have observed acting of the Bergman projection on the weighted Lebesgue spaces with image in spaces of analytic functions, Bloch and Besov spaces. We obtained the estimate for the norm of the Bergman projection onto the Besov spaces, when 1 < p < ∞, and this estimate is asimptoticaly sharp for p → +∞. We have found norm for the special Hilbert case p = 2. We have also considered some weak type inequalities for the norm of the Bergman projection onto Besov space B1 .We ob- served the adjoint operator for the Bergman projection fot the mentioned case B1. Precisely, in this case we obtained two-sided bounds of norm for the adjoint oper- ator. It was represented a method for determinig the norm of the Bergman type projection defined in weighted Lebesgue spaces and with image in Bloch space in the unit ball B in Cn. Specially, we have considered the norm of the integral operator induced with Green function in a context of the weak solutions for the Poisson’s equation with ho- mogenuous condition in the unit ball in higher dimensions. The norm for the men- tioned operator was determined for the Hilbert case and when operator is defined on L∞−spaces (and also L1−case, which is dual).Final results are related to estimating the norm of gradient for the weak solutions of Poisson’s equation. http://hdl.handle.net/123456789/5839 USAVRŠAVANjE MODELA TESNO DVOJNIH SISTEMA SAGLASNO SA REZULTATIMA POSMATRANjA VISOKE PRECIZNOSTI Čeki, Atila The dissertation presents a program for modeling binary systems, Infinity, which combines the traditional principles of modeling (Roche geometry, von Zeipel’s law of temperature distribution and so forth) with a novel geometrical representation of the system. The study of binary stars has a long tradition in Serbia, mostly due to the research done by dr Gojko Đurašević. The program presented in the dissertation is based on his program for modeling of active close binary systems with an accretion disk (Đurašević, 1991) and brings several significant improvements that can be classified in two categories: The first group of improvements has to do with generalizing the model. First and foremost, the new program is applicable to systems with eccentric orbits, and is capable of computing and solving the inverse problem for radial velocities. It also features a new algorithm for determining which elements of the stellar surface are visible during the eclipse. The new algorithm allows for computing the light curves with greater precision using a method of ‘adaptive subdivision’ in the region of eclipse. The geometrical aspects of modeling the accretion disk are also improved and now include the possibility to create both conical and toroidal disks. The second group of improvements is related to the modeling of physical processes on the binary system components. The most significant change is in the way the emergent flux is calculated. Namely, the Đurašević program assumes that the elements of stellar surface produce blackbody radiation, while the new program assigns a one-dimensional Kurucz stellar atmosphere model calculated under the LTE assumption. Improvements are also made in the treatment of the influence of photometric filters on the calculated light curve, and the list of supported filters is extended. A binary system is described by the Roche model, which assumes that the motion and shape of the components are determined by the superposition of gravitational and centrifugal forces, while all the other interactions are neglected. The stars are treated as point masses (under the assumption of total central condensation), so their internal structure is not considered. If the system contains an accretion disk, it is assumed that its mass can be neglected, and that its presence doesn’t influence the dynamics of the system and the shape of the stars. The surface of the components is represented with a polygonal mesh. The stars are approximated with a geodesic, and the accretion disk with an axially-symmetric mesh. The new algorithm for solving the problem of visibility detection, proposed in the dissertation, is based on the so called inverse painter’s algorithm that originates from the field of computer graphics. The most important property of this algorithm is that, other than the polygonal meshes representing the components, it doesn’t require any other information about the geometry of the system. This makes the modeling program easily extendable to more exotic geometric configurations such as systems with eccentric or circumbinary disks, planetary and multiple systems, and so forth. It is assumed that within a single element of stellar surface all the physical quantities – local effective temperature, gravitational acceleration, emergent flux and radial velocity – are constant. Its radiation is affected by gravity darkening, limb darkening and the reflection effect. Gravity darkening is described using von Zeipel’s law, and the limb darkening can be calculated using one of the five available approximations (linear, logarithmic, square-root, quadratic and nonlinear fourth-order approximation). The reflection effect, as well as the temperature distribution along the accretion disk, are calculated using the approach from the Đurašević program (Đurašević, 1991). The local amount of radiation can additionally be affected by active regions that are represented with dark or bright circular spots. The emergent flux is computed as the sum of fluxes from all the surface elements visible in the given moment, and the radial velocity is calculated from the superposition from three kinds of motion: the proper motion of the system center of mass, the motion of the center of the star around the center of mass due to orbital revolution, and the motion of elementary surface around the center of the star due to rotation. Synthetic light and radial velocity curves obtained in this way can be directly compared to observations. The new program solves the inverse problem – estimating the optimal model parameters to fit the given observations – using the Nelder-Mead simplex algorithm to minimize the sum of squares of deviations between the observed and calculated light and radial velocity curves. The validity of the program and the proposed improvements is done by analyzing several binary systems and comparing the results with published solutions obtained with the Đurašević program. http://hdl.handle.net/123456789/5838 EXTREME VALUES IN SEQUENCES OF INDEPENDENT RANDOM VARIABLES WITH MIXED DISTRIBUTIONS Ehfayed, Shneina This thesis has been written under the supervision of my mentor Prof. Dr. Pavle Mladenović at the University of Belgrade, Faculty of Mathematics in the academic year 2012-2013. The title of this thesis is “Extreme values in sequences of independent random variables with mixed distributions”. For good survey of the field, see Resnick, S. I. [25] and Samorodnitsky, Taqqu [27]. The thesis is divided into two chapters. Chapter 1 is divided into 7 sections. In this chapter, we focus on classical results in extreme value theory. We discuss maxima and minima in the first section, univariate extreme value theory in the second section, max-stable distributions in the third section, peaks over threshold models in the fourth section, domain of attraction of the extremal type distributions in the fifth section, tails in the sixth section and tail equivalence in the seventh section. Chapter 2 is divided into 9 sections. In this chapter, we discuss mixed distributions in the first section, mixture of normal distributions in the second section, mixture of Cauchy distributions in the third section, stable distributions in the fourth section, properties of stable random variables in the fifth section, infinitely divisible distributions in the sixth section. Sections 7 and 8 contain the main results for this dissertation. Mixtures of stable distributions are described in the seventh section and mixtures of an infinite sequence of independent normally distributed variables in the eighth section. Conclusion and future research are in the ninth section. http://hdl.handle.net/123456789/5837 MODELIRANJE ZVEZDANIH OSCILACIJA U TESNO DVOJNIM ZVEZDAMA Latković, Olivera The dissertation deals with modeling of stellar oscillations in close binary systems. This topic is импортант not only in for the study of binary stars, but also for asteroseismo-logy. One of the key problems of asetroseismology is the “mode identification” of stellar oscillations. Mode identification is the process of associating an oscillation frequency measured from photometric or spectroscopic observations with a spherical harmonic of degree and order which describes the spatial characteristics of the mode. The main obstacle for this crucial step between the collection and interpretation of data is the symmetry inherent to the patterns of oscillations that makes the periodic changes in the brightness and the radial velocities of surface elements cancel each other out, resulting in diminishing of observed amplitudes. Low sensitivity of both photometry and spectroscopy to high degree modes ( and in photometric and spectroscopic observations, respectively), is one of the consequences of this phenomenon, making the detection and identification of oscillations a difficult task. A method for mode identification by grid-fitting spherical harmonics to photometric light curves of pulsating stars inside eclipsing binary systems is developed in the dissertation. The eclipse of the pulsating star by its binary companion has visible effects on the observed properties of the oscillations because the eclipse spoils the symmetry of oscillation pattern, producing a modulation of frequency, amplitude and phase unique for each mode that can be recognized and fitted. Fitting, however, requires a model of a binary system with pulsating components. The only existing implementation of the method of fitting spherical harmonics to light curves uses a very simple model of the binary, with ideally spherical stars in circular orbits, and does not account for any of the proximity effects. Moreover, the oscillations are represented as sinusoidal perturbations of an arbitrary surface brightness distribution. This dissertation uses a much more detailed model, based on Roche geometry that includes all the important proximity effects. Modeling of stellar oscillations is improved by representing them as direct perturbations of local tempe- v rature and gravitational potential and including the effects of tidal distortion and rotation on the frequencies and spatial properties of the modes. The scientific goal of the dissertation is to develop a reliable method – the method of fitting spherical harmonics – for mode identification on pulsating stars in close binary systems, in order to enable asteroseismological studies of their interiors. The basic assumption is that a close binary system can be represented using the Roche model, which is based on the approximation of total central condensation. In other words, it is assumed that the contribution of outer layers of the star to its total mass can be neglected. With this approximation, it is possible to treat the gravitational interaction of the components as the interaction of point masses. Within the Roche model, the shape of the star is determined by the Roche potential that corresponds to the superposition of the gravitational and the centrifugal force. The temperature distribution across the surface is determined by the von Zeipel law that defines the dependence of flux on the effective gravitational acceleration. Stellar oscillations are then modeled as perturbations of the local values of the Roche potential and temperature. It is assumed that the oscillations are adiabatic, of small amplitudes, and that they can be represented using spherical harmonics. It is also assumed that the symmetry axis of the oscillations will coincide with the dominant geometrical axis (usually with the rotation axis, or in the case of tidally distorted stars, with the line connecting the centers of the components). The proposed method of mode identification on pulsating components of close binary systems is implemented as a part of the modeling program Infinity. Infinity is based on the Roche model and accounts for all the phenomena that influence the formation of light and radial velocity curves, from limb darkening and surface temperature distribution caused by departures from spherical shape, to the reflection effect, irregularities in surface brightness and the presence of the accretion disc. Stellar oscillations can be modeled on one or both components of the system. The method is verified by means of tests using synthetic data. In other words, the results of fitting are compared to the stellar and orbital parameters that were used to generate vi the synthetic light curves, or the fake “observations”. The tests encompass both favorable and unfavorable configurations (total and partial eclipses, approximately spherical and tidally distorted stars, modes of low and high degrees and so forth). The results are finally compared to the results of a similar technique of mode identification – the method of eclipse mapping. From the obtained results, we can conclude that the proposed method of mode identification is reliable (i.e. provides correct identification) for approximately spherical stars in detached binary systems. The results for tidally distorted stars are encouraging, but show that before an attempt is made to apply the method on real observation of that nature, it will be necessary to include the simultaneous fitting of frequency, amplitude and phase for each trial mode into the procedure of mode identification, and to try to solve the problem of ambiguous identifications. This will be the focus of my future work.