http://hdl.handle.net/123456789/1 http://hdl.handle.net/123456789/5798 EKSTEREMALNI PROBLEMI BRAUNOVOG KRETANJA I DRUGIH SLUČAJNIH PROCESA Jovalekić, Milica Let M be a maximum and let N be a minimum of the non-negative martingale X1, X2, . . . , Xn. It is well known, that if X1 = 1, then γ(‖M ‖1) ≤ E (Xn log Xn) and γ(‖N ‖1) ≤ E (Xn log Xn) , where γ(x) = x − 1 − log x, for all x > 0. In this thesis, we prove the analogue of this result in the case when 1 < p < ∞, by proving that δp (‖M ‖p p ) ≤ ‖Xn‖p and δp (‖N ‖p p ) ≤ ‖Xn‖p, where δp(x) = ( 1 − 1 p ) x 1 p + 1 p x 1 p −1, for all x > 0. We also obtain a probabilistic proof of the fact min ρ∈D(Qn) ∫ Qn dx1 . . . dxn ρ (x1, . . . , xn)p−1 ∏n j=1 xαj +1 j = n∏ j=1 ( p p − αj − 1 )p , where p > 1, αj < p − 1 for j = 1, . . . , n and D (Qn) is family of all densities on the n-dimensional unit cube Qn = (0, 1)n in Rn. This provides the proof of the multidimensional weighted Hardy inequality. Namely, if f : Rn + → (0, ∞) is a measurable function, p > 1 and αj < p − 1 for j = 1, . . . , n, then ∫ Rn + n∏ j=1 xαj j Hnf (x)p dx ≤ n∏ j=1 ( p p − αj − 1 )p ∫ Rn + n∏ j=1 xαj j f (x)p dx, where Hnf (x) = 1 x1 . . . xn ∫ x1 0 · · · ∫ xn 0 f (t) dt, is a multidimensional Hardy operator, x = (x1, . . . , xn) ∈ Rn +, t = (t1, . . . , tn) and dt = dt1 . . . dtn. Let B(t) be a standard planar Brownian motion and r(θ) be the length of the projection of B[0, 1] on the line generated by the unit vector eθ = (cos θ, sin θ), where 0 ≤ θ ≤ π. We nd the common distribution function F of the random variables r(θ). Namely, we prove that F(x) = 8 ∞∑ n=1 ( 1 x2 + 1 (2n − 1)2π2 ) exp ( − (2n − 1)2π2 2x2 ) , for every x > 0. As immediate consequence, lower bound for the expected diameter of the set B[0, 1], better than known, is obtained. Namely, it is known that Ed ≥ 1.601, where d is the diameter of the set B[0, 1]. In this thesis we show Ed ≥ 1.856. Sat, 01 Jan 2022 00:00:00 GMT http://hdl.handle.net/123456789/5797 BIPROIZVODI U MONOIDALNIM KATEGORIJAMA Zekić, Mladen 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. Fri, 01 Jan 2021 00:00:00 GMT http://hdl.handle.net/123456789/5796 OCENE RASTA GRADIJENTA ZA FUNKCIJE DOBIJENE UOPŠTENIM PREZENTACIJAMA PAUSONOVOG TIPA Mutavdžić, Nikola In this PhD thesis we investigate bounds of the gradient of harmonic and harmonic quasiconformal mappings. We also discuss such bounds for functions that are har- monic with respect to the hyperbolic metric or certain other metrics. This research has been motivated by some recent results about Lipschitz-continuity of quasiconformal map- pings that satisfy the Laplace gradient inequality. More precisely, the mappings we consider are solutions of the Dirichlet problem for the Poisson equation and can be considered as a generalization of harmonic mappings. Besides the ball, we also work with general domains on which solutions of the Dirichlet problem are defined, as well as general codomains. Finally, we announce new results that have been formulated for regions of C1,α-smoothness, both as the domain and the codomain. Besides presenting the main results, we give an overview of general notions from differential geometry and recall some of the properties of hyperbolic metric in an n-dimensional ball. We also state properties of harmonic and sub-harmonic functions with respect to the hyperbolic metric, which are analogous to some classical results from the theory if harmonic functions and Hardy’s theory. It turns out that the gradients of hyperbolic harmonic functions behave differently from those of euclidean harmonic functions. A similar conclusion is obtained for the family of Tα-harmonic functions. Namely, unlike the space of harmonic functions, the solution of the Dirichlet problem in the space of Tα-harmonic functions is shown to be Lipschitz-continuous when so is the boundary function. In addition, we investigate Hölder- continuity of the solution of the Dirichlet problem for the Poisson equation in the euclidean and hyperbolic metric. We will present versions of the Schwarz lemma on the boundary for pluriharmonic map- pings in Hilbert and Banach spaces. These results will follow from the version of the Schwarz lemma for harmonic mappings from the unit disc to the interval (−1, 1) without the assump- tion that the point z = 0 maps to itself. Furthermore, we show a version of the boundary Schwarz lemma for harmonic mappings from a ball to a ball, not necessarily of the same dimension. The proof uses a version of the Schwarz lemma for multivariable functions, first considered by Burget. This result is obtained by integrating the Poisson kernel over so-called polar caps. The assumption that point z = 0 maps to itself is again not needed, thus yielding a generalization of a recent result by D. Kalaj. At the end of this section, it is demonstrated that the analogous result is false in the case of hyperbolic harmonic functions. In a certain sense, this means that the Hopf lemma is not valid for hyperbolic harmonic functions. Amongst various versions of the Schwarz lemma, we have been investigating bounds of the modulus for classes of holomorphic functions f on the unit disc whose index If fulfils cer- tain geometric conditions. These classes are a generalization of the star and α-star functions, previously investigated by B. N. Örnek. Our method is based on using Jack’s lemma and can be applied in certain more general cases. As an illustration, we derive the sharp bounds for the modulus of a holomorphic function f with index If whose codomain is a vertical strip, as well as bounds for the modulus of the derivative of f at point z = 0. Moreover, we give a bound for the rate of growth of the modulus of holomorphic functions on disk U that map point z = 0 to itself and whose codomain is a vertical strip. Sun, 01 Jan 2023 00:00:00 GMT http://hdl.handle.net/123456789/5795 UTICAJ INTERAKCIJA UDALJENIH GALAKSIJA NA NJIHOVO NETERMALNO RADIO ZRAČENJE Pavlović, Mrina The studyofgalaxiesthroughhighredshiftsarekeytounderstandingthe evolutionofgalaxiesthroughcosmictimes.Assuchobjectsareverydifficultto observedirectly,theyaremainlyexaminedusingempiricallyderivedtoolssuchas the numerouscorrelationsbetweentheirdifferentparametriccharacteristics,one of thembeingthelinearrelationshipbetweenfar-infraredandradioemissionin star-forming galaxies,namedtheFarInfrared-Radio(FIR)Correlation.Although the correlationwasconsideredtobestableintermsoflinearity,recentworks, whichincludegalaxiesathighredshifts(0 < z< 6), showedalargedeviation from thecorrelationinthesesystems.Thegoalofthisdoctoraldissertationis an examinationofthephysicalprocessesthatleadtothiskindofbehavior.As a possiblecauseofthisevolution,wewillassumeforthefirsttime,andexamine interactionsbetweengalaxies(collisionsandcloseapproaches).Interactionsbe- tweengalaxiesleadtotheformationofshockwavesonlargescalesthatcanlead to changesintherelationshipbetweeninfraredandradioemissions.Ourhypoth- esis wastestedinseveralstadiumsandthemainresultsareasfollows:1.We developedmodelsoftheevolutionoftheFIRcorrelationwithredshiftasfunctions of thegalaxyinteractionrate.Wetestedthemodelsonasampleofgalaxieswith an alreadydeterminedmorphologyseparatelyfordiscgalaxiesandforgalaxies that haverecentlybeenorarecurrentlyinteracting-irregulargalaxies. 2. Inasmallsampleof34galaxiesthatwetookfrompaperMiettinenetal.(2017), it wasshownthatthereisanindicationthattheinteractionbetweengalaxiescan beresponsiblefortheevolutionofthecorrelationwiththeredshift. 3. Thenextanalysiswasperformedonamuchlargersampleofstar-forming galaxies takenfromCOSMOSfield,wherewedidnotfindanyevolutionofcorre- lation withtheredshift.Also,itwasshownthatthemeanvalueofthecorrelation parameter islowerinirregulargalaxiesthanindiskgalaxies. Although recentobservationsindicatedanevolutionoftheFIRcorrelation with redshift,theresultsofthisresearchfailedtoreproducethatevolutionand showedthattheFIRcorrelationisstablewithredshift.However,itwasalso shownthatduetotheinteractionofgalaxies,theevolutionoftheFIRcorrelation is possibleiftherepresentationofinteractingsystemsinthesampleishigher. Sat, 01 Jan 2022 00:00:00 GMT