http://hdl.handle.net/123456789/12 http://hdl.handle.net/123456789/5800 POLINOMIJALNA ENTROPIJA ZA MORSOVE GRADIJENTNE SISTEME I LOGISTIČKO PRESLIKAVANJE Perić, Milan This thesis presents a method for calculating the polynomial entropy of the topolog- ical dynamic system with finitely many non-wandering points. A special coding is adapted for such systems. Thanks to this coding the polynomial entropy can be bounded by the number of specific mutually singular points in the closures of stable manifolds of non-wandering points. This method was applied to Morse gradient systems. It is shown that the polynomial entropy of the Morse gradient system is bounded by n(F ) − 1, where n(F ) is the number of different Morse indices of critical points of the Morse function F. If Morse gradient systems on mani- folds of dimension n has only critical points of indices 0 and n, it is proved that the polynomial entropy is equal to 1, and if the system has critical points of indices 0, n/2 and n, it is proved that polynomial entropy is equal to 2. The polynomial entropy for different parameter values in logistic map has also been calculated, and it has been shown that the polynomial entropy distinguishes the systems of low complexity with drastically different behaviours, which cannot be distinguished by the topological entropy. The example of the homeomorphism of the con- nected compact metric space that is not equicontinuous and with vanishing polynomial entropy is also given. Fri, 01 Jan 2021 00:00:00 GMT http://hdl.handle.net/123456789/5799 UTAPANJA PROSTORA HARMONIJSKIH FUNKCIJA SA MEŠOVITOM NORMOM U OGRANIČENIM OBLASTIMA U Rn Jovanović Spasojević, Tanja In this thesis, subjects of consideration are the embeddings theorems of weighted Bergman spaces in Lp-spaces, as well as embeddings theorems of harmonic mixed norm spaces. The first part of the thesis generalizes the theorems of embeddings Bergman spaces into Lp(μ)-spaces, where μ is a Borel measure on a given domain. They have been earlier studied on domains such as unit ball and upper half-space. Generalization refers to bounded domains Ω ⊂ Rn with C1 boundary. This embedding will be valid to any p > 0, whenever the measure of the spaces Lp satisfies the Carledon condition. Reverse the direction will be valid only in case if p > 1 + α+2 n−2 . The second part of the dissertation also generalizes the embeddings theorems of mixed norm spaces of harmonic functions on a unit ball, where the generalization is applied to the domain Ω ⊂ Rn with C1 boundary. However, in addition we are obtaining another important result relating to the limitation of the maximum operators in the mixed norm on the general domain for the class of QNS functions. Sat, 01 Jan 2022 00:00:00 GMT http://hdl.handle.net/123456789/5798 EKSTREMALNI 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