Šiame darbe buvo analizuojama Niutono fraktalų Žulija aibės. Dažniausiai Žulija ir užpildytų Žulija aibių vaizdai gaunami, panaudojant "pabėgimo laiko" algoritmą. Norėdami šį algoritmą naudoti kompleksinio daugianario šaknų vizualizacijai, turime nurodyti iteracijų skaiči��, algoritmo tikslumą, žingsnį bei kompleksiniu Niutono metodu rasti daugianario šaknis. Taikant Niutono metodą, buvo susidurta su pradinių taškų parinkimo problema. Tyrimo metu patvirtinta, kad pakanka Niutono iteracinę funkciją taikyti taškams z, kurių modulis 2. Darbe buvo pasiūlytas šaknų lokalizacijos srities nustatymo būdas. Naudojant PL-algoritmą, pasirinktu žingsniu pereiname visus taškus, kurie patenka į šią sritį. Taip gauname Niutono-Rafsono fraktalus ir lygiagrečiai analizuojame Žulija aibes bei užpildytas Žulija aibes. / Julia sets and filled Julia sets of Newton‘s fractals are analyzed in this work. The Escape Time Algorithm provides us with a means for "seeing" the filled Julia sets of Newton‘s fractals, but roots, (zeros) of the polynomial under investigation should be known. The Newton‘s method for finding roots of an algebraic equation is well known. Here in the paper the complex Newton method for finding roots of a complex polynomial is presented. The main difficulties, associated with implementation of this method in practice, are discussed, namely: construction of the set of initial points (first approximations of the roots), finding the basin of attraction for a particular root and so forth. Some experimental results are presented.

Inspired by Happel's question, whether the exchange graph and the simplicial complex of tilting modules over a quiver algebra are independent from the multiplicities of multiple arrows in the quiver, we study quantitative aspects of Fomin and Zelevinsky's quiver mutation rule. Our results turn out to be very useful in the mutation-infinite case for understanding combinatorial structures as the cluster exchange graph or the simplicial complex of tilting modules, which are governed by quiver mutation. Using a class of quivers we call forks we can show that any such quiver yields a tree in the exchange graph. This allows us to provide a good global description of the exchange graphs of arbitrary mutation-infinite quivers. In particular we show that the exchange graph of an acyclic quiver is a tree if (and in fact only if) any two vertices are connected by at least two arrows. Furthermore we give classification results for the simplicial complexes and thereby obtain a partial positive answer to Happel's question. Another consequence of our findings is a confirmation of Unger's conjecture about the infinite number of components of the tilting exchange graph in all but finitely many cases. Finally we generalise and conceptualise our results by introducing what we call "polynomial quivers", stating several conjectures about "polynomial quiver mutation", and giving proofs in special cases.