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The work in this paper is motivated by I. Banič and A. Taranenko’s recent paper, where they introduced a new notion, the span of a graph. Their goal was to solve the problem of keeping the safety distance while two players are moving through a graph and they presented three different types of graph spans, depending on the movement rules. We observe the same goal, but give a different approach to that problem by directly defining the maximal safety distance for different movement rules two players can take. This allowed us to solve several problems, prove some relations between different graph spans, and calculate the span values for some classes of graphs.

In this paper, we observe the possibility that the group \(S_{n}\times S_{m}\) acts as a flag-transitive automorphism group of a block design with point set \(\{1,\ldots ,n\}\times \{1,\ldots ,m\},4\leq n\leq m\leq 70\). We prove the equivalence of that problem to the existence of an appropriately defined smaller flag-transitive incidence structure. By developing and applying several algorithms for the construction of the latter structure, we manage to solve the existence problem for the desired designs with \(nm\) points in the given range. In the vast majority of the cases with confirmed existence, we obtain all possible structures up to isomorphism.

In this paper, we observe the possibility that the group (S_{n}times S_{m}) acts as a flag-transitive automorphism group of a block design with point set ({1,ldots ,n}times {1,ldots ,m},4leq nleq mleq 70). We prove the equivalence of that problem to the existence of an appropriately defined smaller flag-transitive incidence structure. By developing and applying several algorithms for the construction of the latter structure, we manage to solve the existence problem for the desired designs with (nm) points in the given range. In the vast majority of the cases with confirmed existence, we obtain all possible structures up to isomorphism.

The aim of this research is to develop efficient techniques to construct flag-transitive incidence structures. In this paper we describe those techniques, present the construction results and take a closer look at how some types of flag- transitive incidence structures relate to arc- transitive graphs.

U ovoj disertaciji izložena su istraživanja iz nekoliko područja teorije kompleksnih mreža. Definirane su poopćene verzije mrežnih deskriptora, kao što su transmisija, međupoloženost, vršna produktivnost i vršna profitabilnost koje uzimaju u obzir pretpostavku da u mreži vrhovi na manjim udaljenostima komuniciraju znatno više nego oni na većim udaljenostima. Proučavane su minimalne i maksimalne vrijednosti tih deskriptora i analizirane gornje i donje ograde tih vrijednosti. Nadalje, predložena je modificirana verzija Girvan-Newmanovog algoritma za detektiranje zajednica u mrežama, koja smanjuje broj operacija i dovodi do bržeg uočavanja strukture zajednica. U posljednjem dijelu su analizirane mreže s distribuiranim ključevima i proučavana njihova sigurnost na napad neprijateljskih agenata. Uz dvije različite pretpostavke o djelovanju agenata na mrežu određuju se minimalni brojevi vrhova u mreži i ključeva potrebnih da bi mreža bila sigurna. In this thesis several areas of theory of complex networks are explored. Generalized versions of network descriptors such as transmission, betweenness centrality, networkness and network surplus, which assume that the ammount of communication in the network is greater between vertices which are at smaller distances than that that are on greater distances, are defined. Minimal and maximal values of these descriptors are studied and lower and upper bounds are obtained. Further, a modified version of Girvan-Newman algorithm for community detection is proposed, which reduces the number of operations compared to the original and leads to faster community detection. In the last part, networks with distributed keys are analyzed and their safety under the attack of enemy agents is studied. Under two different assumptions on the behavior of agents in the network, minimal number of vertices in the network and minimal number of distributed keys needed to secure the network, are determined.

Problem trgovackog putnika, skraceno TSP(Traveling Salesman Problem), je jedan od najpoznatijih i najproucavanijih problema kombinatorne optimizacije. Njegov matematicki model je traženje Hamiltonovog ciklusa najmanje težine u težinskom grafu. Ovim radom se daje uvid u prirodu TSP-a te složenost i metode njegovog rješenja.

The topological index MID06 (Molecular Identification Number 06) has been proposed by Chang-Yu Hu and Lu Xu in their paper “Developing Molecular Identification Numbers by an All-Paths Method”. It has been tested successfully on a large number of alkane isomers and molecules containing heteroatoms. Here we analyze the degeneracy of MID06 for alkanes, and we demonstrate that there are many more alkanes then possible MID06 identifiers. Although MID06 has very good discriminative properties, it has degenerate values for certain alkane pairs.

Cilj ovog rada je detaljno pojasniti nastavnu tehniku razrednog glasanja. Sve njene komponente su pojedinačno objašnjene. Niz čimenika utječe na kvalitetu same provedbe glasanja, kao što su vrednovanje, upotreba tehnologije, duljina trajanja i u koliko krugova se glasa. Nakon detaljne obrade svih čimbenika, zaključak je da je tehnika efikasna i primjenjiva na satu matematike, te da je povratna informacija učenika pozitivna. U radu je takoder predočeno nekoliko skupina pitanja iz različitih područja matematike koja se mogu primjeniti u nastavi., The aim of this paper is to explain in detail the teaching technique known as class voting. All its components are explained individually. A number of factors affect the quality of the voting itself, such as evaluation, use of technology, length of time and number of voting rounds. After a detailed analysis of all factors, the conclusion is that the technique is efficient and applicable in the mathematics class, and that the feedback from the students is positive. The paper also presents several groups of questions from different areas of mathematics that can be applied in class.

Teorija grafova je danas iznimno popularna grana matematike sa širokim primjenama. Najčešće primjene su u rješavanju problema pronalaska najkraćih puteva kako bi se minimizirali razni troškovi pa je korisno pogledati različite algoritme pretrage koje imamo u stablima. Najpoznatiji algoritmi su Jarnik-Prim algoritam i Boruvka-Kruskal kojima je jedina razlika u redoslijedu odabira bridova (”puteva”), a kao rezultat daju jednako stablo., Graph theory is lately an extremely popular branch of mathematics with wide applications. The most common applications are in solving problems of finding the shortest paths to minimize various costs, so it is useful to look and understand different types of search algorithms that are used in trees. The most famous algorithms are Jarnik-Prim algorithm and Boruvka-Kruskal. The only difference between these two is in the order selection of edges (”paths“) but the result is the same tree in both cases.

Cilj ovog rada je prikazati kako se s godinama obrazovanja učenika proširuje njihovo znanje o obujmu. Istražen je koncept i tip zadataka koji se proteže kroz školovanje učenika, iz njihovih udžbenika. Obrazovanje o obujmu kreće sa obradom obujma tekućine, nakon toga uči se o obujmu tijela i to znanje se primjenjuje prvo na kocki i kvadru, a nakon toga na svim uspravnim geometrijskim tijelima. Zatim se uči o obujmu geometrijskih tijela primjenjujući pri tome načelo Cavalierijev princip. Osim na geometrijskim tijelima, obujam se tada primjenjuje i na rotacijskim tijelima., This paper aims to show how the student’s knowledge of the volume expands with the years of education. The concept and type of tasks that extend throughout the education of students were investigated from students’ textbooks. Education about volume starts with processing the volume of a liquid after which you learn about the volume of solids and this knowledge is applied first to the cube and cuboid and then to all right geometric solids. Then they learn about the volume of geometric solids by applying Cavalieri’s principle. In addition to geometric solids, the volume is also applied to solids of revolution.

Cilj ovog rada bio je objasniti primjenu metode pražnjenja koja ima važnu ulogu u teoriji grafova i korištena je u slavnom dokazu teorema o četiri boje. Metoda se provodi kroz dva koraka, dodjeljivanje naboja vrhovima, bridovima ili stranama grafa te premještanje tih naboja po određenom skupu pravila. Kroz razne primjere u planarnim grafovima i bojenjima smo pokazali kako navedena metoda funkcionira., The aim of this work was to explain the application of discharging method, which has an important role in graph theory and was used in the famous proof of the four color theorem. The method is implemented through two steps, assigning charges to the vertices, edges or faces of the graph and moving these charges according to a certain set of rules. Through various examples in planar graphs and colorings, we have shown how the specified method works.

Slučajni skupovi igraju ključnu ulogu u modeliranju brojnih procesa u biologiji, medicini i znanosti o materijalima. U prvom dijelu rada koji se odnosi na teroijske rezultate definiran slučajni zatvoreni skup u \(\mathbb{R}^{d}\) kao funkcija nad vjerojatnosnim prostorom te su navedeni ilustrativni primjeri. Potom je definiran pojam funkcionala kapaciteta te su istražena njegova svojstva. U središnjem poglavlju rada dokazan je Choquetov teorem koji nam govori o nužnim i dovoljnim uvjetima da bi funkcional bio funkcional kapaciteta jedinstvenog slučajnog zatvorenog skupa, a time i da na jedinstven način odreduje distribuciju pripadajućeg slučajnog zatvorenog skupa. U završnom dijelu rada napravljena je simulacijska studija čiji cilj je bio istražiti koliko dobro funkcional kapaciteta opisuje realizaciju slučajnog skupa te može li se mjera sličnosti bazirana na funkcionalu kapaciteta iskoristiti za klasteriranje realizacija različitih modela. Rezultati istraživanja pokazali su da funkcional kapaciteta vrlo dobro razlikuje odgovarajuće realizacije pa se može koristiti kao mjera sličnosti za klasteriranje., Random sets play an essential role in modelling several phenomena in biology, medicine and material science. The first part of the paper, which refers to the theoretical results, a random closed set in \(\mathbb{R}^{d}\) is defined as a function over the probability space and illustrative examples are given. Then the concept of capacity functional was defined and its properties were investigated. In the central chapter of the paper Choquet’s theorem is proved. It tells us what are the necessary and sufficient conditions for a functional to be a capacity functional of a unique random closed set, and thus to uniquely determine the distribution of the corresponding random closed set. In the final part of the paper a simulation study was made the goal of which was to investigate how well the capacity functional describes the realization of a random set and whether the similarity measure based on the capacity functional can be used for clustering the realizations of different models. The results of the research showed that the capacity functional distinguishes the corresponding realizations very well, so it can be used as a measure of similarity for clustering.

Cilj je ovog diplomskog rada predstaviti model slobodne metode učenja podrškom primjenjive na rješavanje problema neprekidne kontrole, točnije dvonožnog hodanja. U tu je svrhu prvo opisana struktura i proces učenja neuronskih mreža. Definiramo i osnovne koncepte učenja podrškom, vrste algoritama te dajemo uvid u BipedalWalker-v3 radno okruženje dvonožnog robota u kojem implementiramo algoritme. Zatim prezentiramo TD3 algoritam počevši od DDPG algoritma kao osnove istog te ARS algoritam. Konačno, opisujemo implementaciju u Pythonu i rezultate oba algoritma., The goal of this thesis is to introduce model free reinforcement learning methods applicable on solving continuous control problems, i.e. bipedal walking. For that purpose, the neural networks structure and learning process is first defined. We also define the basic concepts and algorithm types of reinforcement learning and give insight into the BipedalWalker-v3 environment in which we implement the algorithms. Subsequently, we present the TD3 algorithm (starting with the DPPG algorithm as its basis) and the ARS algorithm. Finally, we describe the Python implementation and results of both algorithms.

U ovom radu smo obradili osnove teorije dizajna počevši s incidencijskom strukturom te projektivnom i afinom ravninom i njihovim svojstvima. Nakon toga smo definirali pojmove blok dizajna i Steinerovog sistema. Promatrali smo izomorfizme i potprostore incidencijskih strukutra te smo spomenuli familiju Hadamardovih dizajna i njihove primjene. Zadnja dva poglavlja posvećena su primjeni teorije dizajna u dizajnu igre ’Spot It!’ te vlastite igre s matematičkim pojmovima., In this thesis we explained basic notions of design theory such as incidence structure, affine and projective geometry and their properties. After that, we defined the notion of a block design and that of a Steiner system. We have seen examples of isomorphisms and subspaces of incidence structures and mentioned family of Hadamard designs and it’s applications. Last two chapters were devoted to design theory application in designing game ’Spot It!’ and our own mathematical game.

U ovom diplomskom radu smo proučavali pojam Hamiltonovog ciklusa. Definirali smo osnovne pojmove korištene u teoriji grafova te pojmove zatvorenja i žilavosti grafa. Obradili smo nužne i dovoljne uvjete koji nam olakšavaju pronalazak Hamiltonovog ciklusa u grafu. Sve definirane pojmove smo potkrijepili primjerima kako bi si vizualizirali sami pojam., In this master’s thesis we have been studying Hamiltonian cycles in graphs. We defined basic terms used in graph theory and terms of closure and toughness of graph. We processed sufficient and neccessary conditions for Hamiltonicity. We have supported all the defined concepts with examples in order to better visualize the concept itself.

Teoremi nepotpunosti Kurta Gödela su jedni od važnijih rezultata u matematičkoj logici. U ovom radu su opisani apstraktni sistemi na koje se odnosi prvi Gödelov teorem nepotpunosti. Posebna pozornost je dana Peanovoj aritmetici s i bez potenciranja koje su uvedene aksiomatski i dokazano je da su one nepotpune. Osim toga, opisana je i Gödelova numerizacija koja igra ključnu ulogu u dokazivanju nepotpunosti., Godel’s incompleteness theorems are among the most important results of Mathematical Logic. In this thesis abstract systems to which the first incompleteness theorem can be applied are described. In particular, Peano arithmetic with and without exponentiation is axiomatically introduced and its incompleteness is shown. In addition, Godel’s numbering has been described. It plays an important role in the proof of the incompleteness theorem.

Glavni cilj ovog rada je analiza nepristranih igara. U tu svrhu objašnjene su neke strategije koje se javljaju u kombinatornim igrama. Kako bi pokazali posebna svojstva koje imaju nepristrane igre, definirane su pozicije i opcije u igri, te suma igara. Poseban naglasak stavljen je na igru nim pomoću koje je definirana vrijednost pod nazivom nimberi. To je klasa beskrajno malih vrijednosti koje se ne ponašaju kao nijedna vrijednost s kojima smo se do sada susreli, a poprimaju je nepristrane igre. Jedan od najbitnijih rezultata ovog rada je Sprague-grundy teorem po kojem je svaka nepristrana igra ekvivalenta nim-hrpi. Zbog toga igra nim ima veliku ulogu u analizi nepristranih igara., The main objective of this master’s thesis is the analysis of impartial games. For this purpose, some strategies that occur in combinatorial games are explained. In order to show the special properties that impartial games have, positions and options in the game are defined, as well as the sum of games. Special emphasis is placed on the game nim by which is defined the value nimbers. It is a class of infinitesimals that do not behave like any values that we have yet encountered. They occur naturally as the values of the impartial game. One of the most important results of this master’s is Sprague-grundy theorem according to which any impartial game is equivalent to a nim-heap. Therefore, nim plays a mayor role in the analysis of impartial games.

Vjerojatnosna metoda je snažan alat u kombinatorici i teoriji grafova. Osnovni princip vjerojatnosne metode je sljedeći. Želimo pokazati da postoji neki kombinatorni objekt koji zadovoljava neko svojstvo P. Da bismo to pokazali, promatramo proizvoljni objekt iz dobro odabranog vjerojatnosnog prostora i računamo vjerojatnost da taj objekt zadovoljava svojstvo P. Ako dobijemo da je ova vjerojatnost veća od nula, zaključujemo da objekt s traženim svojstvom P postoji. Ako ne postoji objekt koji zadovoljava svojstvo P, onda će vjerojatnost koju smo izračunali biti nula. U ovom radu pokazali smo kako primjenom vjerojatnosne metode možemo riješiti neke od problema ekstremalne kombinatorike., The probabilistic method is a powerful tool in Combinatorics and Graph Theory. The general principle of the probabilistic method is the following. We want to show the existence of a combinatorial object satisfying a given property P. To do this, we consider a random object in a well chosen probability space and we compute the probability that such an object satisfies the property P. If we show that this probability is greater then 0, then we deduce that an object with property P exists. Indeed, if no object were satisfying property P, then the probability would be zero. In this work, we showed how to solve some of the extremal combinatorics problems by applying the probabilistic method.

Cilj ovog rada bio je predstaviti Erdos-Renyi-ev model slučajnog grafa. Za obraditi taj model, prvo je obrađen proces grananja - nađena je veza preživljavanja i izumiranja u procesu grananja i izvedeni su dokazi za binomni i Poissonov proces grananja. Nakon toga, uveden je pojam Erdos-Renyi-evog slučajnog grafa i nađena je njegova veza sa Poissonovim procesom grananja. Pronađene su ograde za veličinu clustera, a posebno je promatran graf u subkritičnom području, i za taj slučaj nađena je granica najveće povezane komponente. Za kraj je predstavljeno još nekoliko modela slučajnih grafova i njihove primjene., We represent Erdos-Renyi random graph in this master’s thesis. Firstly, we represent branching process - relation between survival and extinction is found and proofs for binomial and Poisson branching processes are given. Secondly, we introduce term Erdos-Renyi random graph and we related it with Poisson branching process. We found restrictions for cluster size and we especially studied graph in subcritical regime - in this case we found a bound for maxi mal cluster. In the end, we represent few different models od random graphs and its applications.

U ovom diplomskom radu naglasak je na starim i novim problemima bojanja planarnih grafova. U radu su definirani osnovni pojmovi teorije grafova, bojanje grafova te planarnost. Također su iskazani razni teoremi i prikazane su veze i posljedice istih. Teorija grafova je grana matematike u kojoj se većina pojmova, definicija, teorema, te razni problemi i njihova rješenja mogu prikazati slikom. U skladu s tim, tekst u radu je potkrijepljen slikama kako bi se tema rada što zornije prikazala., In this master’s thesis, the emphasis is on the old and new problems of coloring planar graphs. The thesis defines the basic concepts of graph theory, coloring of graphs and planarity. Various theorems were also shown and the connections and consequences of the same were presented. Graph theory is the branch of mathematics in which most concepts, definitions, theorems, and various problems and their solutions can be depicted in a picture. Accordingly, the text in the thesis is backed up by the pictures in order to show the work theme as plain as possible.

Moduli nad domenama glavnih ideala mogu se prikazati kao direktna suma konačno mnogo cikličkih modula, i to na dva različita načina, preko invarijantnih faktora ili elementarnih djelitelja. Ako za domenu glavnih ideala uzmemo konkretno prsten polinoma F[x], možemo promatrati posebne blok-dijagonalne matrice, odnosno linearne operacije čije su matrični prikazi blok-dijagonalni; i to na dva načina (kao i gore), preko invarijantnih faktora ili preko elementarnih djelitelja. Koristeći invarijantne faktore dolazimo do racionalne kanonske forme, a koristeći elementarne djelitelje dolazimo do Jordanove kanonske forme. Racionalna kanonska forma je jednaka za sve slične matrice, pa izmedu ostalog može poslužiti za provjeru sličnosti dvije matrice, a iz Jordanove kanonske forme se, između ostalog, odmah može pročitati karakteristični i minimalni polinom matrice., Modules over principal ideal domains can be as a direct sum of a finite num ber of cyclic modules, in two different ways: using invariant factors or ele mentary divisors. If we take specifically the ring of polynomials F[x] as the principal ideal domain, we can observe special block-diagonal matrices, or li near operators whose matrix representations are block-diagonal, in two diffe rent ways (like above), using invariant factors or elementary divisors. Using invariant factors, we obtain rational canonical form, and using elementary divisors, we obtain Jordan canonical form. Rational canonical form iz the same for all similar matrices, so it is useful to check if two matrices are simi lar. Jordan canonical form is useful because the characteristic and minimal polynomials of a matrix can be read immediately from it.

Ovaj rad se bavi proučavanjima zanimljivih veza teorije grafova i teorije grupa. Središnja tema su Cayleyjevi grafovi. Proučavaju se neka njihova svojstva kao što su razni tipovi tranzitivnosti i uvjeti povezanosti te posebice njihove pune grupe automorfizama. Također smo definirali G-grafove i proučili neka njihova karakteristična svojstva te smo na kraju odredili vezu G-grafa i Cayleyjevog grafa., This work deals with connections between group theory and graph theory. Main topic are Cayley graphs. Their properties like transitivity, conditions for graph connectivity and especially their full automorphisms group are being studied. We also defined G-graphs and analyzed their characteristic properties and, at the end, we consider a relation between Cayley graphs and G-graphs.

Rad se bavi pozitivnim aspektima primjene grupnog rada u nastavi matematike i istodobno daje praktične savjete kako umanjiti one negativne. Cilj je različitim nastavnim tehnikama i metodama strukturirati grupni rad tako da njegova realizacija vodi ka uspješnom suradničkom učenju. Rad sadrži nekoliko ideja i primjera konkretnih materijala za provedbu grupnog rada u nastavi matematike za osnovnu školu., This thesis explores positive aspects of implementing group work in teaching mathematics, and also to give practical advice how to lessen the negative ones. Aim is to structure group work using various teaching techniques and methods, so that it results in successful collaborative learning. Thesis contains some ideas and examples of teaching preparations designed to implement group work in math teaching in primary school.

Rad ne sadrži sažetak.

Rad ne sadrži sažetak.

U radu se ističe važnost vrednovanja učenika, s posebnim naglaskom na vrednovanje tijekom obrade nekog gradiva radi promicanja njihovog znanja. Vrednovanje se provodi kroz cijelo obrazovanje učenika, pa ipak ovdje smo posebnu pažnju posvetili formativnom vrednovanju kao onom koje ima ulogu da na vrijeme učenicima, roditeljima i nastavnicima pokaže koje su se poteškoće pojavile u razumijevanju matematičkih sadržaja.

U radu je obrađena kreativnost kao važna kompetencija nastavnika matematike s naglaskom na upotrebu igrara, projekata i mentalnih mapa u nastavi matematike.

U radu su izloženi različiti pristupi poučavanju konika u nastavi matematike s posebnim naglaskom na prednosti i nedostatke sintetičkog i analitičkog pristupa.

Ovaj rad se prvotno temelji na obrazovnoj strategiji otkrivanja i metode istraživanja, koja se u drugom dijelu primjenjuje na konkretnim problemskim zadacima. U metodi istraživanja, posebna se pažnja posvećuje metodičkim i didaktičkim načelima kako bi učenik vlastitim sposobnostima mogao riješiti problem. Učenicima se u nastavnom procesu prepušta da istraže sve što je moguće postavljanjem hipoteza i donošenjem zaključaka. Takav pristup je izazov i zabava i učinit će da učenici povjeruju u svoju sposobnost za uspjeh u matematici. Aktivnosti koje se pojavljuju u ovom radu namijenjene su za istraživanje, uživanje i poticanje znatiželje među učenicima kako bi pronašli načine za rješavanje problema koji se javljaju u stvarnom životu, a ne samo u udžbenicima. Uglavnom je to rad u skupinama jer interakcija učenika međusobno i zajedno s nastavnikom čini te aktivnosti bogatijima., This work is primarily based on the educational strategy of discovery and the methods of research based learning, which in the second portion is applied to the solving of concrete mathematical tasks. In the investigational method, special attention is payed to the methodical and didactical principles so that the student could solve a problem with his own capabilities. In the teaching the proces, the student is allowed to investigate and research everything that is possible by proposing hypotheses and coming to conclusions. This approach is therefore both fun and challenging and will increase a student's self-confidence in his problem solving capabilities and his capacity for succes in math. The acctivities proposed in this essay are ment to increase the amount od research and add enjoyment, as well as encourage curiosity amongst the students so that they can find ways to solve problems that they may encounter in real life, not just those in the textbooks. It is mainly done in groups because the interaction of students with each other and together with the teacher enriches the quality of these activities.

U radu je dan pregled različitih manipulativnih materijala i njihove primjene u nastavi matematike.