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Igranje je prirodan način na koji djeca uče. To je proces kroz koji istražuju, razvijaju kreativnost i motivaciju. Igranjem dijete povezuje trenutno znanje s novim pojmovima te se susreće s novim situacijama. Jedna od igara koja je vrlo korisna za učenje novih pojmova kao i za povezivanje već naučenog gradiva je i igra Matoboj. Budući da je igra izmijenjena uz dodatak sličica i primjenu matematike u svakodnevnici, potrebno je provesti testiranje kako bi se vidjelo sadašnje stanje igrice. S obzirom na to da se igra Matoboj pokazala kao vrlo koristan alat za razvoj brojnih aspekata kod učenika, u ovom će radu biti prikazano testiranje uz povratne informacije o samoj igri., Playing is a natural way for children to learn. It is a process through which they explore, develop creativity and motivation. By playing, the child connects current knowledge with new concepts and encounters new situations. One of the games that is very useful for learning new concepts as well as for connecting already learned material is the Mathduel game. Since the game has been modified with the addition of thumbnails and the application of mathematics in everyday life, it is necessary to conduct testing to see the current state of the game. Given that the Mathduel game has proven to be a very useful tool for the development of numerous aspects in students, this paper will present testing with feedback on the game itself.

Cilj ovog istraživanja bio je ispitati razlike u samoefikasnosti u procesu učenja, subjektivnoj vrijednosti zadatka i ispitnoj anksioznosti učenika između predmeta Hrvatskog jezika i Matematike, kao i razlike u procjenama učenika nastavničkog humora i podržavajućeg odnosa nastavnika prema učenicima između tih predmeta te ispitati prediktivnu valjanost ovih varijabli za završne ocjene iz tih predmeta. Istraživanje je provedeno putem on-line upitnika na uzorku od 319 učenika (25.7% mladića) koji pohađaju srednjoškolski program opće gimnazije u Gradu Zagrebu. Rezultati pokazuju nekoliko značajnih razlika između Hrvatskog jezika i Matematike u motivacijskim i nastavničkim varijablama. Kombinacijom svih korištenih varijabli u posljednjem koraku hijerarhijske regresijske analize objašnjeno je 46% ukupne varijance zaključne ocjene iz Matematike te 35% ukupne varijance zaključne ocjene iz Hrvatskog jezika. Samoefikasnost u procesu učenja i podržavajući odnos nastavnika su pozitivni prediktori, a ispitna anksioznost negativan prediktor zaključne ocjene u oba predmeta. Subjektivna vrijednost zadatka značajan je pozitivan prediktor zaključne ocjene u Matematici dok u Hrvatskom jeziku nije pokazan značajan doprinos u objašnjenju zaključne ocjene. Prikladan i neprikladan stil humora nisu se pokazali statistički značajnima u objašnjenju ukupne varijance zaključne ocjene iz oba nastavna predmeta. Dobiveni rezultati upućuju na donekle različit odnos motivacijskih čimbenika kod ova dva nastavna predmeta te na komplicirani odnos nastavničkog humora s obrazovnim ishodima kojeg je potrebno dodatno istražiti. The aim of this study was to examine differences in self-efficacy in the learning process, subjective task value, and test anxiety among students between the subjects of Croatian language and Mathematics, as well as differences in students' perceptions of instructional humor and supportive teacher-student relationships between these subjects. Furthermore, the study aimed to investigate the predictive validity of these variables on final grades in these subjects. The research was conducted using an online questionnaire on a sample of 319 students (25.7% males) attending a grammar school in the city of Zagreb. The results show several significant differences between the Croatian language and Mathematics in motivational and teacher-related variables. By combining all used variables in the final step of hierarchical regression analysis we explained 46% of the total variance in the final grade in Mathematics and 35% of the total variance in the final grade in the Croatian language. Self-efficacy in the learning process and supportive teacher-student relationships were positive predictors, while test anxiety was a negative predictor of the final grade in both subjects. Subjective task value was a significant positive predictor of the final grade in Mathematics, but it did not show a significant contribution to explaining the final grade in the Croatian language. Appropriate and inappropriate humor styles were not statistically significant in explaining the overall variance of the final grade in both subjects. The obtained results indicate a somewhat different relationship between motivational factors in these two subjects and a complex relationship between instructional humor and educational outcomes that needs further investigation.

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.

U ovom diplomskom radu prikazana je obrazovna reforma iz 1972. godine, s naglaskom na promjene u nastavi matematike nižih razreda osnovne škole. Na samom početku ukratko su opisani društveni i ekonomski razlozi koji su uvjetovali pokretanje promjena u odgojno-obrazovnom sustavu. Opisana je reforma New Math koja je tijekom 1960-ih godina unijela promjene u nastavni plan i program matematike u SAD-u. New Math pokretom se u nastavu matematike uvode novi sadržaji, kako bi se njen plan i program obogatio te kako bi u skladu pratio tehnološki napredak. Zatim su prikazani socio-ekonomski uvjeti u Socijalističkoj Republici Hrvatskoj, koji u doveli do izmjene i unaprjeđenja plana i programa iz 1958. te stvaranja novog plana i programa 1972. godine. Opisane su i objašnjene promjene koje je novi plan i program donio u nastavu matematike nižih razreda osnovne škole. Navedene su njegove odrednice, definirani cilj i zadaci te prikazan raspored gradiva i način rada u nastavi. Istaknute su promjene koje su se javile novim planom i programom te su uspoređeni i analizirani zadaci iz udžbenika i zbirki zadataka. Na samom kraju plan i program iz 1972. uspoređen je s današnjim kurikulumom, odnosno odgojno-obrazovnim ishodima današnje nastave matematike. Također, analizirani su i uspoređeni primjeri zadataka iz udžbenika i ostalih nastavnih materijala iz 1972. s današnjima. Cilj ovog diplomskog je objasniti zašto je došlo do promjena u planu i programu nastave matematike te kakve su one bile. This diploma thesis presents the educational reform from 1972 with an emphasis on teaching mathematics in the lower grades of elementary school. At the very beginning, the social and economic reasons that led to the initiation of changes in the educational system, are briefly described. Also, the New Math reform, that took place in 1960s has been presented and briefly described. With the New Math movement, new contents and methods are introduced into the teaching of mathematics, in order to enrich its curriculum and to keep pace with technological progress. Furthermore, the socio-economic conditions in the Socialist Republic of Croatia are presented. Those conditions led to the modification and improvement of the plan and program from 1958 and the creation of a new plan and program in 1972. The changes that the new plan and program brought up to the teaching of mathematics in the lower grades of elementary school are described and explained. Furthermore, the new plan and program’s determinants are listed, the goal and tasks are defined and the layout of the material and the way of working in class are shown. The changes derived by the new plan and program were highlighted, and math problems from textbooks were compared and analyzed. At the very end, the plan and program from 1972 was compared with today's curriculum. Also, task examples from textbooks and other teaching materials from 1972 were analyzed and compared with today’s materials. The aim of this thesis is to explain why and how the changes in the mathematic curriculum occurred in the first place there were changes in the mathematics curriculum and how the impact of those changes is represented in today’s teaching of mathematics.

Polinom kao jedan od najfundamentalnijih pojmova u gotovo svim područjima matematike, ima istaknutu ulogu u predtercijarnom i tercijarnom matematičkom obrazovanju. Upravo to ga kvalificira da bude idealan objekt institucijskog i epistemološkog istraživanja o didaktičkim procesima. Potreba za takvim istraživanjem inducirana je: 1. na globalnoj razini, malobrojnim istraživanjima u kojima je polinom glavni objekt istraživanja te neodgovorenim pitanjima o razlozima različitog načina definiranja pojma polinoma u predtercijarnom obrazovanju [13,221]; 2. na lokalnoj razini, recentnim promjenama u znanju za podučavanje u predtercijarnom obrazovanju i mogućim efektima tih promjena na tercijarno obrazovanje. Odabrani teorijski okvir, Antropološka teorija didaktike (ATD), nudi alate koji omogućavaju istraživanje epistemoloških i institucijskih aspekata znanja za podučavanje. Stoga se alatima ATD-a analizira tranzicija prema studiju matematike, kroz studij, kao i inverzna tranzicija, ona koju nastavnici matematike prolaze kada počinju podučavati matematiku u predtercijarnom obrazovanju. Dakle, analizira se dvostruki Kleinov prekid i tranzicija kroz studij matematike s obzirom na pojam polinoma. Pojam matematičke prakseologije koristi se za modeliranje i analiziranje matematičkih sadržaja i aktivnosti. Diseminacija matematičkih prakseologija analizira se i modelira pojmom didaktičke prakseologije. Didaktička transpozicija, koja je istodobno alat i metodologija, omogućava analiziranje znanja unutar obrazovnih institucija u odnosu na znanje discipline matematike. Prvo se analiziraju geneza i razvoj pojma polinoma u disciplini matematike, tj. razvoj znanja o pojmu polinoma koja epistemološki legitimiraju znanja koja se podučavaju u predtercijarnim i tercijarnim obrazovnim institucijama. Uzima se u obzir i reforma Nove matematike koja je inducirana promjenom u disciplini matematike. Posebno se istražuje realizacije reforme u hrvatskom gimnazijskom obrazovnom sustavu, kao i njeni rudimenti i mogući efekti na aktualni obrazovni sustav. Dokumenti koji propisuju znanje za podučavanje u hrvatskom predtercijarnom obrazovanju podvrgnuti su prakseološkoj analizi, kao i najkorišteniji gimnazijski udžbenici u zadnjih dvadesetak godina, kako bi se dobio uvid u sekundarno-tercijarnu tranziciju (prvi Kleinov prekid) od kad je uveden Bolonjski proces. S obzirom na pojam polinoma, analizira se obrazovanje matematičara i nastavnika matematike na Prirodoslovno-matematičkom fakultetu Sveučilišta u Splitu. Promatra se i mogući efekti drugog Kleinov prekida na buduće nastavnike matematike. The polynomial, as one of the most fundamental notions in almost all areas of mathematics, has a prominent role in pre-tertiary and tertiary mathematics education. That qualifies it to be an ideal object of institutional and epistemological research on didactic processes. The need for such research was induced by: 1. at the global level, the few studies in which the polynomial is the main object of research, and the unanswered questions about the reasons for the different ways of defining the notion of a polynomial in pre-tertiary education [13,221]; 2. at the local level, recent changes in knowledge for teaching in Croatian pre-tertiary mathematics education and the possible effects of these changes on university mathematics education. The selected theoretical framework, the Anthropological Theory of the Didactic (ATD), offers tools that enable the investigation of epistemological and institutional aspects of knowledge for teaching. Therefore, ATD tools are used to analyse the transition towards the study of mathematics, through the study, as well as the inverse transition, the one that mathematics teachers go through when they start teaching mathematics in pre-tertiary education. Thus, Klein’s double discontinuity, and transition through the study of mathematics, are analysed concerning the notion of a polynomial. A mathematical praxeology is a tool that is used for modelling and analysing mathematical contents and activities. The dissemination of mathematical praxeologies is analysed and modelled by the notion of a didactic praxeology. A didactic transposition, which is both a tool and a methodology, enables the analysis of knowledge within educational institutions concerning the knowledge of the discipline of mathematics. First, the genesis and development of the notion of a polynomial in the discipline of mathematics are analysed, i.e. the development of the knowledge about the notion of a polynomial that epistemologically legitimises the knowledge taught in pre-tertiary and tertiary educational institutions. The reform of New Mathematics, which was induced by a change in the discipline of mathematics, is also taken into account. The implementation of the reform in the Croatian high school education system is especially investigated, as well as its rudiments and possible effects on the current education system. Documents prescribing knowledge for teaching in Croatian pre-tertiary education were subjected to praxeological analysis, as well as the most used secondary textbooks in the last twenty years, as would gain an insight into the secondary-tertiary transition (Klein’s first discontinuity) since the Bologna process was introduced. The education of mathematicians and mathematics teachers at the Faculty of Science, University of Split is analysed considering the notion of a polynomial. The possible effects of Klein’s second discontinuity on prospective pre-tertiary mathematics teachers are also observed.

Broj 𝜋 s razlogom se može nazvati čudesnim brojem. Već stotinama i tisućama godina fascinira matematičare, a i znanstvenike općenito, svojom zagonetnom beskonačnošću. Što točnije ga odrediti bio je primarni cilj. Kako doći do što točnije formule za izračun, kako ga primijeniti na poljima matematike i arhitekture, problemi su koji su golicali umove znanstvenika kroz povijest. Isto je i danas, kada postoji svojevrsna računalno-algoritamska utrka koja je istovremeno i sprint i maraton – u što kraćem roku doći do što većeg broja točnih decimala koje se sada već broje u stotinama milijarda. There is a reason why number 𝜋 is called the magical number. For more than a couple of hundreds and thousands of years it fascinates mathematicians, and scientists alike with its enigmatic infinity. The prime goal is the more exact value of it. How to get to the more correct formula and how to apply it in a field of math and architecture are problems which intrigued the minds of scientists through the history. Same is today, when there is a kind of computer-algorithm race which at the same time is sprint and marathon – in the least time possible to get to a greater number of decimals which are now already counted in hundreds of billions.

Ovaj rad predstavlja implementaciju edukacijske aplikacije za vježbanje matematičkih zadataka za djecu koja pohađaju četvrti razred osnovne škole. Cilj aplikacije je pomoći djeci sa savladavanjem gradiva na zabavan način. U prvom poglavlju rada opisani su svi zahtjevi koje aplikacija zadovoljava. U drugom poglavlju opisane su slične aplikacije koje već postoje. Nakon toga prikazan je model rješenja aplikacije. U četvrtom poglavlju bavimo se arhitekturom aplikacije i korištenim tehnologijama. Nakon toga objašnjeno je kako su sve funkcionalnosti implementirane te kako lokalno pokrenuti izvorni kod. This paper presents the implementation of an educational application for practicing word problems in mathematics. The aim of the application is to help children master the subject matter in a fun and engaging way. First chapter of the paper describes all the requirements that the application meets. In the second chapter, similar applications that already exist are described. After that, a model of the application's solution is presented. In the fourth chapter, we deal with the architecture of the application and the technologies used. After that, it is explained how all the functionalities were implemented and how to run the source code locally.

Autor se u radu bavi integralnom biografijom Mirka Danijela Bogdanića (Virovitica, 1760. – Budim, 1802.), jednoga od najznamenitijih hrvatskih prirodoslovnih znanstvenika tijekom druge polovice XVIII. stoljeća u Habsburškoj Monarhiji. Donosi se sažeti pregled dosadašnjih istraživanja o Bogdaniću te povijesni kontekst u razdoblju tijekom kojega je živio i radio. Posebno se analizira njegovo obiteljsko porijeklo po očevoj strani važno za njegov napredak kao i važnost rodnoga grada Virovitice osobito sredinom XVIII. stoljeća. U životopisu se interpretira i raščlanjuje Bogdanićeva mladost, obrazovanje, stručna i profesorska karijera, inicijativa za pokretanje prvih pretpreporodnih novina na hrvatskomu jeziku, astronomsko-kartografske ekspedicije, socijalno-psihološki profil te smrt. U drugom se poglavlju interpretiranju i analiziraju Bogdanićeva objavljena i neobjavljena djela s najvećim naglaskom na latinskoj matematičkoj raspravi i sintezi svjetske povijesti staroga Istoka na hrvatskomu jeziku. Naposljetku je poglavlje o recepciji na Bogdanića poslije njegove smrti do današnjega doba. O njemu je do danas pisano u nizu inozemnih periodičkih publikacija te inozemnih i hrvatskih enciklopedija. Bogdanićev primjer potvrđuje da je i u okviru biografike bio slabo zastupljen u hrvatskoj historiografiji. U ovomu se radu Bogdanićev život i rad proučavaju u kontekstu razvoja povijesti znanosti u Hrvata u širem okviru Habsburške Monarhije tijekom XVIII. stoljeća. Temeljem dosadašnjih znanja i novostečenih spoznaja dokazuje se i objašnjava da je Mirko Danijel Bogdanić jedan od najznačajnijih hrvatskih prirodoslovnih znanstvenika XVIII. stoljeća nesvećeničkoga i neplemićkoga porijekla.

Konstrukcija blokovnih dizajna s određenim dopustivim parametrima se često pokušava izvesti za određeni skup parametara uz pretpostavljanje nekih dodatnih ograničenja na strukturu dizajna kako bi pretraživanje bilo računalno izvedivo. Prirodno ograničenje je pretpostavka da određena grupa automorfizama djeluje na dizajn. Jedna od metoda konstruiranja blokovnih dizajna s pretpostavljenom grupom automorfizama je metoda koja koristi orbitne matrice, a sastoji se od dvaju koraka: konstrukcije orbitnih matrica za pretpostavljenu grupu automorfizma i konstrukcije blokovnih dizajna za orbitne matrice dobivene na ovaj način (ovaj korak se naziva indeksiranje orbitnih matrica). Indeksiranje orbitnih matrica obično se provodi metodom iscrpnog pretraživanja. Međutim, ponekad iscrpno pretraživanje nije izvedivo jer postoji previše slučajeva koje bi trebalo provjeriti. Slično, jako regularni grafovi s određenim dopustivim parametrima se mogu konstruirati korištenjem orbitnih matrica za pretpostavljenu grupu automorfizma. Genetski algoritam je metoda pretraživanja koja se koristi u računarstvu za pronalazak točnih ili približno točnih rješenja za probleme optimizacije i pretraživanja. Genetski algoritam oponaša prirodnu evoluciju, odnosno temelji se na optimizaciji populacije, podskupa cijelog prostora pretraživanja. Kao i u prirodi, populacija se sastoji od jedinki koje se mogu razmnožavati i nad kojima mogu djelovati određene mutacije te se na taj način stvaraju nove jedinke s boljim ili lošijim svojstvima od prethodnih. Cilj algoritma je usmjeriti populaciju ka stvaranju boljih jedinki što može rezultirati pronalaskom optimalnih rješenja zadanog problema. U ovoj doktorskoj disertaciji opisujemo korištenje genetskog algoritma u koraku indeksiranja orbitnih matrica za konstrukciju blokovnih dizajna i jako regularnih grafova s pretpostavljenom grupom automorfizama. Korištenjem ovog pristupa konstruirali smo nove blokovne dizajne, točnije nove Steinerove sustave s parametrima S(2,5,45) i nove simetrične dizajne s parametrima (71,15,3) te nove jako regularne grafove s parametrima (96,19,2,4) i (96,20,4,4). Construction of block designs with certain admissible parameters is often attempted for a particular set of parameters with the assumption of some additional constraints on the design structure in order to make the search computationally feasible. A natural constraint is the assumption that a given group of automorphisms acts on the design. One of the methods for constructing block designs with a prescribed automorphism group is the method that uses so called orbit matrices. It consists of two steps: construction of orbit matrices for the given automorphism group and construction of block designs for the orbit matrices obtained in this way (this step is called "indexing of orbit matrices"). Indexing is usually performed by exhaustive search. However, sometimes exhaustive search is not feasible because there are too many cases to check. Similarly, strongly regular graphs with certain admissible parameters can be constructed using orbit matrices for the prescribed automorphism group. Genetic algorithms are search methods used in computing whose objective is to find exact or approximate solutions to optimization and search problems. A genetic algorithm mimics natural evolution, that is, it is based on optimizing a population (a subset of the entire search space). As in nature, the population consists of individuals that can reproduce and that can be affected by certain mutations, thus creating new individuals with better or worse properties than the previous ones. The goal of the algorithm is to direct the population towards creating better individuals, which can result in finding optimal solutions to a given problem. In this doctoral dissertation, we describe the use of a genetic algorithm in the step of indexing of orbit matrices for the construction of block designs and strongly regular graphs with a prescribed automorphism group. Using this approach, we have constructed new block designs, namely new Steiner systems with parameters S(2,5,45), new symmetric designs with parameters (71,15,3) and new strongly regular graphs with parameters (96,19,2,4) and (96,20,4,4).

Zadaci riječima su logički strukturirane govorne cjeline koje sadrže kvantitativne podatke u različitim odnosima i vezama te zahtjev da se iz poznatih uvjeta i podataka pronađe nepoznati broj ili veličina, a upravo su oni tema ovog diplomskog rada. Rad je podijeljen na dva dijela, a to su teorijski i istraživački dio. Teorijski dio odnosi se na davanje dubljeg uvida u rješavanje matematičkih zadataka, s posebnim naglaskom na zadatke riječima, dok se praktični dio odnosi na ispitivanje stavova učitelja osnovnih škola u Zadru prema spomenutim zadacima. Na samom početku teorijskog dijela rada pobliže se opisuje razlika između tradicionalne i suvremene nastave matematike nakon čega slijedi pobliže upoznavanje sa pojmom matematičkog zadatka. Nadalje, opisane su i karakteristike različitih vrsta matematičkog zadatka. Glavninu diplomskog rada zapravo zauzimaju zadaci riječima u nastavi matematike te njihova podjela. Također, susrest ćemo se i sa problematikom koja obuhvaća zadatke riječima, a koja se odnosi većinom na učenike te učitelje. U istraživačkom dijelu rada glavni cilj bio je ispitati stavove učitelja osnovnih škola u Zadru prema zadacima riječima u nastavi matematike te njihovo mišljenje o određenim aspektima takvih zadataka. Dobiveni odgovori su raznoliki, no učitelji se većinom zauzimaju za zadatke riječima. Također, u odgovorima se može izvući jedan od većih problema današnjice, a to je problem novih generacija o kojem se više govori u samom radu. Word problems are logically structured speech units that contain quantitative data in various relationships and connections and the requirement to find an unknown number or quantity from known conditions and data, and they are the subject of this thesis. The paper is divided into two parts, namely the theoretical part and the research part. The theoretical part refers to giving a deeper insight into solving mathematical problems, with a special emphasis on word problems, while the practical part refers to the examination of the attitudes of primary school teachers in Zadar towards the aforementioned problems. At the very beginning of the theoretical part of the paper, the difference between traditional and modern mathematics teaching is described in detail, followed by a closer introduction to the concept of a mathematical task. Furthermore, the characteristics of different types of mathematical tasks are described. The main part of the thesis is actually word assignments in mathematics classes and their division. Also, we will encounter the problem that includes word tasks, which mostly concerns students and teachers. In the research part of the work, the main goal was to examine the attitudes of primary school teachers in Zadar towards word tasks in mathematics classes and their opinion on certain aspects of such tasks. The received answers are diverse, but teachers mostly focus on word tasks. Also, one of the major problems of today can be extracted from the answers, which is the problem of new generations, which is discussed more in the paper itself.

U ovom radu opisana je teorijska pozadina matrica i izrađena mobilna aplikacija za rad s matricama. Navedene su neke od postojećih aplikacija za rad s matricama i njihovi nedostatci. Nakon detaljnog opisa matematičke teorije matrica, opisana je izrada mobilne aplikacije. Mobilna aplikacija obuhvaća unos željenih dimenzija matrica, prikaz matrica, unos vrijednosti u matrice, računanje regularnosti, inverza, zbrajanje, množenje i množenje skalarom. Ispitana je funkcionalnost aplikacije na matricama raznih dimenzija. In this paper, the theoretical background of matrices is described and a mobile application for matrix operations is created. Some of the existing applications for working with matrices and their shortcomings are listed. After a detailed description of the mathematical theory of matrices, the process of developing a mobile application is described. The mobile application includes entering the desired dimensions of matrices, displaying matrices, entering values into matrices, calculating regularity, inverse, addition, multiplication and scalar multiplication. The functionality of the application was tested on matrices of various dimensions.

Matematičke igre su fenomen koji je postao vrlo popularan u poučavanju matematike, ali usprkos svojoj popularnosti i brojnim prednostima, značajan broj nastavnika ih još uvijek izbjegava. Matematička igra Matoboj za osnovnu školu timska je igra, nastala s ciljem poticanja učeničkog povezivanja i sistematizacije određenih matematičkih koncepata. Osim toga, korištenjem Matoboja u nastavi matematike, potiče se učenike na suradnju i diskusiju, a posljedica je razvoj kritičkog razmišljanja i uvažavanja tuđeg mišljenja. Prva verzija igre nastala je 2017. i popraćena je različitim testiranjima i istraživanjima. U diplomskom radu prezentirana je revizija igre, tj. revizija kartica potrebnih za igranje igre., Mathematical games are a phenomenon that has become very popular in the teaching of mathematics, and despite of their popularity and numerous advantages, a significant number of teachers still avoid them. Mathematical game MathDuel for elementary school is a team game, created with the aim of encouraging student connection and systematization of certain mathematical concepts. In addition, by using MathDuel in mathematics classes, students are encouraged to cooperate and discuss, which results in the development of critical thinking and respect for others' opinions. In 2017, the first version of the game was created, accompanied by various testing and researches, and the revision of the game, was presented in the thesis.

U radnji poopćavamo Weierstrassovu reprezentacijsku formulu na plohe u prostoru Minkowskog. Weierstrassova parametrizacija je lokalna konformna parametrizacija plohe kojom se svaka minimalna ploha u \(\mathbb{E}^3\) može reprezentirati parom \((f,g)\) kompleksnih funkcija, gdje je \(f\) holomorfna i \(g\) meromorfna funkcija. U prvom dijelu uvodimo osnovne pojmove i rezultate o plohama u prostoru Minkowskog s posebnim naglaskom na svjetlosne plohe i dualne funkcije, koje ćemo koristiti kao alat za reprezentaciju svjetlosnih ploha. Analiziramo parametrizacije za prostorne i vremenske plohe u \(\mathbb{M}^3\) koje su već pronađene te za formulu McNertney pokazujemo da reprezentira sve regularne prostorne plohe ako koristimo funkcije \(f\) i \(g\) koje nisu nužno holomorfne. Pronalazimo Weierstrassovu parametrizaciju za svjetlosne plohe pomoću para \((f,g)\) dualnih funkcija, gdje je \(f\) holomorfna i \(g\) meromorfna funkcija. Sve svjetlosne plohe u \(\mathbb{M}^3\) su minimalne. Koristeći vezu s dualnim brojevima, pronalazimo svjetlosni analogon katenoida i helikoida te općenito asociranu familiju i adjungiranu plohu svjetlosne plohe. Pokazujemo 1-1 korespondenciju izmedu konformnih parametrizacija maksimalne/minimalne plohe i konformnih parametrizacija jedinične sfere. U drugom dijelu sve Weierstrassove parametrizacije poopćavamo na 2-plohe u \(\mathbb{M}^4\) analogno kao u euklidskom prostoru. Za prostorne plohe je to već napravljeno ranije pomoću kompleksnih funkcija, za vremenske plohe koristimo kompleksne funkcije \((q,r)\) dviju dualnih varijabli i realne funkcije \((f,g)\), a za svjetlosne plohe koristimo dualne funkcije \((\rho, f,g)\). Za svaku reprezentaciju pronalazimo klasu funkcija koje reprezentiraju maksimalne/minimalne plohe. Parametrizaciju za prostorne plohe poopćavamo na 2-plohe u \(\mathbb{M}^n\). U teoriji relativnosti pronalazimo lokalnu parametrizaciju tzv. zarobljenih ploha u Schwarzschildovom prostoru, najjednostavnijem modelu prostor-vremena s crnom rupom. In this work we generalize the Weierstrass representation formula for surfaces in the Minkowski space. The Weierstrass parametrization is a local conformal parametrization of a surface which represents every minimal surface in \(\mathbb{E}^3\) by a pair of complex functions \((f,g)\), where \(f\) is holomorfic and \(g\) a meromorfic function. In the first part we introduce the basic terms and results on surfaces in the Minkowski space with focus on lightlike surfaces and dual functions, which we will use as the tool for representing lightlike surfaces. We analyse the parametrizations for spacelike and timelike surfaces which are already known and show that McNertney’s formula represents all regular spacelike surfaces when the functions \((f,g)\)) are not necessarily holomorphic. We find the Weierstrass representation for lightlike surfaces using a pair \((f,g)\) of dual functions, where \(f\) is holomorphic and \(g\) a meromorphic function. Every lightlike surface in \(\mathbb{M}^3\) is minimal. Using the link with dual numbers, we find the lightlike catenoid and helicoid and we construct the entire associated family and the adjoint surface of a lightlike surface. We show a 1-1 correspondence between the conformal parametrizations of a maximal/minimal surface and the conformal parametrizations of a unit sphere. In the second part we generalize every Weierstrass parametrization for 2-surfaces in \(\mathbb{M}^4\) analogously as in the Euclidean space. For spacelike surfaces this is already done by using complex functions, for timelike surfaces we use complex functions \((q,r)\) of two real variables and real functions \((f,g)\) and for lightlike surfaces we use dual functions \((\rho, f,g)\). For each representation we find the class of functions which represent maximal/minimal surfaces. We generalize the parametrization for spacelike surfaces for 2-surfaces in \(\mathbb{M}^n\). In general relativity we find a local parametrization of the so-called trapped surfaces in the Schwarzschild space, the simplest model of spacetime containing a black hole.

U sklopu ovog završnog rada razvijena je mobilna igra za vježbanje matematike koja se težinom prilagođava igraču i web stranica koja omogućava jednostavnu izradu postavki igre i detaljni pregled rezultata pojedinog učenika u nekoj igri. Navedena igra i web stranice čine jedan jednostavni sustav. Unutar ovog rada detaljno je opisano što ovaj sustav sve nudi te su također opisani i najznačajniji implementacijski detalji. As part of this thesis, there were developed a mobile game for practicing mathematics that is adaptive to the player’s abilities and a website that allows easy creation of game settings and a detailed overview of the results of each student in a game. Mentioned game and website make one simple system. Within this paper, it is described what this system offers in details, just like the most important implementation details.

Struktura izračunljivosti na metričkom prostoru je skup nizova s određenim svojstvima koji nam omogućava da klasične koncepte teorije izračunljivosti prenesemo u proizvoljan metrički prostor. U disertaciji će poseban naglasak biti na strukturama izračunljivosti koje su maksimalne s obzirom na inkluziju te strukturama izračunljivosti koje sadrže gust niz, to jest separabilnim strukturama izračunljivosti. Svaka separabilna struktura izračunljivosti je maksimalna, no obrat općenito ne vrijedi. U radu će se precizno opisati maksimalne strukture na euklidskom prostoru \(\mathbb{R}^n\) te će biti dokazano da je svaka maksimalna struktura izračunljivosti na nekom podskupu od \(\mathbb{R}^n\) zapravo izometrična slika strukture izračunljivosti koja se sastoji od izračunljivih nizova u \(\mathbb{R}^n\), to jest kanonske strukture izračunljivosti. Iz te veze maksimalnih i kanonskih struktura slijedit će da je svaka maksimalna struktura izračunljivosti na \(\mathbb{R}^n\) separabilna. Nadalje, u disertaciji ćemo se fokusirati i na pronalaženje uvjeta pod kojima je separabilna struktura izračunljivosti na metričkom prostoru jedinstvena ili jedinstvena do na izometriju. Pri tome ćemo se ograničiti na efektivno kompaktne metričke prostore. Općenito pitanje je povlači li efektivna kompaktnost metričkog prostora njegovu izračunljivu kategoričnost. Poznato je da je odgovor potvrdan u slučaju kada postoji samo konačno mnogo izometrija tog prostora. U disertaciji ćemo taj rezultat proširiti dokazom da ako dva efektivna separirajuća niza dijele izračunljiv skup za koji postoji samo konačno mnogo izometrija prostora koje ga fiksiraju, tada ta dva niza moraju biti ekvivalentna. Osim toga, dokazat ćemo da je orbita izračunljive točke pri djelovanju grupe izometrija koizračunljivo prebrojiv skup te ćemo to iskoristiti kako bismo dokazali da su određeni efektivno kompaktni podskupovi euklidskog prostora izračunljivo kategorični. Iz tih rezultata slijedit će da na podskupovima od \(\mathbb{R}^2\) i \(\mathbb{R}^3\) efektivna kompaktnost zaista povlači izračunljivu kategoričnost. Na kraju ćemo se baviti prostorima koji su efektivno kompaktni čim na njima postoji efektivan separirajući niz, to jest kategorički efektivno kompaktnim prostorima. Pokazat ćemo kako je pitanje kategoričke efektivne kompaktnosti povezano s pitanjem povlači li izračunljiva prebrojivost skupa njegovu izračunljivost te ćemo dokazati kategoričku efektivnu kompaktnost rubova otvorenih omeđenih konveksnih skupova u \(\mathbb{R}^n\) A computability structure on a metric space is a set of sequences with certain properties that enables us to introduce classical concepts of computability theory into an arbitrary metric space. In the dissertation, special emphasis will be on computability structures that are maximal with respect to inclusion and computability structures that contain a dense sequence, i.e. separable computability structures. Any separable computability structure is maximal, but the converse is in general not true. In the dissertation, maximal computability structures on Euclidean space \(\mathbb{R}^n\) will be described precisely and it will be proved that any maximal computability structure on a subset of \(\mathbb{R}^n\) is actually an isometric image of the computability structure which consists of computable sequences in \(\mathbb{R}^n\), i.e. canonical computability structure. From that bond between maximal and canonical computability structures, it will follow that any maximal computability structure on \(\mathbb{R}^n\) is separable. We will also focus on finding the conditions under which a separable computability structure on a metric space is unique or unique up to an isometry. In doing so, we will limit ourselves to effectively compact metric spaces. A general question is whether effective compactness of a metric space implies its computable categoricity. It is known that this implication is true for metric spaces which have only finitely many isometries. In the dissertation we will expand that result with the fact that if two effective separating sequences share a computable set which has the property that there are only finitely many isometries of the underlying space which fix that set, then the sequences have to be equivalent. Moreover, we will prove that the orbit of a computable point under the isometry group is a co-computably enumerable set and we will use this to prove that certain effectively compact subsets of Euclidean space are computably categorical. From these results, it will follow that in \(\mathbb{R}^2\) and \(\mathbb{R}^3\) effective compactness implies computable categoricity. Lastly, we will examine spaces which are effectively compact if they contain at least one effective separating sequence, i.e. categorically effectively compact spaces. We will show how the question of categorical effective compactness is connected to the question whether computable enumarability of a set implies its computability and we will show categorical effective compactness for boundaries of open bounded convex subsets of \(\mathbb{R}^n\).

U skladu s najnovijim aktivnostima u EU može se zaključiti da prethodno prihvaćene strateške odrednice EU energetske politike koje se očituju u smanjenju emisija stakleničkih plinova, iskorištavanju obnovljivih energetskih izvora i otpadne topline, poboljšanju energetske učinkovitostikako u industriji tako i u zgradarstvu, ne samo da se nastavljaju već i intenziviraju. Ključne aktivnosti su definirane u Energy Road Map 2050. Jedno je sigurno: EU legislativa jasno definira put razvoja energetskih postrojenja temeljenih na obnovljivim energetskim izvorima, posebice mikro/malih/srednjih snaga (1-10-5000 kW) za distribuiranu proizvodnju (eng. distribuited production) električne energije ali istovremeno i toplinske energije u kombi postrojenjima (eng. cogeneration power system). U tom kontekstu je značajno stimulirati proizvođače koji su istovremeno i potrošači energije (eng. prosumers), tj. milione malih investitora koji proizvode energiju za vlastite potrebe u svojim mikro i malim postrojenjima (eng. micro and small scale power system), prodavajući pri tome višak energije u centralnu mrežu. Sumirajući prethodna razmatranja može se zaključiti da distribuirana proizvodnja energije temeljena na obnovljivim energetskim izvorima (DES/RES), mikro mreže (eng. micro grids) te napredne mreže (eng. smart grid) s primjenom naprednih digitalnih tehnologija kao „digitalnog oblaka“ (eng. digital cloud) bit će ključni element energetske politike u EU u bliskoj budućnosti. To znači da hibridni sustavi koji integriraju solarnu i geotermalnu energiju, energiju biomase i vjetra, otpadnu toplinu, dizalice topline i skladištenje energije kod energetski plus zgrada, radeći u naprednim mrežama su izazov koji stoji pred nama. Pošto su solarna, geotermalna i energija iz biomase, te otpadna toplina pretežito nisko i srednjetemperaturni energetski izvori (do 350 °C) za proizvodnju mehaničke energije odn. električne energije a kod kogeneracijskih postrojenja i toplinske energije, u Rankineovom ciklusu vodu odn. vodenu paru kao radni fluid potrebno je zamijeniti s lakohlapljivim organskim fluidom pa imamo organski Rankineov ciklus (eng. Organic Rankine Cycle – ORC). Zbog niske temperaturne razine ORC ima relativno nisku termodinamičku iskoristivost pa je veoma bitna iskoristivost pojedinih komponenti, pumpe, isparivača, kondenzatora, a posebno je kritičan ekspander koji proizvodi mehanički rad. Kao ekspanderi se koriste turboekspanderi i volumni ekspanderi. Volumni ekspanderi su vijčani, pužni, spiralni, klipni i s rotirajućim krilcima, dok su turboekspanderi radijalna i aksijalna turbina. Treba napomenuti da se radijalne turbine znatno više koriste od aksijalnih. U ovoj doktorskoj disertaciji u središtu istraživanja je inovativna, aksijalna, višestupanjska turbina male snage (50 kW). Provodit će se numeričke simulacije strujanja na različitim geometrijama strujnog aparata turbine kako bi se utvrdio utjecaj geometrijskih parametara (parcijalnosti privoda, koraka i širine statorskih i rotorskih rešetki, itd.) na strujne karakteristike odn. unutarnju (izentropsku) iskoristivost turbine. Dobivene spoznaje rezultirat će preporukama za konstruiranje mikro i aksijalnih turbina malih snaga za primjenu u postrojenjima s ORC-om, čija će iskoristivost biti jednaka ili veća od radijalnih turbina. Imajući istovremeno u vidu nove tehnologije izrade (3D printanje) to će doprinijeti širem korištenju aksijalnih turbina. The energy crisis is a challenge for sustainable development. It is caused by the reduction in fossil fuel reserves and significant fluctuations in fuel prices, which are frequently caused by geopolitics and environmental issues (increasing global warming caused by greenhouse gases, air pollution caused by various pollutants, ozone layer damage, acid rains, etc.) because of the rapid consumption of fossil fuels. Low- and medium-temperature heat sources, such as renewables (solar, geothermal, biomass, and ocean thermal energy) as well as waste heat from various industrial plants and processes (such as petrochemical plants, gas turbines, and internal combustion engines (ICEs)), are widely available worldwide. The organic Rankine cycle (ORC) can play an important role in utilizing these heat resources, which is similar to a steam Rankine cycles but uses an organic fluid. The ORC working fluid is very important because it affects the efficiency, operating conditions, economic viability, and environmental efficiency of the entire system. Therefore, the selection of a suitable working fluid has been the objective of many investigations. After 40 years of development, the ORC has proven to be a reliable and adaptable technology in many applications for renewable energy source conversion and waste heat utilization. The key component of the ORC is the expander, which is available in different types: volumetric (piston, scroll, and screw) and turboexpanders (axial and radial inflow and outflow turbines). Axial turbines are not receiving much attention as an option for expanders in small ORCs. Hence, this study focuses on the design of an innovative small-scale, multistage, axial turbine in an ORBC (Organic Rankine Bottoming Cycle) for ICE (Internal Combustion Engine) waste heat utilization, which is competitive with volumetric expanders in terms of efficiency. The high rotational speed of axial turbines, which is their most criticized characteristic, is overcome by using partial admission. The main aim of this research is to develop an innovative small axial action–reaction multistage turbine with partial admission intended for an ORC. It is known from the turbine theory that the rotational speed has a positive effect on the expansion ratio and a negative effect on the height of blades, that is, the mean diameter. Therefore, in most cases, small turbines have high speeds and small mean diameters, which limit their application in ORCs. If the design of the flow part with partial admission is adopted, a satisfactory blade height can be achieved, and the mean diameter can be increased to reduce the rotational speed. In this way, the limiting factor for small turbine application in ORCs is eliminated. CFD analysis is used to investigate the characteristics of the organic fluid flow in the small organic turbine, with the aim of increasing the efficiency of energy conversion. Methods The main aim of this dissertation is to develop an innovative small axial action–reaction multistage turbine with partial admission intended for an ORC, which is comparable in terms of efficiency to existing volumetric expanders. The design process consists of the following steps. Mean diameter 2D preliminary design (PD). The PD has many repetitive calculations to find the relevant geometric characteristics of the nozzle and moving blade cascades, as well as the aero and thermodynamic performances of all turbine stages. This process is known as mean diameter 2D modeling, which is based on flow analysis of the turbine mean diameter and neglects flow property variations in the spanwise direction. The calculation equations and procedure, that is, the mathematical model for this approach, was developed using the Microsoft Excel® software as an in-house 2D code with the input parameters. The Microsoft Excel® software was connected to the REFPROP® database to determine the thermodynamic properties at characteristic points of the turbine stages. Subsequently, the geometric and gas dynamic characteristics of the newly designed axial, action–reaction, multistage turbine with partial admission were determined. Blade geometry generation. Nonstandard profiles were used to design the nozzle and moving blades of single-turbine stages. These profiles were obtained using the original software developed by the University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture. Based on the obtained geometric parameters, the aerodynamically ideal profiles of all nozzle and moving blades were designed using the previously mentioned original software. It is an analytical method of profile design automated by the Matlab® code. The contours of the concave and convex sides of the blade profiles are polynomials of the fifth degree. Modeling and meshing of the nozzle and moving blade cascades. The aim of this step is to create and mesh the stator and rotor cascades from 2D models based on the previous step. The tool for this is the ANSYS Gambit® module, which is used for discretization, that is, dividing the passage, inlet, and outlet fluid domains into a suitable number of elements. Based on the nonstandard designed profiles, the 3D nozzle and moving blade cascades were modeled. The first stator cascade inlet and last rotor cascade outlet were modeled as pressure inlet and pressure outlet, respectively. The hubs, cases, and blades were modeled as walls, while the inlets and outlets of other cascades were set as interfaces. The internal cascade volume was set as a fluid. 3D CFD methodology. The limitation of the previously presented 2D mean diameter PD model of an innovative turbine is that it does not consider the flow inside the stator/rotor passages, which has an influence on providing efficient expansion through the passage. An accurate evaluation of the achievable small ORC turbine performance entails experimental data that are currently lacking and costly in terms of a prototype. Therefore, there is a need for a more advanced technique, such as 3D CFD analysis, to obtain more accurate predictions regarding the performance of a small ORC turbine. In this study, the simulations of steadystate 3D viscous, single-phase, compressible flow in both the nozzle and moving blade cascades of each stage in the entire volume (in the flow part) of the innovative axial turbine were performed using the ANSYS Fluent 16® solver. An element-based finite volume method was adopted to solve the 3D Reynolds-averaged Navier– Stokes equations with a k–ω based shear-stress transport (SST) turbulence model. The k–ω based SST turbulence model has the ability to capture the turbulence closure based on an automatic wall function treatment by identifying the nondimensional distance y + , which allows smooth shifting between the wall function formulation and low Reynolds number through computational grids without loss of accuracy. It produces a highly accurate prediction by the inclusion of transport effects in the flow separation prediction into the formulation of the turbulent viscosity (eddy viscosity). The SST model is a combination of the k–ω model (near the wall) and k–ε model (in the outer portion of the boundary layer ). A topology with a first-order upwind advection scheme was chosen because it is numerically stable. For all runs, the average value of y + was kept approximately at unity. If there is any information on the inlet turbulence, the medium turbulence intensity (intensity = 5%) is the recommended option. All CFD simulations were carried out with the convergence criteria in the range of 10−2 to 10−5 for all residual values and a time scale of 0.5/Ω. The solutions were obtained when the convergence criteria were satisfied. 3D CFD simulations of the nozzle and moving blade cascades. To achieve high isentropic efficiency of the turbine, the aerodynamic characteristics of the blade cascades were checked by numerical flow simulations. Then, based on the simulation results, they were improved by changing the geometries of the nozzle and moving blades. Modeling and meshing of the entire turbine flow volume. After improving the aerodynamic characteristics of the cascades, the partial admission (nozzle) and full (rotor) cascades connected with other dimensions of the turbine stages from the 2D PD were used to form the entire turbine flow volume. The nozzle cascades were all centered, with the Z axis being the centerline. The stator cascade interfaces are arcs, whereas the rotor cascade interfaces are annuluses. When ANSYS Fluent 16® connects them, the stator arc and equivalent arc portion of the rotor annulus are transformed into a single interface, while the rest of the annulus is converted into a wall. The entire flow volume (domain) was meshed using ANSYS Gambit®, and the simulation domain contained 12×106 cells. 3D CFD simulations of the entire turbine flow volume. Using the innovative turbine model, 3D CFD simulations were performed for the entire flow space (volume). The applied boundary conditions were the total temperature, total pressure, flow direction, and rotational speed as the inlet conditions. A rotational adiabatic wall was chosen for the blade, hub, and shroud surfaces. The static pressure was chosen as the output boundary condition. The results show that the quality of energy conversion based on the efficiencies of each cascade, each stage, and the entire turbine can be calculated. The simulations were computationally demanding in terms of RAM, CPU load, and hard drive space required for storing the ANSYS Fluent 16® case and data files. The comparison of 2D PD and 3D average differences of the Mach number, static temperature and pressure range from 2.2 to 13.1% for the stators and from 0.3 to 13.2% for the rotors. Generally, the 3D CFD simulation results agree well with the values obtained from the 2D PD. CFD verification. Regarding the output power and efficiency, owing to the lack of experimental data in small axial turbines working with organic fluids (high-density working fluids), the current 3D CFD simulation results were compared against 2D mean diameter PD results at the operational conditions, to assess the reliability of the PD model and provide an overall evaluation of the 3D CFD model. The maximum difference in terms of power is 6.8%, while that in terms of isentropic efficiency is 9.8%. The calculated parameters are in good agreement and these results are consistent with those which were obtained by other researchers. Results To develop an innovative and efficient small-scale axial multistage turbine with partial admission, an in-house hybrid design methodology, in which 3D CFD analysis in combination with a 2D mean diameter PD code for the calculation of the basic expander geometry, was used. Real gas formulations of isopentane as the working fluid were obtained using the REFPROP® software. Subsequently, 3D steady-state simulations were performed with the next results: 1. The CFD analysis shows that discretization, that is, mesh quality, has a significant impact on the simulation results. Full stator and rotor cascades must be meshed because periodic boundary conditions cannot be used for flow simulation in a partial admission turbine. 2. The simulations are computationally demanding in terms of RAM, CPU load, and hard drive space required for storing the ANSYS Fluent case and data files. 3. Simulations in ANSYS Fluent 16® must be started with isopentane as an ideal fluid, and after it reaches satisfactory convergence, real gas EOS should be enabled. If the simulations are started with real gas EOS, they will crash. 4. The 3D CFD simulation results show good agreement with the 2D PD values with respect to the turbine isentropic efficiency (9.8%) and power output (6.8%). In addition, the values of pressure, static temperature, entropy, and Mach number (speed of sound) at the inlet and outlet of each turbine stage generally agree well with the 2D results. The exception is the Mach number, which show significant discrepancies, especially in rotor cascades owing to the appearance of large values of local Mach numbers. However, it confirms that 3D CFD simulations can capture better the flow physics than the 2D PD. 5. Although this is a subsonic turbine, supersonic flow occurs locally throughout the flow space. Therefore, additional losses due to supersonic flow are expected locally. In addition, entropy increases in areas that are not supplied with fresh steam, which is a result of windage and segment losses of energy in partial admission stages. 6. The pathlines of 3D CFD simulations indicate that the flow in the turbine is not axisymmetric but 3D. In addition, local vortices are present at the flow border areas. 7. The turbine exhibited an isentropic efficiency of 74.8%, power output of 60.35 kW, and ORC thermal efficiency of 11.18%. These results highlight the potential of using a multistage axial turbine to enhance the performance of a small-scale ORC system. Conclusion Waste heat recovery technologies are important for further minimizing fuel consumption and CO2 emissions of ICEs. In this dissertation, an ORC is integrated into a 537 kW natural gas engine to evaluate the possibility of generating electricity by recovering the engine exhaust heat (from exhaust gases and engine cooling water). ORC systems are currently regarded as among the most potent candidates for recovering engine exhaust energy and converting it to electricity. The critical aspects for maximizing the efficiency of ORC systems are the selection of the working fluid and expander design. The main goal of this dissertation is to develop an innovative and efficient small-scale axial multistage turbine with partial admission. To achieve this goal, an in-house hybrid design methodology, in which 3D CFD analysis in combination with a 2D mean diameter PD code for the calculation of the basic expander geometry, was used. The dissertation resulted in a new type of small axial multistage organic turbine with partial admission, which is competitive with existing volume expanders in terms of efficiency. A more detailed and extensive CFD study were conducted to determine the influence of the geometric parameters of turbine stages (pitch of cascades, width of cascades, etc.) on the turbine isentropic efficiency, which will be used to attempt to obtain further efficiency gains from the design. Scientific contribution The scientific contribution of the dissertation includes: 1) Development of a numerical model for the investigation of the influence of the geometrical parameters of the flow part on the flow characteristics in the organic turbines; 2) Methodology for design small organic turbines with recommendations for achieving high isentropic efficiency; 3) A new type of small axial action-reaction multistage turbine with partial admission, intended for ORC, which in terms of efficiency is competitive with existing volume expanders.

Svakodnevno se susrećemo s raznim preprekama i zadatcima koje svjesno ili nesvjesno rješavamo uz pomoć matematike. Matematičko modeliranje u nastavi važan je interaktivan proces u kojem se učenici upoznavaju s problemima iz stvarnoga svijeta i svojega okruženja. Čitavo obrazovanje polako teži boljem obrazovanju učenika o temi rješavanja problema, uključujući matematičko modeliranje, ali to još nije praksa u nižim razredima osnovnih škola. Naše istraživanje opisuje što je zapravo matematičko modeliranje u osnovnoj školi te njegove prednosti i poteškoće, sa stajališta učitelja razredne nastave. Istraživanje je provedeno na 1000 učitelja razredne nastave iz Hrvatske i Slovenije. Rezultati istraživanja ne pokazuju gotovo nikakve razlike prema zemlji podrijetla, iako smo ih očekivali budući da je Slovenija prethodno uvela matematičko modeliranje u svoje kurikule. Rezultati također ukazuju na to da su učitelji prilično neobrazovani po pitanju matematičkoga modeliranja i poteškoća koje bi mogle nastati njegovim uvođenjem u učionice. Također se može izvijestiti da su učitelji željni i spremni za obrazovanje o navedenoj temi te u većini pokazuju interes za dodatno, prijeko potrebno, obrazovanje o matematičkom modeliranju.

U ovoj disertaciji razmatramo model jednodimenzionalnog toka viskoznog toplinski provodljivog realnog mikropolarnog plina kojeg karakterizira generalizirana jednadžba stanja. Iz konstitutivnih jednadžbi mehanike fluida i zakona očuvanja izvodimo pripadni početno-rubni problem s homogenim rubnim uvjetima, prvo u Eulerovim, a zatim u masenim Lagrangeovim koordinatama. Izvedeni model zatim primjenjujemo u slučaju toka i termalne eksplozije reaktivnog fluida. Za oba modela konstruiramo niz aproksimativnih rješenja korištenjem Faedo-Galerkinove metode, opisujemo algoritam za numericko rješavanje i diskutiramo rezultate numeričkih testova. Kako bismo potvrdili znacaj novih modela, ispitujemo utjecaj mikropolarnosti i genera- ˇ lizirane jednadžbe stanja na ponašanje fluida. Naposljetku, za svaki od modela pokazujemo da ima jedinstveno rješenje lokalno po vremenu, a zatim i na proizvoljnom vremenskom intervalu. In this dissertation, we consider a model of the one-dimensional flow of a viscous and thermally conductive real micropolar fluid characterized by a generalized equation of state. From the constitutive equations of fluid mechanics and the conservation laws, we derive the corresponding initial-boundary value problem with homogeneous boundary conditions, first in Euler and then in Lagrangian mass coordinates. The derived model is then applied to the case of flow and thermal explosion of a reactive fluid. For each model, we first show that it has a solution locally in time. We prove this result using a constructive technique based on the Faedo-Galerkin projection. In particular, we show that the constructed sequence of approximate solutions is bounded, and then we obtain convergence (on a subsequence) using classical compactness theorems. Next, we show that the governing initial-boundary problem has at most one solution. Based on the theorems on local existence and uniqueness just proved, we then prove that the problem under consideration has a solution in any finite time interval by applying the principle of expansion. In proving this result, we bound the solutions independently of the finite time interval chosen, using the generalized energy method and the Kazhikov representation of the mass density. Finally, using the semi-discretized approximate systems constucted in the proof of the local existence theorem, we develop a fully discretized numerical scheme for the initial-boundary value problems considered. We perform several numerical tests to confirm the validity of the numerical method and the model itself. We do this by discussing the stabilization properties of the solution. To confirm the significance of the new models, we also investigate the influence of micropolarity and the generalized equation of state on the fluid behavior.

Matematika je znanost koja je nezaobilazna u svim područjima umjetnosti i dizajna. Od samih početaka razvoja civilizacija Egipta i Grčke pa sve do danas, koristi se u arhitekturi, slikarstvu, kiparstvu te modnom i tekstilnom dizajnu. Kroz povijest inspirirala je razne tehnike tkanja, pletenja, prošivanja, raznih vezova pa do izrade turskih i drugih tepiha kao i ćilim. Naglasak ovog rada bit će na op-artu, umjetničkom pravcu koji je nastao 60-ih godina 20. stoljeća. To je razdoblje u povijesti umjetnosti koje koristi poznavanje geometrije, matematičkih pravilnih oblika i optike te djelovanje boje na čovjekov vid kako bi se dobila iluzija kretanja i treperenje. Matematički obrasci glavni su element u stvaranju umjetničkih djela op-arta te će se prikazati i njihova primjena u tekstilnom i modnom dizajnu., Mathematics is a science that is indispensable in all areas of art and design. From the very beginnings of the development of the civilizations of Egypt and Greece until today, it is used in architecture, painting, sculpture and fashion and textile design. Throughout history, it has inspired various techniques of weaving, knitting, quilting, various embroideries, and even the production of Turkish and other carpets as well as rugs. The emphasis of this work will be on op-art, an artistic direction that arose in the 60s of the 20th century. It is a period in the history of art that uses knowledge of geometry, mathematical regular forms and optics and the effect of color on human vision to create the illusion of movement and flicker. Mathematical patterns are the main element in the creation of op-art works of art, and their application in textile and fashion design will also be shown.

Hrvatski astronom, matematičar, geodet, povjesničar i preporoditelj Mirko Danijel Bogdanić (Virovitica, 5. studenoga 1760. – Budim, 31. siječnja 1802.) ostavio je veliki trag u hrvatskoj ranomodernoj povijesti i povijesti egzaktnih znanosti u Hrvata. Jedan je od najpoznatijih prirodoslovnih matematičara Habsburške Monarhije, kojega je znameniti Ivan Paskvić proglasio prvim matematičarem Monarhije. Iako je ostao upamćen kao voditelj astronomsko-kartografske ekspedicije za izradu geografske karte Ugarske i susjednih zemalja Jánosa Lipszkog te pokretač prvih pretpreporodnih novina na hrvatskome jeziku, o Bogdaniću u hrvatskoj historiografiji do danas ne postoje integralna i interdisciplinarna znanstvena istraživanja. U ovom će se izlaganju temeljem raščlambe Bogdanićeva karijernog životopisa istražiti njegov odgojno-obrazovni razvoj od pučkoškolca i učenika zagrebačkoga kolegija do jednoga od najpoznatijih profesora Peštanskoga sveučilišta krajem XVIII. stoljeća. Posebna pažnja u izlaganju pridat će se interpretaciji i raščlambi razvoja pučkoga školstva u ranomodernomu dobu u hrvatskim zemljama u okrilju franjevačkoga i isusovačkoga kulturnog kruga te kruga znamenitih hrvatskih učenjaka na peštanskome sveučilištu u drugoj polovici XVIII. stoljeća. Time će se na jedinstvenom slučaju prikazati intelektualnoga ishodišta jednoga kontinentalnoga hrvatskoga učenjaka u kontekstu političke, društvene, kulturne i intelektualne povijesti hrvatskih zemalja u Stoljeću razuma.

U ovoj disertaciji proučavamo neke afine verteks algebre na nivoima bliskim dopustivima. Poseban naglasak je na prostoj afinoj verteks algebri \(L_{-5/2} (sl(4))\). Određujemo eksplicitnu formulu za singularni vektor konformne težine četiri u univerzalnoj afinoj verteks algebri \(V^{-5/2} (sl(4))\) i dokazujemo da generira maksimalni ideal u \(V^{-5/2} (sl(4))\). Klasificiramo ireducibilne \(L_{-5/2} (sl(4))\)– module u kategoriji \(\mathscr{O}\) i određujemo pravila fuzije za ireducibilne module u kategoriji jakih modula \(KL_{-5/2}\). Dokazujemo i da je \(KL_{-5/2}\) poluprosta tenzorska kategorija. U dokazima koristimo eksplicitnu formulu za singularni vektor i Zhuovu teoriju. Također koristimo svojstva konformnog ulaganja \(gl(4) \hookrightarrow sl(5)\) na nivou \(k = -5/2\) [6] i teoriju afinih \(\mathcal{W}\)-algebri. Pritom dokazujemo da je \(k = -5/2\) kolapsirajući nivo obzirom na subregularni nil-potentni element \(f_{subreg}\) i dokazujemo određene rezultate o iščezavanju i neiščezavanju funktora hamiltonske kvantne redukcije \(H_{f_{subreg}}\). U drugom dijelu disertacije dajemo novi pristup klasifikaciji ireducibilnih modula u kategoriji \(\mathscr{O}\) za proste afine verteks algebre na nivoima bliskim dopustivima. Uvodimo pojam skoro dopustivih i glavnih skoro dopustivih težina i pokazujemo kako se pomoću njih mogu opisati ireducibilni moduli u kategoriji \(\mathscr{O}\) za verteks algebru \(L_{-5/2}(sl(4))\). Pokazujemo da se takav pristup može primijeniti i za verteks algebre \(L_{-1}(sl(n))\), za \(n \geq 3\), čija je klasifikacija ireducibilnih modula u kategoriji \(\mathscr{O}\) određena u [12]. Za navedene verteks algebre konstruiramo široku familiju modula za koje vrijedi Kac-Wakimotova formula karaktera. Na kraju proučavamo verteks algebru \(L_{-7/2}(sl(6))\) i navodimo slutnju u slučaju višeg ranga. In this thesis we study certain affine vertex algebras beyond admissible levels. Our primary focus is the simple affine vertex algebra \(L_{-5/2} (sl(4))\). We determine an explicit formula for the singular vector of conformal weight four in the universal affine vertex algebra \(V^{-5/2} (sl(4))\)and show that it generates the maximal ideal in \(V^{-5/2} (sl(4))\). We classify irreducible \(L_{-5/2} (sl(4))\)–modules in the category \(\mathscr{O}\), and determine the fusion rules between irreducible modules in the category of ordinary modules \(KL_{-5/2}\). We also prove that \(KL_{-5/2}\) is a semi-simple tensor category. In our proofs we use an explicit formula for singular vectors and Zhu’s theory. We also use properties of conformal embedding \(gl(4) \hookrightarrow sl(5)\) at level \(k = -5/2\) [6] and the theory of affine \(\mathcal{W}\) –algebras. We show that \(k = -5/2\) is a collapsing level with respect to the subregular nilpotent element \(f_{subreg}\) and prove certain results on vanishing and non-vanishing of the quantum Hamiltonian reduction functor \(H_{f_{subreg}}\). In the second part of this thesis we give a new approach to the classification of irreducible modules in the category \(\mathscr{O}\) for simple affine vertex algebras beyond admissible levels. We introduce the notion of almost admissible and principal almost admissible weights and show how they can be used to describe irreducible modules in the category \(\mathscr{O}\) for vertex algebra \(L_{-5/2}(sl(4))\). We also show that such an approach can be applied to vertex algebras \(L_{-1}(sl(n))\), for \(n \geq 3\), whose classification of irreducible modules in the category \(\mathscr{O}\) was determined in [12]. For the above algebras, we construct a large family of modules to which the Kac-Wakimoto character formula applies. In the end, we study the vertex algebra \(L_{-7/2}(sl(6))\) and give the conjecture for the higher-rank cases.

Fokus ove teze su modularne krivulje i modularne forme za neku Fuchsovu grupu prve vrste \(\Gamma\), posebno za grupu \(\Gamma_0(N)\). Proučavamo Weierstrassove i \(n\)–Weierstrassove točke, \(n \in \mathbb{N}\), na krivulji \(\mathfrak{R}_{\Gamma}\) u jeziku modularnih formi. Za danu modularnu krivulju \(\mathfrak{R}_{\Gamma}\) i paran cijeli broj \(m \geq 4\) želimo dati efektivan algoritam za provjeru jeli kasp \(a_{\infty}, m/2\)–Weierstrassova točka na \(\mathfrak{R}_{\Gamma}\). Uvodimo prirodno poopćenje pojma Wronskiana kaspidalnih modularnih forma. Proučavamo Wronskiane kanonskih baza prostora \(M_m(SL_2(\mathbb{Z}))\). Proučavamo biracionalna preslikavanja \(X(1) \longrightarrow \mathbb{P}^2\) i računamo jednadžbe dobivenih krivulja. Razvijen je algoritam u SAGE-u koji funkcionira za sve krivulje tipa \(X_0(N)\), genusa \(g \geq 3\) koje nisu hipereliptičke. Kao posljedicu tog algoritma izračunali smo jednadžbe svih kanonskih krivulja tipa \(X_0(N)\), genusa \(3 \leq g \leq 5\), koje nisu hipereliptičke. Razvijen je algoritam za račun Wronskiana linearno nezavisnih modularnih formi. U SAGE-u su izračunati Wronskiani kanonskih baza prostora \(M_m(SL_2(\mathbb{Z}))\), za parne \(m = 12,14,16,\dots,108,110,120\). Temeljem toga dokazan je teorem o vrijednosti tih Wronskiana za bilo koji parni \(m\), do na neku ne–nul konstantu \(\lambda\). Za \(m = 12t\) iskazana je slutnja o vrijednosti konstante \(\lambda\) do na predznak. Dani su numerički primjeri računa u SAGE-u kojima smo dobili jednadžbe ravninskih krivulja \(\mathcal{C} \subseteq \mathbb{P}^2\) biracionalno ekvivalentnih krivulji \(X(1)\). We are interested in modular curves and modular forms for some Fuchsian group of the first kind, especially for the group \(\Gamma_0(N)\). We are studying Weierstrass and \(n\)–Weierstrass points, \(n \in \mathbb{N}\), on curve \(\mathfrak{R}_{\Gamma}\) in the language of modular forms. For a given modular curve \(\mathfrak{R}_{\Gamma}\) and an even integer \(m \geq 4\), we want to give an effective algorithm for checking whether cusp \(a_{\infty}\) is a Weierstrass point on \(\mathfrak{R}_{\Gamma}\). We have introduced a natural generalization of the usual notion of the Wronskian of cuspidal modular forms. We are studying the Wronskians of the canonical bases of the spaces \(M_m(SL_2(\mathbb{Z}))\). We are studying the birational maps \(X(1) \longrightarrow \mathbb{P}^2\) and calculate the equations of the obtained curves. An algorithm has been developed in SAGE that works for all curves of type \(X_0(N)\), of the genus \(g \geq 3\), that are not hyperelliptic. As a consequence of this algorithm, we calculated the equations of all canonical curves of type \(X_0(N)\), genus \(3 \leq g \leq 5\), which are not hyperelliptic. An algorithm for the calculation of the Wronskian of linearly independent modular forms has been developed. In SAGE, we have calculated Wronskians of canonical bases for \(M_m(SL_2(\mathbb{Z}))\), for even \(m = 12,14,16,\dots,108,110,120\). Based on this, the theorem on the value of these Wronskians for any even \(m\), up to some non - zero constant \(\lambda\), has been proved. For \(m = 12t\) we made a conjecture about the value of the constant \(\lambda\) up to the sign.

U radu su autorice istaknule nekoliko svojih primjera korelacije nastave povijesti i nastave matematike, s posebnim osvrtom na organizaciju školskog događaja, znanstveno-povijesnog kampa HiSTEMory kao rezultata takve suradnje. Ukazuju i primjerom potvrđuju da je realizacijom takvoga događaja vrlo uspješno ostvarena zajednička suradnja među učenicima osnovnih škola, učiteljima povjesničarima i matematičarima, lokalnom zajednicom i drugim institucijama koje su prepoznale vrijednost ovakve ideje. Autorice navode primjere znanstveno-povijesnih radionica održanih na HiSTEMory-ju koje su potaknule učenike na sudjelovanje, edukaciju i dodatno istraživanje. Takva aktualizacija učenika u procesu učenja razvija brojne vještine kojima će uspješnije rješavati probleme, donositi odluke, formirati osobnost, razvijati interes za interkulturalizam, znanost, medijsku i financijsku pismenost i drugih vještina suvremenog čovjeka.

Pomorstvo kao djelatnost je skup radnji koje su usko povezane sa morem i brodovima. međutim ono ne uključuje samo plovidbu i navigaciju već povezuje navigaciju, brodogradnju, trgovinu, ekonomiju, menadžment i kompanije. Matematika kao znanost je prisutna u pomorstvu od njegovog početka. Služi za precizne izračune pozicija, opterećenja brodske konstrukcije, izgradnju luka i terminala te još mnogo toga. Cilj ovog rada je pojasniti i prikazati praktičnu primjenu matematičkih funkcija u nekim od svakodnevnih obaveza modernog pomorca. Njihovom primjenom rješavaju se problemi vezani za navigaciju, izgradnju luka i gibanje brodova. Maritime science as an activity is a set of activities that are closely related to the sea and ships. however, it does not only involve navigation but connects navigation, shipbuilding, trade, economics, management and companies. Mathematics as a science has been present in maritime affairs since its inception. It is used for precise calculations of positions, loads of ship construction, construction of ports and terminals and much more. The aim of this paper is to clarify and present the practical application of mathematical functions in some of the daily duties of a modern seafarer. Their application solves problems related to navigation, port construction and ship movement.

Predmet istraživanja doktorske disertacije su samoortogonalni i LCD kodovi konstruirani iz slabo p-samoortogonalnih 1-dizajna. U prvom dijelu disertacije uvest će se osnovni pojmovi i tvrdnje teorije grupa, teorije dizajna, teorije grafova i teorije kodiranja. U drugom dijelu disertacije opisat će se postojeće metode dobivanja binarnih samoortogonalnih kodova iz slabo samoortogonalnih 1-dizajna proširivanjem matrice incidencije, orbitne matrice dizajna i podmatrica orbitnih matrica te će se postojeće metode proširiti i generalizirati. Uvest će se metode konstrukcije samoortogonalnih kodova iz slabo p-samoortogonalnih dizajna nad proizvoljnim konačnim poljem. U trećem dijelu disertacije uvest će se konstrukcija LCD kodova nad proizvoljnim konačnim poljem iz slabo p-samoortogonalnih 1-dizajna, bazirana na proširenju incidencijske matrice, orbitne matrice dizajna i podmatrica orbitnih matrica dizajna. Dodatno, analizirat ce se pod kojim uvjetima će proširenje matrice incidencije t-dizajna i matrice susjedstva jako regularnog grafa generirati LCD kod. U zadnjem poglavlju razvijene metode potkrijepit će se konkretnim primjerima i djelomičnim klasifikacijama te će se opisati i analizirati svojstva dobivenih samoortogonalnih kodova. Priložit će se primjeri konkretnih samoortogonalnih i LCD kodova i djelomične klasifikacije te će se opisati i analizirati njihova svojstva. Za sve navedene konstrukcije koristit će se programski paket GAP ([28]) i njegov paket DESIGN ([41]) te programski paket Magma ([6]). The main subject of the thesis are self-orthogonal and LCD codes constructed from weakly self-orthogonal 1-designs. First part of dissertation will be introduction to group theory, design theory, graph theory, and coding theory. In the second part, we will describe known methods of construction binary self-orthogonal codes from weakly self-orthogonal 1-designs obtained by using extended incidence matrix, orbit matrices, and submatrices of orbit matrices of a design as a generator matrix of a code. Known methods will be extended and generalized in order to obtain self-orthogonal codes over arbitrary finite field. We will describe methods of construction of self-orthogonal codes from weakly p-self-orthogonal designs. In the third part of dissertation we will develop a method of construction of LCD codes from weakly p-self-orthogonal 1-designs, using suitable extension of incidence matrix, orbit matrix and submatrices of orbit matrices of 1-designs as generator matrix of a code. Additionally, we will analyse under which conditions the extension of incidence matrix of t-design and adjacency matrix of a strongly regular graph generates an LCD code. In the last part, we will provide examples and partial classification of self-orthogonal and LCD codes constructed using described methods. We will analyse properties of constructed codes using the computational algebra system GAP ([28]) and it’s package DESIGN ([41]) and computational algebra system Magma ([6]).

Učestalost ozljeda prednje ukrižene sveze kod profesionalnih sportaša i sportski aktivne populacije posljednjih godina raste. Kod ozljede u većini slučajeva je operacijski zahvat jedino rješenje za kvalitetan povratak pacijenta svakodnevnim aktivnostima. Trenutačno se koriste dva kirurška načina liječenja puknute sveze, rekonstrukcija i popravak. Iako popravak sveze nije zlatni standard liječenja puknute sveze, postoji veliki potencijal za primjenu te tehnike jer ona, za razliku od rekonstrukcije, iskorištava potencijal cijeljenja sveze nakon proksimalnog puknuća. Prvi cilj doktorskoga rada bio je eksperimentalno in vitro i numerički ispitati tri trenutačno dostupne tehnike popravka prednje ukrižene sveze radi utvrđivanja biomehaničkih nedostataka postojećih rješenja. Eksperimentalno su se biomehanički nedostaci utvrđivali mjerenjem vrijednosti sila i pomaka na kidalici i umaralici te bilježenjem općeg stanja sveze i kostiju nakon ispitivanja. Numeričko ispitivanje provodilo se primjenom programskog paketa Abaqus. Trodimenzionalni modeli ovčjih kadaveričnih koljena dobiveni su rekonstrukcijom geometrije iz podataka računalne tomografije. Izrada prototipa biomehaničke potpore cijeljenju prednje ukrižene sveze, bio je drugi cilj ovoga doktorskog rada. Konstrukcijsko rješenje biomehaničke potpore odabrano je na temelju računalnih simulacija i rezultata eksperimentalnog ispitivanja sklopa bedrena kost – biomehanička potpora – goljenica. Rezultati eksperimentalnog ispitivanja pokazali su da od tri postojeće tehnike, tehnika popravka s unutarnjom potporom ima dobra biomehanička svojstva, usporediva s nativnom prednjom ukriženom svezom. Međutim, sve tri postojeće tehnike imaju biomehaničke nedostatke vidljive kroz povećanje vrijednosti von Mises naprezanja i pomaka na samoj svezi te manjih iznosa najvećih vlačnih sila uspoređujući ih s vrijednostima vlačnih sila nativne sveze. Dinamičkim ispitivanjem utvrđena je pojava proširenja provrta na mjestima dodira implantata s unutarnjom potporom, dok je numeričkim ispitivanjem utvrđena pojava većih iznosa von Mises naprezanja na mjestima dodira bedrene kosti i gumbića postojećih tehnika popravka što je u kliničkoj praksi povezano s otkazivanjem implantata i smanjenjem gustoće kostiju – Stress Shielding. Rezultati istraživanja su pokazali da se može napraviti prototip biomehaničke potpore koji ne djeluje destruktivno na okolna tkiva i nema iste biomehaničke nedostatke kao i postojeća rješenja uz zadržavanje dobrih mehaničkih svojstava. Background: The incidence of anterior cruciate ligament (ACL) injuries in professional and recreational athletes has increased in recent years. The main function of the ACL is to connect the back of the thigh to the front of the tibia, providing stability to the knee joint and limiting rotation during movement. A tear of the anterior cruciate ligament affects the biomechanics of the entire knee joint, increases the risk of secondary injury and osteoarthritis, and limits the patient's quality of life, especially in professional athletes. In most cases where the ACL is significantly damaged, surgery is the only treatment that allows patients to return to their daily activities with good results. Currently, two surgical treatments can be performed in medical practice when an ACL tear is diagnosed. The first treatment, which is now considered the gold standard, is ACL reconstruction, and the second is ACL primary repair. ACL reconstruction involves reconstructing the ligament with a graft of healthy donor tissue, while ACL repair involves fusing the torn remnants of the ligament together to take advantage of the healing potential. Both techniques have certain biomechanical disadvantages. However, the ACL repair technique has more significant biomechanical disadvantages and biological defects in the ligament itself that prevent this healing technique from becoming a new gold standard for the medical treatment of proximal ACL rupture. Despite the biomechanical disadvantages, the primary ACL repair technique has great potential in terms of ligament healing properties. Namely, the union of ligamentous remnants after proximal rupture allows healing and preservation of natural tissue as well as preservation of proprioceptive sensory fibers. Therefore, this topic was selected as a research problem to be addressed in this thesis. The proposed research objective is to develop a biomechanical support that improves the healing of the ACL using a minimally invasive surgical procedure. In order to achieve the proposed goal, three currently available ACL repair techniques must be studied experimentally "in vitro" and numerically to identify the biomechanical shortcomings of the existing solutions and to avoid these shortcomings in the development of a new biomechanical support. The research is based on the following hypothesis: it is possible to develop a biomechanical support for the healing of the anterior cruciate ligament of the knee joint with good mechanical properties, ensuring the stability of the knee and the installation of the support without destructive effects on the surrounding tissues. Materials: Experimental studies were performed on fresh knee joint sheep carcasses. The use of animal by-products was approved by the Veterinary and Food Safety Directorate of the Ministry of Agriculture, Zagreb, Croatia, under class: UP / I-322-01 / 20-01 / 32. The specimens were supplied in one piece; the hind legs with all the supporting tissue. There were 35 specimens in total, divided into five groups of seven specimens each. The specimens were prepared for testing in predefined groups before freezing to prevent deterioration of the material properties of the ACL. All soft tissues were removed from the specimens; the only connective tissue remaining in the joint was the ACL. The fibula was also removed from the specimens, leaving only the femur and tibia as bone tissue. Finally, a femur - intact ACL - tibia (FATC) was obtained. After the speciman preparations, it was necessary to adjust the length of the bones for the experimental tests. Therefore, the femur and tibia were cut 20 cm from the joint capsule and drilled with a Φ 6 drill to fit into the mechanical holders constructed according to the available literature to perform experimental tests on the FATC. After the initial preparation of the FATC, techniques to repair the ACL were incorporated into the complex. Rupture of the ACL on the specimens for the ACL repair groups was performed at the proximal level. Five groups of specimens were formed: - Group 0 - FATC with intact ACL; - Group 1 - FATC with proximally dissected ACL repaired using an end-to-end suture ACL repair technique; - Group 2 - FATC with proximally dissected ACL repaired with a primary ACL repair using an Internal Brace bridging technique [1]; - Group 3 - FATC with proximally dissected ACL repaired with a non-absorbable suture using a bridging technique with a distally placed spring - "dynamic bridge". - Group 4 - FATC with proximally dissected ACL repaired with a new biomechanical support. The first test group 0 served as a control group. In group 1, the ACL was repaired using the end-to-end ACL repair technique with 0.2 mm thick Krakow-style medical sutures at the ACL stumps. The proximally dissected ACL was sutured with three self-locking sutures longitudinally on the side of the ligament to its midpoint, then the suture was moved to the other side of the ligament and three more sutures were placed proximally. In group 2, the repair technique was performed with a braided suture tape installed in the FATC by passing a 2.5-mm-thick polyethylene suture through a drilled tibial tunnel, the dissected ACL, and the drilled femoral tunnel and fastening it to the lateral cortical part of the femur and the medial metaphyseal part of the tibia with four-hole buttons. The four-hole buttons, 12 mm in length and 2 mm in diameter, were made of steel and cut with a laser. In group 3, the ACL was repaired with non-absorbable suture in a bridging fashion with a distally placed spring - "dynamic bridge", where the spring mechanism is uniquely designed according to the literature data. Self-locking is provided by a bearing ball to prevent damage to the suture material. The installation of the dynamic bridge repair solution required the creation of two tunnels in which the polyethylene suture is placed using the "dynamic bridge" technique. The support is attached to the lateral part of the femur with a button and a slightly larger hole is drilled in the tibia to place the spring system, through which the polyethylene brace is passed and attached to the end of the system with a bearing ball and screw. Group 4 is a group in which a new biomechanical support was applied to a proximally dissected ACL and developed as part of the doctoral research. Numerical testing setup: For FATC three-dimensional (3D) reconstruction, it was necessary to collect the sheep knee CT scans. To obtain CT scans, the sheep knee joint cadaver was processed in the radiology laboratory on a Siemens SOMATOM Definition Edge ultra-computed tomography (Siemens Healthcare GmbH), in Sestre milosrdnice Clinical Hospital Center (Clinical Department of Radiology, Zagreb, Croatia). Data from CT were imported into Mimics (Materialise, Leuven, Belgium), and models were further refined in SolidWorks 2018. For mesh generation, C3D10: A 10-node square tetrahedral finite element type was used. The load applied to the finite element model corresponds to the performed experimental tensile loading testing. For the numerical tests, the femur was loaded with a tensile force of 100 N for all groups except group 1 for which a force of 10 N was applied. The entire tibia was fixed (U1=U2=U3=UR1=UR2=UR3=0) for all groups. Therefore, to simplify the calculations, isotropic, homogeneous, and elastic material properties were assigned to all parts, including the bone. Five test models were created: - Model 0 corresponds to experimental group 0; - Model 1 corresponds to experimental group 1; - Model 2 corresponds to experimental group 2 [1]; - Model 3 corresponds to experimental group 3 - Model 4 corresponds to experimental group 4 Methods: Experimentally, the biomechanical deficits were determined by measuring the forces and displacements on the static and servo-hydraulic testing machine on sheep cadavers and by recording the general condition of the ligament and bones after testing. Experimental testing was divided into three types. The first type was the cyclic loading test, which was performed on a servo-hydraulic fatigue testing machine type LFV-50-HH, Walter+Bai (Switzerland). The purpose of the cyclic loading tests was to determine the mechanical properties, such as the stability of the FATC, under native and repaired ACL and the condition of the tissue itself due to exposure to the same cyclic loading regime. Tests were performed under 2000 cycles at a frequency of 1 Hz under an applied force ranging from 5 N - 100 N for group 0, group 2, group 3, and group 4. Since the same force range could not be obtained for group 1, the group was tested with a force in the range of 5 N - 10 N. In a dynamic cyclic loading test, the tibia was tightened while the femur was loaded under limited displacement conditions. The second type of test was an extension and flexion simulation performed on a specially designed biomechanical device attached to the servo-hydraulic testing machine. Tests were performed under 5000 cycles at a frequency of 1 Hz without any load being applied. During experimental testing, the tibia was tightened while the femur could move within the specified limits of 27 ± 4 degrees. Determining the stability of the joint, as well as the effect of ACL repair technique methods on bone fixation after a specified number of cycles, was the goal of dynamic testing on a biomechanical device. The last group was a uniaxial tensile test on a static testing machine Beta 50-5, Messphysik (Austria) with a maximum loading force of 50 kN. In a uniaxial tensile test, the tibia was clamped while the femur was loaded with tensile force at a speed of 200 mm/s until failure. Within this test, force-displacement curves were compared to determine the maximum load to failure (Ft,max) that a given test group could withstand. The mechanical bone holders were used for the tensile load tests. These holders are designed to allow positioning of the axis of the femur and tibia in accordance with the axis of the ACL to avoid the occurrence of varus / valgus and changes in torsional rotation. The specimens were removed from the freezer 24 hours before testing and were moistened with saline (NaCl) throughout the test period. Numerical tests were performed in the Abaqus software package using three- dimensional models of sheep knees obtained by geometry reconstruction from computed tomography images. The new design of biomechanical support was selected based on computer simulations and in vitro experimental testing of the bone-implant complex, while measurement of displacement was performed using the digital image correlation method. The conditions and type of experimental and numerical testing were the same as for testing the existing ACL repair techniques. The biomechanical support was made of thermoplastic polyethylene, circular in shape with three adhering parts on which are the holes for the screws with which the support is fixed to the bone. Results: Comparing all four techniques based on experimental testing results, the highest stress to failure was in group 0, followed by group 2, group 3, and finally ACL repair with suture (group 1). All groups showed satisfactory stability during dynamic testing, except for group 1. In the controlled displacement group, the test was performed within the set displacement limits. In group 2, shear action and notch effect occurred at the site of button suture fixation, but the ligament remained preserved. Based on the collected results of the von Mises stress and displacement values, it can be concluded that the lowest stress and displacement values are those at the native ACL (Model 0). The increase in the value of von Mises stress in the comparison of other models is: 22.4% higher in the end - to - end suture technique (Model 1), 49.736% higher in the Internal Brace technique (Model 2), and 24.542% higher in the internal support technique with a distally placed spring (Model 3). The occurrence of significantly higher stress values 458.3 MPa was observed in Model 2 at the contact of the button, and in Model 3 at the same place, and is 400 MPa. The more significant von Mises stress values are those at the suture of the Model 2 and Model 3 and are 547 MPa and 513.7 MPa, respectively. The maximum displacement value at the ACL in Model 1 is 46.226% higher compared to the ACL value in Model 0, 63.548% higher in Model 2 compared to Model 0 and 71.27% in Model 3 compared to Model 0. The appearance of stresses on the femur at the point of contact with the button is indicative of the occurrence of the notch effect and shear action. These stresses may lead to instability of the knee joint during the rehabilitation process and may be the cause of failure for an ACL repair technique with internal brace augmentation. The occurrence of higher stresses on the button and Internal Brace support may cause the suture to pull out during the application of external forces on the knee during rehabilitation. Based on clinical studies, the ACL usually ruptures at the contact points between the bone and the ligament, or these injuries are usually proximal. Consequently, the results of the numerical analysis showed that the distribution of maximum von Mises stress is at these locations. New biomechanical support maintained the stability of the assembly during the dynamic test. Von Mises stress values on the ACL repaired with new biomechanical support are lower than in any of the techniques tested. The design solution showed good mechanical properties compared to existing solutions. The data obtained using the ARAMIS optical system were not applicable as the paint layer peeled off the ligament during the test. Limitations: The entire study was done on an animal model, although the animal model is more acceptable for preclinical studies. The original implant for the DIS technique could not be obtained, so a replica of this ACL repair solution was made. Numerical models are simplified. The ACL is assigned the property of linear elasticity. Since the properties of existing anterior cruciate ligament repair techniques and the impact on surrounding tissues were examined as part of the rehabilitation process where the forces on the knee are of smaller amounts, such a simplification was acceptable. Although the experimental 1DOF ACL tests do not have great clinical significance, the proposed studies can be used for comparison with relevant clinical findings and can be a basis for further identification of biomechanical deficiencies of the tested techniques. Conclusion: Based on the results of the experimental test performed, it is possible to determine the biomechanical flaws of ACL repair techniques through static and dynamic tests. The results of the dynamic test showed that the end-to -end suture repair technique is not a technique that contributes to the stability of the femur-repaired anterior cruciate ligament-tibia complex, while the repair technique Internal Brace and internal support with a spring system provide satisfactory stability of the FATC. The dynamic study showed the occurrence of the notch effect of the femur and tibia in the Internal Brace and DIS technique. This phenomenon is associated with the risk of implant failure in scientific and clinical studies. Fibrous damage in ACL is visible after a dynamic test in all repair techniques. The results of a numerical study of the existing techniques showed that the von Mises stress values at the ACL increase with the repair techniques compared with the native ACL. The critical stress is at the contact between the femur and the button in the Internal Brace technique. Clinical evidence suggests that this site is the cause of potential implant failure. In addition, the occurrence of greater stress at this site is the cause of Stress Shielding, resulting in the weakening of the bone tissue. The design of the new biomechanical support has a lower tensile load compared to other repair techniques. However, based on dynamic testing, the design was found to have satisfactory stability. In addition, the design of the biomechanical support reduces the von Mises stress on the ligament itself and precludes the occurrence of large values of von Mises stress on the femur. The design of the biomechanical support precludes the drilling of a bone tunnel and the placement of implants in the bone marrow, which is the cause of Stress Shielding. In addition, the notch effect on the femur was not noticed. It was found that it is possible to design biomechanical support that has satisfactory mechanical properties in terms of sufficient stability of the knee joint and stress reduction for the ACL itself and, unlike existing solutions, does not have a destructive effect on other surrounding tissues. Future work: The biomechanical support prototype can be improved in various ways. One way is to look like a fishing net. The arrangement of threads within the net can be made by methods of different types of weaving that differ in the different interconnection of vertical and horizontal threads. Weaving threads can be with or without reinforcement. Different designs of the biomechanical support enable the real potential application of the newly constructed solution, which is the goal of such and similar research, and not just a theoretical presentation of a possible solution. Finally, the application of the newly constructed biomechanical support is not limited to the knee joint, but there is the potential to apply it to other tendons and ligaments of the locomotor system to improve the quality of human life.

Prvih šest godina života su vrlo bitne za razvoj djeteta. Bitne su za razvoj sposobnosti i spremnosti za učenje. Dijete se u tom razdoblju nalazi na putu stvaranja. To je vrijeme kada dijete uči nesvjesno, osjećajima i kretanjem. Djeca u razdoblju do šeste godine života imaju osobinu samorazvoja koju treba potaknuti utjecajem odgovarajuće okoline i određenim didaktičkim materijalima. Dijete rado istražuje, sklapa kontakte i brzo se može prilagoditi novim situacijama. Da bi dijete samostalno, slobodno usvojilo određeno znanje potrebno je prije svega djetetu pružiti osjećaj prihvaćenosti i ljubavi. Djeci treba pružiti mogućnost slobode biranja sadržaja i materijala te aktivnosti kojim će se baviti. To je cilj odgoja djece predškolske dobi prema Mariji Montessori. U odgoju i obrazovanju se mogu uočiti određene razlike danas s obzirom na prijašnje vrijeme. Razlika se uočava u odnosu samog odgojitelja prema djeci. Da bi djeca razvila matematičke sposobnosti Maria Montessori je podredila prostor i vrijeme djetetu. Didaktički materijali i vježbe su prilagođeni da djeca razvijaju matematičke sposobnosti te da upoznaju svijet. Posebno se naglašava razvoj matematičkog razmišljanja. Za nju je važno da dijete stvara konkretna iskustva, pojmove i relacije iz kojih će se kasnije moći lakše razvijati aspekt pojmova, pravila i postupaka. Odgojitelji dobivaju novu ulogu pratioca djece kroz učenje i upoznavanje svijeta u kojem živimo. Cilj je djecu pripremiti za svakodnevni život, da postanu odgovorni i sposobni članovi obitelji i društva. The first six years of life are very important for a child's development. It is essential for the development of abilities and readiness to learn. The child is on the path of creation during this period. It is a time when a child learns unconsciously, by feelings and movement. Children up to the age of six have a trait of self-development that should be encouraged by an appropriate environment and certain didactic materials. The child is happy to explore, make contacts and can quickly adapt to new situations. In order for a child to independently and freely acquire certain knowledge, it is first of all necessary to give the child a feeling of acceptance and love. Children should be given the opportunity to freely choose the content and materials and activities they will engage in. This is the goal of raising preschool children according to Maria Montessori. Upbringing and education is completely different today than it was before. The difference is noticeable in the attitude of the educator himself towards the children. In order for children to develop mathematical ability Maria Montessori subordinated space and time to the child. Didactic materials and exercises are adapted for children to develop mathematical abilities and to get to know the world. The development of mathematical thinking is especially emphasized. It is important for her that the child creates concrete experiences, concepts and relations from which it will be possible to develop the aspect of concepts, rules and procedures later. Educators get a new role of accompanying children through learning and getting to know the world we live in. The goal is to prepare children for everyday life, to become responsible and capable members of family and society

Cilj ovoga diplomskog rada jest prikazati način na koji se kroz povijest gradila metodika i način poučavanja matematike. Na samome početku opisana je metodika matematike, nekada i danas, način na koji je nastajala te kako se prenosila s generacije na generaciju. Osim navedenoga, ukratko je prikazana struktura društva, točnije razdoblje koje je obilježilo djelovanje poznatoga matematičara, Franje Močnika. To razdoblje je početak 19. stoljeća, a popraćeno je mnogobrojnim nacionalnim pokretima u Europi. U to vrijeme, sve je više jačala nacionalna svijest pojedinih naroda te težnja za izgradnjom vlastite kulture. Jačanje nacionalne svijesti dovelo je do Revolucije poznate kao i Proljeće naroda 1848. godine. Bez obzira na teške društvene i političke promjene, Franjo Močnik nastavio je svoj rad usmjeravati ka elementarnoj matematici na Tehničkoj akademiji, gdje je ujedno bio i profesor. Njegov rad i doprinos obrazovanju bio je vrlo značajan. Posebice se isticao ambicijama i upornošću u radu, a ni nakon stečene učiteljske profesije, nije odustajao te je doktorirao filozofiju. Posvetio se pisanju udžbenika za učitelje i učenike te na taj način utjecao na što bolju organizaciju obrazovanja, kao i bolji položaj učitelja. Svrha ovoga diplomskog rada bila je prikazati život Franje Močnika, njegov intelektualni rad te pedagoški utjecaj na pisanje udžbenika. Kako bi se dokazao navedeni utjecaj, u ovome radu, komparativno će se prikazati analiza nekoliko udžbenika kroz prizmu metodike. Na samome kraju, analizom udžbenika vidjet će se kolike su sličnosti, a kolike razlike s današnjim školskim udžbenicima. Through this diploma thesis, an attempt was made to show the way in which the methodology and the way of teaching mathematics have been built throughout history. To begin with, the methodology of mathematics then and today is described, how it was created and how it was passed from one generation to another. In addition to this part, the structure of society in that period is briefly presented, more precisely the period that marked the work of the famous mathematician Franjo Močnik. Namely, this period, the beginning of the 19th century, was accompanied by numerous national movements in Europe. The national consciousness of certain people, the aspiration to build their own culture and the like were growing stronger. Also, the very famous revolution of 1848 broke out. Despite this, Franjo was at that time a professor of elementary mathematics at the Technical Academy. His work and contribution to education was very significant. He stood out with his ambitions and perseverance in his work. After acquiring his teaching profession, he did not give up and received his doctorate in philosophy. After some time, he started writing textbooks, both for students and teachers. He also worked at the University of Moravia. However, Močnik also fought for the best possible organization of education and the best possible position for teachers. The aim of this thesis was to show the life of Franjo Močnik in general, but also his work and the way of writing textbooks. Therefore, an analysis of several of his books on elementary school was done. Through the analysis of these books we can notice both similarities and differences with today’s way of writing textbooks.

U ovom radu istraženo je koliko su funkcije zastupljene u kurikulumu osnovne i srednje škole te na koji način se uvode i obrađuju. Cilj je da se dobije jasan pregled zastupljenosti funkcija u kurikulumu uz komentare usmjerene na matematičku korektnost, postupnost i sistematičnost obrade funkcija na nastavi. Zaključak je da se funkcije ne obrađuju dovoljno u osnovnoj školi dok su u srednjoj školi funkcije velikim dijelom matematički korektno i postupno obrađene., In this paper I'll demonstrate how functions are represented in the primary and secondary school curriculum and how they are introduced and processed. The goal is to get a clear overview of the representation of functions in the curriculum with comments focused on the mathematical correctness, gradual and systematic processing functions in schools. Conclusion is that functions are not covered enough in primary school while in high school functions are largely mathematically correct and gradually processed.

U radu definiramo novu oblikovnu kategoriju \(Sh^{\circledast}\) koju ćemo nazvati kategorijom konačnoga gruboga oblika. Ta će kategorija imati iste objekte kao kategorije oblika i gruboga oblika, ali će morfizmi među tim objektima biti drugačiji. Kategoriju konačnoga gruboga oblika konstruiramo korištenjem teorije inverznih sustava i poliedarskih ekspanzija topoloških prostora, to jest, vanjskim pristupom. Definirat ćemo dva odgovarajuća vjerna funktora, jednoga iz kategorije oblika u kategoriju konačnoga gruboga oblika, a drugoga iz kategorije konačnoga gruboga oblika u kategoriju gruboga oblika. Primjerima ćemo pokazati da ti funktori nisu puni, to jest, da je kategorija konačnoga gruboga oblika prava natkategorija kategorije oblika i prava potkategorija kategorije gruboga oblika. Kategoriju konačnoga gruboga oblika za kompaktne metričke prostore opisat ćemo i \textit{unutarnjim pristupom}. Da bismo to postigli, restringirat ćemo klasu objekata na skup svih zatvorenih podskupova Hilbertove kocke \(Q\), a teoriju inverznih sustava zamijeniti teorijom \(\epsilon\)-neprekidnih funkcija. Stoga ćemo prvo detaljno razraditi teoriju \(\epsilon\)-neprekidnosti definirajući osnovne pojmove i dokazujući najvažnija svojstva \(\epsilon\)-neprekidnih funkcija i relacije \(\epsilon\)-homotopije. Potom generaliziramo Borsukove fundamentalne i aproksimativne nizove te Sanjurjove približavajuće nizove definirajući \(\circledast\)-fundamentalne, \(\circledast\)-aproksimativne te \(\circledast\)-približavajuće nizove redom. Na skupovima \(\circledast\)-fundamentalnih, \(\circledast\)-aproksimativnih i \(\circledast\)-približavajućih nizova definirat ćemo odgovarajuće relacije ekvivalencije čije će klase biti morfizmi novih kategorija \(Sh^{\circledast}_f\), \(Sh^{\circledast}_a\) i \(InSh^{\circledast}\) redom. Kategoriju \(InSh^{\circledast}\) nazvat ćemo \textit{kategorijom unutarnjega konačnoga gruboga oblika}. Nadalje, definiramo tri odgovarajuća funktora, jedan među kategorijama \(Sh^{\circledast}|_Q\) (restrikcija kategorije \(Sh^{\circledast}\) na zatvorene podskupove od \(Q\)) i \(Sh^{\circledast}_f\), jedan među kategorijama \(Sh^{\circledast}_f\) i \(Sh^{\circledast}_a\) te jedan među kategorijama \(Sh^{\circledast}_a\) i \(InSh^{\circledast}\). Dokazat ćemo da će tako definirani funktori biti kategorijski izomorfizmi, što će značiti da je \(InSh^{\circledast}\) unutarnja reinterpretacija kategorije konačnoga gruboga oblika zatvorenih podskupova Hilbertove kocke. Uz to, definirat ćemo vjeran funktor među kategorijama \(InSh\) i \(InSh^{\circledast}\) koji objekte drži fiksnima, a svakom morfizmu unutarnjega oblika pridružuje inducirani morfizam unutarnjega konačnoga gruboga oblika. Stoga ćemo postojeću Sanjurjovu kategoriju \(InSh\) unutarnjega oblika smatrati pravom potkategorijom nove kategorije \(InSh^{\circledast}\) unutarnjega konačnoga gruboga oblika. Konačno, dokazat ćemo da unutarnji konačni grubi oblik ne ovisi o ulaganju metričkoga prostora u Hilbertovu kocku \(Q\), odnosno, da su svaka dva smještenja proizvoljnoga kompaktnoga metričkoga prostora istoga unutarnjega konačnoga gruboga oblika. Time ćemo klasifikaciju po unutarnjem konačnom grubom obliku proširiti na cijelu klasu \(\mathcal{MC}pt\) svih kompaktnih metričkih prostora i pritom dokazati da su kompaktni metrički prostori \(M\) i \(M'\) istoga konačnoga gruboga oblika ako i samo ako su \(M\) i \(M'\) istoga unutarnjega konačnoga gruboga oblika. In this thesis we will define a new shape category \(Sh^{\circledast}\) of topological spaces called the finite coarse shape category. Category \(Sh^{\circledast}\) will have the same class of objects as the categories of shape and coarse shape, but morphisms will be different. We will contruct the finite coarse shape category by using polyhedral expansions and inverse systems theory, i.e., using an external approach. Two appropriate faithful functors will be defined, one of them from the shape category \(Sh\) to the finite coarse shape category \(Sh^{\circledast}\) and the other one from the finite coarse shape category \(Sh^{\circledast}\) to the coarse shape category \(Sh^*\). We will prove by the examples that those functors are not full, i.e., that the finite coarse shape category is a proper subcategory of the coarse shape category and that it contains the shape category as its proper subcategory. The coarse shape category of metric compacta will also be given an intrinsic approach. To achieve that, we will restrict the class of objects to the set of all closed subsets of the Hilbert cube \(Q\) and replace the inverse systems theory by the theory of \(\epsilon\)-continuous functions. Therefore we shall give a detailed overview of the \(\epsilon\)-continuity theory by defining the main notions and proving the most important properties of the \(\epsilon\)-continuous functions and \(\epsilon\)-homotopy. We will generalise Borsuk's fundamental and approximative sequences and Sanjurjo's proximate sequences by defining so called \(\circledast\)-fundamental, \(\circledast\)-approximative and \(\circledast\)-proximate sequences, respectively. On the sets of the \(\circledast\)-fundamental, \(\circledast\)-approximative and \(\circledast\)-proximate sequences between any two closed subsets of the Hilbert cube \(Q\) we will define an appropriate equivalence relations, classes of which will be morphisms of the new categories \(Sh^{\circledast}_f\), \(Sh^{\circledast}_a\) and \(InSh^{\circledast}\), respectively. Category \(InSh^{\circledast}\) will be called the intrinsic finite coarse shape category}. Futhermore, three new functors will be defined, one of them between the categories \(Sh^{\circledast}|_Q\) (the restriction of the \(Sh^{\circledast}\) category to the set of all closed subsets of \(Q\)) and \(Sh^{\circledast}_f\), the second one between \(Sh^{\circledast}_f\) and \(Sh^{\circledast}_a\) and the third one between the categories \(Sh^{\circledast}_a\) and \(InSh^{\circledast}\). We will prove that each of these three functors is an isomorphism, which means that category \(InSh^{\circledast}\) is an intrinsic reinterpretation of the finite coarse shape category \(Sh^{\circledast}|_Q\) of all closed subsets of \(Q\). We will also define a faithful functor from \(InSh\) to \(InSh^{\circledast}\) which is an identity on the set of objects and which associates each intrinsic shape morphism with the induced intrinsic finite coarse shape morphism. Hence, Sanjurjo's intrinsic shape category \(InSh\) may be considered as a proper subcategory of the constructed intrinsic finite coarse shape category \(InSh^{\circledast}\). In the last section we will prove that the intrinsic finite coarse shape does not depend on the embedding of a compact metric space into the Hilbert cube \(Q\), i.e., that every two embeddings of any compact metric space have the same intrinsic finite coarse shape. That will allow us to extend the classification by the intrinsic finite coarse shape to the whole class \(\mathcal{MC}pt\) of metric compacta. Finally, we will prove that two compact metric spaces \(M\) and \(M'\) have the same finite coarse shape if and only if \(M\) and \(M'\) have the same intrinsic finite coarse shape.

Iako STEM, preciznije ICT sektor, robotika i elektrotehnika za žensku populaciju nisu više strani pojmovi i sve više žena odlučuje se upravo za taj pravac u karijeri, znatna razlika u interesima među spolovima i dalje je prisutna. Kako bi se pronašao uzrok, a time i potencijalno rješenje, u ovom radu ispitana su četiri cilja: utvrditi u kojoj mjeri spol utječe na interese iz tri školska predmeta (matematike, informatike i tehničke kulture), utvrditi utjecaj najuže obitelji na stavove učenika iz navedena tri predmeta, provjeriti postoje li stereotipi prema spolu među učenicima kada se radi o programiranju te kako ženska figura u obitelji utječe na njihove stavove, odnosno umanjuje li stereotipe ukoliko postoje, provjeriti postoje li stereotipi prema spolu među učenicima kada se radi o robotici te kako ženska figura u obitelji utječe na njihove stavove, odnosno umanjuje li stereotipe ukoliko postoje. U radu je provjereno i kako razmišljaju dječaci, a kako djevojčice kada se radi o konfiguracijama robota, kakvi ih oblici više privlače, odnosno odbijaju, kako bi se dobila okvirna slika za buduće radionice kojima je cilj privući što više djevojčica i potaknuti njihov interes. Even though STEM, more precisely the ICT sector, robotics, and electrical engineering are no longer unknown for the female population and more and more women are choosing this path in their careers, a significant difference in interests between the sexes is still present. To find the cause, and thus the potential solution, this paper examines four goals: the influence of gender on interests for three school subjects (mathematics, informatics, and technical culture), to determine the influence of the family on students' attitudes from these three school subjects, to check whether there are gender stereotypes among students when it comes to programming and how the female figure in the family affects their attitudes, or whether it reduces stereotypes if there are any, to check whether there are gender stereotypes among students when it comes to robotics and how the female figure in the family influences their attitudes, whether it diminishes stereotypes if they exist. The paper also examines how boys and girls think about robot configurations, what shapes attract or repel them more, to get a rough picture for future workshops aimed at attracting as many girls as possible and stimulating their interest.

Matematika, kao bitan alat u pomorskoj navigaciji, ima jako široku primjenu u pomorstvu, posebice u navigaciji koja se bavi vođenjem broda najsigurnijim, najkraćim i najpovoljnijim putem. Od prvih matematičkih izračuna navigacije do danas prošlo je mnogo godina, a samo ta činjenica pokazuje neizostavno polazište u pomorskoj navigaciji. Bez matematike nezamislivo je proučavati stabilnost broda kao i ucrtavanje planova putovanja broda. U pomorstvu se matematika može promatrati kroz primjenu geometrije i trigonometrije u astronomskoj i terestričkoj navigaciji kao i u proračunima koji se koriste prilikom izrada navigacijskih karata. U ovom završnom radu analizirat će se matematički izračuni navigacije na primjeru otoka Visa gdje se odvija plovidba po loksodromi jer se radi o obalnoj navigaciji.

Osmišljena je i razvijena aplikacija za učenje vektora u programskom jeziku Java. Uvedeni su pojmovi i operacije s vektorima na razini osnovne i srednje škole. Aplikacija omogućuje interaktivno učenje, vježbanje i strukturiranu provjeru konceptualnog i proceduralnog znanja korisnika. An application for learning vectors was designed and developed in the Java programming language. Concepts and operations with vectors at the level of primary and secondary school have been introduced. The application enables interactive learning, practice and structured testing of users' conceptual and procedural knowledge.

Upotreba informacijsko-komunikacijske tehnologije (skraćeno IKT) neizostavna je u suvremenim oblicima učenja i poučavanja te podrazumijeva transfer i upotrebu svih vrsta informacija i sva tehnička sredstva koja omogućavaju korisnicima da koriste informacije te komuniciraju. Nastava postaje zanimljiva, raznolika, dinamična, inovativna i kreativna kada nastavnik uz standardne metoda poučavanja koristi i IKT. U ovom radu prikazana je upotreba informacijsko-komunikacijske tehnologije (IKT) u nastavi matematike sa svrhom postizanja boljeg uspjeha i motivacije kod učenika. Prikazana je primjena alata Quizizz na odabranu nastavnu jedinicu iz matematike u srednjoj školi Ivan Švear u Ivanić-Gradu. Također prikazani su i rezultati ankete provedene nakon primjene kviza u nastavi matematike. Pokazalo se da je kviz odlična motivacijska metoda, a sam alat Quizizz vrlo jednostavan za korištenje kako nastavniku tako i učenicima.

Rad opisuje postupak razvoja web aplikacije MathTVZ za pomoć studentima prve godine stručnog studija informatike pri učenju i praćenju nastavnog gradiva kolegija Matematika I na Tehničkom veleučilištu u Zagrebu. U prvom dijelu rada opisane su sve tehnologije i programski jezici korišteni za razvoj. Središnji dio rada čine poglavlja koja opisuju funkcionalnosti i način programiranja same aplikacije, a na kraju se navode metode testiranja, rezultati izrade aplikacije i mogućnosti nadogradnje, The paper describes the process of developing MathTVZ web application. This application helps first year students of informatics in their process of understanding and learning the teaching material of the course Mathematics I at the University of Applied Sciences in Zagreb. The first part of the paper describes all technologies and programming languages used for development. The central part of the paper consists of chapters that describe the functionality and programming of the application itself, and at the end lists, the testing methods, application development results and upgrade options.

U disertaciji proučavamo familiju logaritamskih \(C_2\)-konačnih algebri verteks operatora \(SF(d)^+\), poznatu pod nazivom simplektički fermioni. Te algebre verteks operatora se pojavljuju kao parni dio (ili tzv. \(\mathbb{Z}_2\)-orbifold) superalgebri verteks operatora \(SF(d)\). Naš pristup ovom problemu bazira se na Zhuovima algebrama (koje je uveo Y. Zhu u [69]). T. Abe je u članku [1] (između ostaloga) odredio ireducibilne reprezentacije algebri verteks operatora \(SF(d)^+\) i izračunao Zhuovu algebru u slučaju \(d = 1\). U prvom dijelu radnje, računamo Zhuovu algebru od SF(d)+ za preostale prirodne d, te pomoću toga dokazujemo slutnju o dimenziji vektorskog prostora one-point funkcija na \(SF(d)^+\), koju su postavili Y. Arike i K. Nagatomo u članku [15]. Također, pokazujemo da je dimenzija Zhuove algebre od \(SF(d)^+\) jednaka dimenziji tzv. \(C_2\)-algebre od \(SF(d)^+\) za takve \(d\) (slučaj \(d = 1\) je riješio T. Abe u clanku [1]). Općeniti problem određivanja algebri verteks operatora za koje vrijedi ta jednakost dimenzija su u članku [40] promatrali M. Gaberdiel i T. Gannon. U drugom dijelu radnje, bavimo se višim Zhuovim algebrama, generalizaciji pojma Zhuove algebre, koju su uveli C. Dong, H. Li i G. Mason u [29]. Računamo prvu Zhuovu algebru za superalgebru verteks operatora \(SF(1)\) i algebru verteks operatora \(SF(1)^+\). Koliko znamo, ovo je prvi slučaj računanja više Zhuove algebre za neku logaritamsku \(C_2\)-konačnu algebru verteks operatora. In this thesis, we study a family of logarithmic \(C_2\)-cofinite vertex operator algebras \(SF(d)^+\), known as symplectic fermion VOA (of rank \(d\)). Those VOAs are an even part (or, equivalently, \(\mathbb{Z}_2\)-orbifold) of vertex operator superalgebras \(SF(d)\). Our approach is through studying Zhu’s algebras (introduced by Y. Zhu in [69]). T. Abe in [1] determined irreducible representations of \(SF(d)^+\) and calculated Zhu’s algebras in case \(d = 1\). In the first part of this thesis, we calculate Zhu’s algebra of \(SF(d)^+\) for remaining natural \(d\), and we use this to prove a conjecture on the dimension of vector space of one-point functions on \(SF(d)^+\), posed by Y. Arike and K. Nagatomo in [15]. Also, we show that the dimension of Zhu’s algebra of \(SF(d)^+\) is equal to the dimension of the \(C_2\)-algebra of \(SF(d)^+\) for those \(d\) (the case \(d = 1\) was done by T. Abe in [1]). General problem of determining for which VOAs that equality of dimensions holds was introduced M. Gaberdiel and T. Gannon in [40]. The second part of this thesis is concerned with higher Zhu’s algebras, a generalization of the concept of Zhu’s algebra, introduced by C. Dong, H. Li and G. Mason in [29]. We calculate the first Zhu’s algebra for the vertex operator superalgebra \(SF(1)\) and vertex operator algebra \(SF(1)^+\). As far as we know, this is the first calculation of a higher Zhu’s algebra for some logarithmic \(C_2\)-cofinite VOA.

Evoluta dane krivulje je geometrijsko mjesto središta zakrivljenosti te krivulje, a istovremeno to je i omotaljka njezinih normala. Od 1665. godine kada je matematičar Huygens uveo pojam evolute, do danas o evolutama konika u euklidskoj ravnini napisani su brojni radovi. U ovom doktorskom radu proučavaju se svojstva evoluta uz isticanje njihovih karakteristika vezanih za Plückerove formule (red, razred, broj dvostrukih točaka/tangenata, broj šiljaka, infleksionih točaka/pravaca) u sedam od ukupno devet projektivno-metričkih ravnina. Rad je podijeljen na pet poglavlja u kojima je napravljen opsežan pregled osnovnih pojmova i svojstava evoluta konika u euklidskoj, pseudoeuklidskoj, kvazihiperboličnoj, kvazieliptičnoj i projektivno proširenoj hiperboličnoj ravnini. U radu je detaljno proučeno kako položaj konike prema apsolutnoj figuri utječe na red i razred njezine evolute. Pri istraživanju u projektivnim modelima projektivno-metričkih ravnina koriste se metode sintetičke geometrije koje omogućuju konstruktivnu obradu krivulja u dinamičkim računalnim programima The Geometer’s Sketchpad i Geogebra. Pri analitičkom istraživanju evoluta koriste se programi Wolfram Mathematica i Demos. Cilj je rada sistematizacija činjenica vezanih za istaknute projektivno-metričke ravnine te znanstveni doprinos pri klasifikaciji evoluta i njihovoj konstruktivnoj obradi. The locus of centers of osculating circles is called the evolute of a curve. The evolute is also the envelope of the family of normal lines to the curve. From 1665, when the mathematician Huygens introduced the notion of the evolute of a conic in the Euclidean plane, to the present day numerous papers have been written. In this doctoral thesis, the properties of evolutes are studied with emphasis on their characteristics related to Plücker formulas (order, class, number of double points/tangents, number of cups, inflection points/lines) in seven out of a total of nine projective-metric planes. The thesis is divided into five chapters in which an extensive review about basic concepts and properties of the evolute of a conic is made in the Euclidean plane, pseudo-Euclidean plane, quasi-hyperbolic plane, quasi-elliptic plane and projective extended hyperbolic plane. The thesis studies in detail how the position of the conic towards the absolute figure affects the order and class of its evolute. The syntehetic and analytical methods have been used in the studies. The computer programs used in the research are The Geometer’s Sketchpad, Geogebra, Wolfram Mathematica and Demos. The main aim of this thesis is to systematize the facts related to prominent projective-metric planes and scientific contribution to the classification of evolutes and their constructive processing.

Nastavnici matematike se često u nastavi susreću s pomanjkanjem interesa i motivacije učenika za matematiku. Kako istraživanja pokazuju da rad nastavnika matematike utječe i na rad i motivaciju učenika, na nastavnicima je velika odgovornost da svojim pristupom i zalaganjem obogate nastavni proces novim sadržajima i aktivnostima koji bi bili motivirajući učenicima. U članku se opisuju šifre primjerene za rad u osnovnoj školi te predlažu nastavni sadržaji iz matematike u koje bi se te šifre mogle uključiti.

Pomorstvo i pomorske djelatnosti obavljaju se na više od tri četvrtine površine svijeta te su jedna od većih industrija današnjice. Od samih početaka brodostrojarstvo i matematika su povezane znanosti. Upotreba i održavanje brodova, te strojeva i sustava na njima, doveli su do velike ovisnosti o matematičkim načelima ili modelima. Međutim, ti modeli teško dolaze u obliku „čiste“ matematike, već su ugrađeni u znanosti koje se, opet, razvijaju zbog matematike. Ovaj rad je imao za cilj istražiti korisnost matematike u obrazovanju brodostrojara. U radu se raspravlja o različitim znanostima tj. predmetima u kojima se brodostrojari susreću s uporabom i primjenom matematike. U radu se također identificiraju posebne vrste ili područja matematike koje brodostrojari koriste kako bi savladali gradivo te se adekvatno obrazovali.