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U ovom radu je pokazano kako se poznavanjem osnovnih pojmova i činjenica iz linearne algebre mogu dokazati zanimljive, često i važne nejednakosti iz različitih matematičkih disciplina. Primjerima su ilustrirane neke primjene u algebri, kombinatorici, kombinatornoj geometriji te teoriji dizajna. Pri dokazivanju samih nejednakosti korištene su poznate činjenice poput svojstava ranga matrice, dimenzije potprostora vektorskih prostora, kriterij postojanja netrivijalnog rješenja homogenog sustava linearnih jednadžbi te Cauchy-Schwarzove nejednakosti. Zastupljen je i poopćeni oblik Binet-Cauchyjevog teorema o determinanti umnoška matrica, koji je takoder koristan u primjenama. Medu rezultatima dokazanima u radu obuhvaćene su Fisherova nejednakost za kombinatorne dizajne te gornja meda veličine ekviangularnog skupa pravaca u d-dimenzionalnom euklidskom prostoru, što pripada problemima kombinatorne geometrije koji već desetljećima zaokupljaju matematičare i o kojima su posljednjih godina postignuti neki značajni rezultati. In this paper, it is shown how some interesting, often important inequalities from various mathematical disciplines can be proved by using basic concepts and facts from linear algebra. Some applications in algebra, combinatorics, combinatorial geometry and design theory are illustrated by examples. Several well-known statements on the properties of the rank of a matrix, dimensions of subspaces of vector spaces, the criterion of the existence of a non-trivial solution of a homogeneous system of linear equations and the Cauchy-Schwarz inequality were used in the proofs. A generalized form of the Binet-Cauchy theorem on the determinant of the product of matrices is also included, as it is useful in applications. Among the results proved in the paper are Fisher’s inequality for combinatorial designs and the upper limit of the size of the equiangular set of directions in the d-dimensional Euclidean space. The latter result belongs to the topics in combinatorial geometry that have been investigated by mathematicians for decades and about which some significant results have been achieved in recent years.

SET je iznimno zabavna misaona igra kartama najmijenjena djeci i odraslima. Igrajući SET mnogi ne bi naslutili da se u pozadini igre krije matematika. Osim što je zabavna i vrlo zarazna, igra se povezuje s mnogim matematičkim problemima iz područja afine geometrije, linearne algebre, teorije skupova, kombinatorike i vjerojatnosti. Ovaj se rad temelji na geometrijskoj interpretaciji igre SET pomoću konačnog vektorskog i afinog prostora \(\mathbb Z_3^4\). Tokom igre pojavljuju se mnoga pitanja do čijih se odgovora dolazi matematičkim “alatima”. Glavni je problem rada doći do odgovora na jedno od tih pitanja. U matematičkoj interpretaciji SET-a rješavanje postavljenog problema svodi se na pronalasku maksimalne veličine podskupa točaka unutar afinog prostora \(\mathbb Z_3^4\) koji ne sadrži pravac, takozvana 4-kapica. Općenito, maksimalna veličina \(n\)-kapice u \(\mathbb Z_3^n\) poznata je za \(n\leq6\) , a za ostale dimenzije moguće je odrediti samo gornju granicu i to pomoću teorema koji iznosimo u radu. Za prve četiri dimenzije pomoću igre SET pronalazimo primjer maksimalne \(n\)-kapica, a zatim matematički dokazujemo da je veličina pronađene n-kapice zaista maksimalna moguća. Osim toga, rad se bavi i nekim osnovnim svojstvima afinog prostora \(\mathbb Z_3^n\) i posebno geometrijom u \(\mathbb Z_3^n\) za \(n\) = 1, 2, 3, 4. Posljednje poglavlje posvećeno je primjeni igre SET u nastavi matematike. SET je igra kartama pa otvara mnoga kombinatorna i vjerojatnosna pitanja koja možemo postaviti učenicima srednje škole i potaknuti ih na logičko i apstraktno razmišljanje. Isto tako, SET predstavlja konkretan model skupa pa iznosimo i način na koji bi učenici osnovne i srednje škole pomoću zabavne igre mogli uspješno savladati osnovne operacije sa skupovima. SET is an extremely fun thoughtful card game intended for children and adults. What many wouldn’t assume while playing SET is the fact that mathematics is hidden in the core of the game. Apart from being fun and highly addictive, the game associates with many mathematical problems from the field of affne geometry, linear algebra, set theory, combinatorics and probability. This paper is based on the geometric interpretation of the game SET with the help of finite vector and affne space \(\mathbb Z_3^4\). During the course of the game, many problems arise for which the answers are found by mathematical ”tools”. The main problem of the paper is getting an answer to one of those questions. In the mathematical interpretation of SET, solving a set problem comes down to finding the maximum value of a subset of points inside affne space \(\mathbb Z_3^4\) which doesn’t contain a line, the so-called 4-cap. Generally, the maximum value of an \(n\)-cap in \(\mathbb Z_3^n\) is known for \(n\leq6\), whereas for other dimensions it’s possible only to determine the upper limit, which is done with the help of the theorem we’re stating in the paper. For the first four dimensions with the help of the game SET we are coming across an example of a maximum \(n\)-cap, then we mathematically prove that the value of found n-cap is indeed the highest possible. Besides that, the paper deals with some basic properties of affne space \(\mathbb Z_3^n\) and especially the geometry in \(\mathbb Z_3^n\) for \(n\) = 1, 2, 3, 4. The last chapter is dedicated to the application of the game SET in mathematics classes. SET is a card game so it opens many combinatorial and probabilistic questions which can be presented to high school students and encourage them to think logically and abstractly. SET also presents a concrete model of sets so we are presenting a way in which elementary and high school students would be able to grasp basic set operations with the help of a fun game.

Tema rada je kombinatorni problem kompletiranja setova od k elemenata, pri čemu se elementi razlikuju po n parametara, a po svakom parametru su razvrstani u m razreda. Pri kompletiranju setova smiju se spajati samo elementi iz susjednih razreda pojedinog parametra. Rad se bavi definiranjem problema, kategoriziranjem i opisivanjem ključnih pojmova vezanih uz kompletiranje setova, a potom se iznosi rješenje za višeparametarsko kompletiranje setova, detaljno za dvoparametarski slučaj, a potom i za opći slučaj. U radu je opisana i programska podrška kojom se koristilo u rješavanju problema, odnosno korisnička aplikacija koja je bila izrađena za potrebe rješavanja zadatka. The paper discusses the combinatorial problem of completing of sets of k elements, wherein the elements differ by n different parameters, and m different classes for each parameter. When completing the sets only elements from the neighboring class of each parameter could be merged together in a set. The paper deals with defining the problem, categorizing and describing the key terms related to the completion of sets, and then presents a solution for multi-parameter completing sets, in detail for the two-parameter case, and then also in the general case. This paper also describes the software which is used to solve this problem, that is user application that was created for the purpose of solving this task

U radu se pomoću teorije informacija i kombinatorike istražuje jesu li financijski vremenski nizovi slučajni. Povrati dionica čine vremenski niz. Nad tim podacima poziva se funkcija autokorelacije. Iz grafova se iščitavaju vrijednosti te se u skladu s tim tumače. Podaci vezani uz ovaj rad ukazuju na prisutnost slučajnosti u povratima koji se gledaju kao vremenski nizovi. Nakon prvotne provjere, vremenske nizove koje čine povrati, pretvaraju se u binarne vremenske nizove. Nad njima se ponovno vrši analiza jesu li slučajni. Ovoga puta to se čini pomoću kombinatorike, odnosno prebrojavanja nula i jedinica. Rezultati su u suprotnosti s rezultatima koje daje autokorelacija. Vremenski nizovi sastavljeni od nula i jedinica nisu slučajni. This paper uses theory of information and combinatorics to analyze if the financial time series are a random process. Returns of the stock make a time series. Over this data, autocorrelation function is called. The results are read and interpreted from the charts. Data from this paper show that returns are a random process. After that, time series made of returns are being converted to binary time series and combinatorics is used to analyze them. After that, zeros and ones are counted. Results are then different than the first ones. Time series that are made of zeros and ones is not a random process.

Ovdje se objavljuje editio princeps Boškovićeva autografa De calculo probabilitatum quę respondent diversis valoribus summę errorum post plures observationes, quarum singulę possint esse erroneę certa quadam quantitate, koji je od 1962. godine pohranjen u Bancroft Library u sastavu University of California, Berkeley, u zbirci Boscovich Papers, s Truhelkinom oznakom No 62 iz 1924. godine i sa sadašnjom signaturom: Carton 1, Part 1: no. 62, Folder 1:79. Transkripcija je popraćena bilješkama i uvodom. U uvodu se pak redom: 1.opisuje rukopis; 2. obrađuje stanje istraženosti, tj. ističe da je rukopis dosad istraživao samo ruski povjesničar znanosti Oskar Šejnin početkom 1970-ih kad je o njemu objavio tri članka; 3. izlaže Boškovićev problem iz teorije pogrešaka i njegovo rješenje. Na te se prve korake nastavljaju istraživanja s ciljem da se: 4. što je preciznije moguće odredi datacija ovoga nedatiranoga Boškovićeva rukopisa; 5. raspravi pitanje o dovršenosti rukopisa; 6. zauzme stav o njegovu značenju unutar Boškovićeva djela i za znanost 18. stoljeća. Matematički problem, u naslovu Boškovićeva spisa prikazan kao »izračunavanje vjerojatnostī koje odgovaraju različitim vrijednostima zbroja pogrešaka nakon više opažaja« uz uvjet da pojedini opažaji mogu odstupati od točne izmjere »za neku određenu vrijednost«, Bošković je strogo matematički formulirao: 1. izrijekom je pretpostavljeno da su pogreške jednako vjerojatne i poprimaju tri vrijednosti 1, 0, –1; 2. traži se omjer vjerojatnostī za pojedine zbrojeve pogrešaka koji mogu nastati »nakon danoga broja n opažajā«; 3. podrazumijeva se da se raspon ukupne pogreške kreće od –n do n i da je taj n konačan. Da bi pripremio rješenje, Bošković je uveo pojam kombinacije r-toga razreda od n elemenata, a zatim je osobito pomno razmotrio prvi slučaj, kad je zbroj pogrešaka jednak nuli i sastavio r-članu formulu za broj svih kombinacija uz uvjet da suma pogrešaka bude 0. Istom metodom sastavio je istovrsne formule za broj kombinacija kad je suma pogrešaka jednaka 1, 2, …, 8. Potom je u tim formulama izostavio članove koji su jednaki nuli. U zadnjem koraku upotrijebio je svoje formule da izračuna broj kombinacija za različite brojeve opažaja n = 1, 2, …, 8. Boškovićev autograf De calculo probabilitatum nije datiran, a već je 1970. Oskar Sheynin upozorio da značenje ovoga rukopisa ovisi o njegovoj dataciji, odnosno o njegovoj povezanosti s geodetsko-kartografskom ekspedicijom za izmjeru duljine luka merijanskoga stupnja na rimskome meridijanu. Ali tekstualna analiza rukopisa ne nudi nijedan podatak u prilog takvoj povezanosti niti omogućuje da se rukopis poveže s Boškovićevom statističkom metodom za izravnanje nesuglasnih opažaja, prvi put primijenjenoj 1757. godine u časopisu bolonjske akademije u članku »De litteraria expeditione per Pontificiam ditionem«, a prvi put objavljenoj 1760. u dopuni »De recentissimis graduum dimensionibus, et figura, ac magnitudine Terrae inde derivanda« uz petu knjigu Stayeva epa Philosophia recentior. Radeći godine 1760. na uzorku od pet odabranih, a nesuglasnih geodetskih opažaja, Bošković je formulirao drukčiji problem – problem za pronalaženje ispravaka uz tri drukčija uvjeta: da razlike između izmjerenih stupnjeva budu razmjerne razlikama između vrijednosti sinus versus udvostručene širine; »da zbroj pozitivnih ispravaka bude jednak zbroju negativnih; da zbroj svih ispravaka, kako pozitivnih tako i negativnih, bude najmanji od svih koji se mogu dobiti ako su očuvana prva dva uvjeta.« Dakle, formulacija problema 1760. godine bitno se razlikuje od formulacije problema u rukopisu De calculo probabilitatis. Zajedničko im je samo to da je broj n mjerenja konačan, zapravo vrlo malen: u rukopisu De calculo probabilitatis n dostiže 8 kao najveću vrijednost, a u geodetskim izvješćima u prvoj primjeni 1757. n = 5, a 1770. n = 9. Da bi se odredila godina ili bar razdoblje u kojem je Bošković napisao svoj rukopis De calculo probabilitatum, u ovom su članku primijenjena dva pristupa, proizašla iz vlastita istraživačkoga iskustva. Prvi je utrt pitanjem: Je li se, kad i kako Bošković susretao s teorijom pogrešaka u geodeziji i geofizici, a ako ne s teorijom pogrešaka, onda barem s postojanjem više različitih mjerenja iste veličine? Jedan odgovor na to pitanje ponudio je sam Bošković kad je 1760. godine podsjetio da je već »od 1738. izražavao sumnju u nepravilnost u mjerenju niza stupnjeva i u obliku Zemlje«. U tom je podsjećanju Bošković uputio na tri svoje rasprave: Dissertatio de Telluris figura (1739), De inaequalitate gravitatis in diversis Terrae locis (1741) i De observationibus astronomicis, et quo pertingat earundem certitudo (1742). Drugi pristup omogućuju obilježja Boškovićeva rukopisa De calculo probabilitatum, koja odudaraju od kasnijega stila Boškovićevih znanstvenih radova. U kolovozu 1740. Bošković je objavio dvije kratke rasprave, koje dokumentiraju lako uočljivu promjenu u njegovu pristupu kompoziciji znanstvenoga teksta. U raspravi De motu corporum projectorum in spatio non resistente posljednji je put tekst organizirao kao niz propozicija, uz koje je još istaknuo uvodne leme i sholije. A u raspravi De circulis osculatoribus prvi je put numerirao paragrafe, tj. podijelio je tekst rasprave na »brojeve «. Od numeriranja odlomaka vrlo je rijetko odstupao, primjerice u podnescima vladarima ili u člancima za neke časopise. U rukopisu De calculo probabilitatum Bošković je primijenio međurješenje, tj. formulirao je problem u obliku propozicije, a potom usredištenim podnaslovima numerirao korake prema njegovu rješenju. Ta dva pristupa utiru put zaključku: vrijeme u kojem je nastao rukopis De calculo probabilitatum treba stegnuti na rano razdoblje 1738–1740. A to znači da uz već uočena dva područja: statističku metodu za izravnanje nesuglasnih opažaja s prvom primjenom u geodeziji (1757, 1760, 1770) i uporabu vjerojatnosti uz odabrane teme u prirodnoj filozofiji (1754, 1755, 1758, 1763), ovaj rukopis nudi uvid u najraniji Boškovićev pristup teoriji vjerojatnosti, koji najbolje reprezentira formulacija Boškovićeva problema iz teorije pogrešaka na početku rukopisa., Here published is the editio princeps of Bošković’s autograph De calculo probabilitatum quę respondent diversis valoribus summę errorum post plures observationes, quarum singulę possint esse erroneę certa quadam quantitate, housed in 1962 at the Bancroft Library within the University of California at Berkeley, in the collection Boscovich Papers, call number: Carton 1, Part 1: no. 62, Folder 1:79. The transcription is accompanied by notes and introduction. The latter contains the following: (1) description of the manuscript; (2) outline of the contents of Bošković’s writing related to the theory of probability, notably the mathematical problem as formulated by Bošković; (3) description of the status of investigation, with regard to the fact that until today Oscar Sheynin, Russian science historian, has been the only scholar to study this manuscript, about which he published three articles in the 1970s. Following in these initial research steps is the investigation primarily aimed at: (4) exact dating of this undated manuscript; (5) establishment of whether the manuscript was completed or not; and, lastly, (6) assessment of its significance within Bošković’s work and within the history of science in the eighteenth century. The mathematical problem, introduced in the title of Bošković’s manuscript as the “calculus of probabilities which correspond to various values of the sum of errors after several observations,” under the condition that single observations can differ from the exact measurement “for a given quantity,” Bošković formulated as follows: “If in a certain series of observations equally probable errors 1, 0, −1 are presupposed for single observations, we ask the ratio of probability for single sums, which sums can thence be obtained after a given number n of observations. Error sums can take on all the values from n up to −n for different combinations. The probabilities for any value will relate as the numbers of combinations, from which the same value is obtained. In order to determine this number for single sums, may the series of these errors be arranged in three lines: I 1, 1, 1, 1, 1, 1, 1, 1 etc. II 0, 0, 0, 0, 0, 0, 0, 0 etc. III −1, −1, −1, −1, −1, −1, −1, −1 etc. Then we ask the numbers of combinations for single values of the sum of errors from 0, 1, 2 etc. up to n. The numbers [of the combinations] for [the sums of errors] −1, −2, −3 etc. can be easily found; these numbers will be the same as the numbers for [the sum of errors] 1, 2, 3 etc., because once those [with the positive sum of errors] are found, these [with the negative sum of errors] will be found too.”1 In his preparation of the solution, Bošković introduced the notion of combination of n objects taken r at a time. With great scrutiny he then examined the first case, when the sum of errors is equal 0, and provided the formula for the number of all combinations under the condition that the sum of errors equals 0. By using the same method, he provided the formulas for the number of combinations when the sum of errors equals 1, 2, …, 8. With regard to these formulas he then examined the “determination of the number of useful terms” (determinatio numeri terminorum utilium), i.e. omitted the terms equal to 0. In the final step he applied his formulas to calculate the number of combinations for different numbers of observations n = 1, 2, …, 8. Bošković’s autograph De calculo probabilitatum is not dated, and as early as 1973 Oscar Sheynin emphasised that the significance of this manuscript rested on its dating: “The manuscript under consideration being undated its evaluation is hardly possible. If it was written in 1750–1753, i.e. during his participation in the arc measurement, or at least before 1756 [read: before the publication of Simpson’s letter ‘On the Advantage of taking the Mean of a Number of Observations, in practical Astronomy’ in Philosophical Transactions], Boscovich should be regarded as the first to use a quantitative stochastic method in the theory of errors and, thus a precursor of T. Simpson.”2 Indeed, the text of the manuscript is not linked with the geodetic and cartographic expedition along the Roman meridian from Rome to Rimini in the period 1750–1752, or with Bošković’s statistical method for correcting discordant observations, for the first time applied in Bošković’s article “De litteraria expeditione per Pontificiam ditionem” (“On the scientific expedition through the Papal State,” 1757) in the journal of the Bologna Academy, and for the first time published in Bošković’s supplement to Book Five of Benedikt Stay’s epic Philosophia recentior on Newton’s and Bošković’s natural philosophy under the title »De recentissimis graduum dimensionibus, et figura, ac magnitudine Terrae inde derivanda« (“On the most recent measurements of longitude of meridian degree and on the shape and magnitude of the Earth derived thereof,” 1760). In 1760, while working on the sample of five geodetic observations, Bošković formulated a different problem with three conditions: “Given the number of degrees [i.e. the number of measurements of meridian degrees], find the corrections which ought to be assigned to the individual observations, whereby the following three conditions ought to be preserved: namely, [1] that diferrences between longitudes of degree be proportional to the differences between the values of sinus versus of duplicate latitude; [2] that the sum of positive corrections be equal to the sum of the negative ones; [3] that the sum of all the corrections, either positive or negative, be the smallest of all sums which can be obtained if the first two conditions are preserved.”3 The formulation of the problem from 1760 departs considerably from that in the manuscript De calculo probabilitatum. The only thing in common is that the number n of measurements is finite, in fact, very small: in the manuscript De calculo probabilitatum n reaches 8 as the biggest value, whilst in the geodetic reports n = 5 in the first application in 1757, and n = 9 in the second one in 1770. In order to establish the year or at least the time-frame in which Bošković wrote his De calculo probabilitatum, in this article two approaches have been employed, both based on my own research experience. The first approach is guided by the question: Did Bošković encounter the theory of errors in geophysics and geodesy, and if so, when and how, and if he did not encounter the theory of errors, was he at least familiar with the existence of different measurements of the same quantity? Bošković himself provided one answer to that question in 1760: »365. Since 1738, in the treatise on the shape of the Earth I have expressed suspicion about the irregularity in measuring the series of degrees and the shape of the Earth, the cause of which I specifically perceive. This suspicion I pursued in the mentioned treatise, as well as in the other two treatises from 1741 and 1742.«4 In his statement, Bošković draws attention to three of his treatises: Dissertatio de Telluris figura (1739), pp. XVIII-XXIII, De inaequalitate gravitatis in diversis Terrae locis (1741), n. 37, and De observationibus astronomicis, et quo pertingat earundem certitudo (1742), n. 63. In them only has Bošković referred to the same problem: uncertainty in measuring the length of a meridian degree, and suspicion about the irregularity of the Earth. The second approach is facilitated by the features of Bošković’s manuscript De calculo probabilitatum, which depart from the later style of Bošković’s scientific works. In August 1740, Bošković published two short treatises which document an easily discernible change in his approach to the composition of a scientific text. The treatise De motu corporum projectorum in spatio non resistente was the last printed dissertation prepared for the public defense at the Collegium Romanum that he structured as a series of propositions, in addition to which he emphasised the introductory lemmas and scholia. In the treatise De circulis osculatoribus Bošković for the first time numbered the paragraphs, i.e. divided his text into ‘numbers.’ He rarely departed from the numeration of paragraphs, for example, in the reports submitted to rulers or in the articles for some journals. In the manuscript De calculo probabilitatum Bošković resorted to a compromise, i.e. he introduced the proposition and numerated the steps to the solution by assigning centred subtitles to them. The genesis of Bošković’s research in geophysics and geodesy on the one, and textual analysis, which suggests that the manuscript originated before the key change in the composition of Bošković’s early treatises in August 1740, on the other hand, pave the path for the following conclusion: the time-frame in which the manuscript De calculo probabilitatum was written should be narrowed to the early period 1738–1740. This implies that, besides the already indicated two approaches―statistical method for correcting discordant observations with the first aplication in geodesy (1757, 1760, 1770), and the application of probablity in the selected topics in his natural philosophy (1754, 1755, 1758, 1763)―this manuscript provides an insight into Bošković’s earliest approach to probability, best represented by the formulation of the problem in the theory of errors at the beginning of the manuscript.

U završnom radu pod nazivom "Utjecaj vremenskih neprilika na planiranje izvođenja građevinskih radova" provedeno je istraživanje s ciljem analize utjecaja vremenskih neprilika na planiranje i izvođenje građevinskih radova. Na temelju meteoroloških podataka za gradove Krk i Pazin, dobivenih od Državnog hidrometeorološkog zavoda (DHMZ), i primjenom kombinatorike i statističke obrade podataka analiziran je broj nepovoljnih dana pri radu gradilišta zbog vremenskim neprilikama. Ovi podaci pružaju kvantitativni uvid u učestalost i trajanje nepovoljnih vremenskih uvjeta te omogućuju bolje planiranje radnih aktivnosti na gradilištima. Uz prikupljanje statističkih podataka, provedena je verifikacija istih. Putem razgovora s voditeljima gradilišta građevinske struke, prikupljene su i verificirane informacije o utjecaju vremenskih neprilika na izvođenje radova. Rezultati istraživanja pružaju vrijedan uvid u utjecaj vremenskih neprilika na planiranje izvođenja radova te se ističe važnost adekvatnog pristupa pri izradi vremenskog plana i upravljanju rizicima uzrokovanim nepovoljnim vremenskim uvjetima., The research conducted in the Final Paper titled "The impact of adverse weather conditions on construction planning and execution" aimed to analyze the influence of weather conditions on the planning and execution of construction projects. Based on meteorological data for the cities of Krk and Pazin, obtained from the State Hydrometeorological Institute (DHMZ), and by applying combinatorics and statistical data analysis, the number of unfavorable working days due to adverse weather conditions on construction sites was analyzed. These data provide a quantitative insight into the frequency and duration of adverse weather conditions, enabling better planning of work activities on construction sites. In addition to collecting statistical data, their verification was also conducted. Through interviews with construction site managers, information regarding the impact of weather conditions on project execution was gathered and verified. The research results provide valuable insights into the influence of weather conditions on project planning and highlight the importance of an adequate approach to weather planning and risk management caused by adverse weather conditions.

Ovaj završni rad istražuje koncept funkcija izvodnica i njihove raznovrsne primjene. Funkcije izvodnice predstavljaju moćan matematički alat koji se koristi za prikazivanje i manipuliranje nizovima brojeva. One se primjenjuju u različitim područjima, uključujući kombinatoriku, teoriju grafova i teoriju brojeva. Rad istražuje kako se funkcije izvodnice mogu koristiti za rješavanje kombinatornih problema, analizu svojstava grafova i proučavanje problema prebrojavanja. Rad ističe svestranost i učinkovitost funkcija izvodnica u rješavanju složenih matematičkih problema te pruža uvid u njihovu široku primjenjivost u različitim područjima. U sklopu rada izrađena je interaktivna edukativna aplikacija i implementiran je model za usitnjavanje novčanica metodom funkcija izvodnica. This final thesis explores the concept of generating functions and their diverse applications. Generating functions are powerful mathematical tools used to represent and manipulate sequences of numbers. They find applications in various areas, including combinatorics, graph theory, and number theory. The paper investigates how generating functions can be utilized to solve combinatorial problems, analyze graph properties, and study counting problems. The thesis highlights the versatility and effectiveness of generating functions in solving complex mathematical problems and provides insights into their broad applicability across different domains. As part of the thesis, an interactive educational application was created and a model for exchanging banknotes for coins using generating function method was implemented.

Catalanovi brojevi javljaju se pri prebrojavanju iznenađujuće mnogo kombinatornih i drugih matematičkih objekata. U ovom radu detaljnije su opisani problem triangulacije konveksnog n-terokuta, problem zagrada, problem binarnih stabala, problem putova u cjelobrojnoj mreži, problem Dyckovih planinskih putova te Bertrandov problem glasovanja. Naposlijetku je pokazano da rješenja tih problema odgovaraju n-tom Catalanovom broju. Radi lakšeg manipuliranja nizom brojeva, izvedena je i odgovarajuća funkcija izvodnica te je tako dobivena formula za računanje općeg člana niza. Konačno, navedene su još neke zanimljive interpretacije Catalanovih brojeva., Catalan’s numbers appear in various combinatorical and other mathematical objects. In this paper convex polygon triangulation as well as balanced parentheses, problem of binary trees, problem of lattice paths, problem of Dyck paths and Bertrand’s ballot problem are being analysed. Ultimately, it was shown that the solution to each problem corresponds to the \(n^{th}\) Catalan number. In order to manipulate these numbers in an easier way, there is a generating function which is used for calculating the general number of the sequence. Finally, there are shown some other interesting interpretations of Catalan’s numbers.

Genetski su algoritmi jedna od metoda evolucijskog računarstva koji imaju široku primjenu. Najčešće se upotrebljavaju za rješavanje kombinatoričkih problema i optimizacijskih problema čija je domena realno područje, no mogu se izuzetno kvalitetno koristiti i za treniranje neuronskih mreža kao alternativa algoritmu propagacije unatrag. Ovaj se rad bavi rješavanjem optimizacijskog problema mapiranja logičkog u fizički FPGA svijet pomoću genetskog algoritma. Pripremljeni su testovi na kojima je izmjerena kvaliteta rada genetskog algoritma. Genetic algorithms are one of the methods used in evolutionary computing that has many applications. They are used for combinatoric and in optimization problems whose domain is continuum but can also give very satisfying results when used for training neural networks as an alternative to backpropagation algorithm. This paper is dealing with the problem of mapping logical model into physical FPGA model with the help of genetic algorithm. Different tests are prepared for measuring algortihm’s quality.

U ovom radu cilj je bio dokazati i vidjeti primjene jednog od važnijih teorema iz područja kombinatorike, Hallovog teorema. U prvom dijelu rada iskazana je njegova kombinatorna formulacija i ekvivalencija u smislu teorije grafova i sparivanja te su dana četiri različita dokaza Hallovog teorema. Prvi iskaz teorema govori o nužnom i dovoljnom uvjetu za postojanje transverzale danih konačnih skupova. Jednostavnom generalizacijom dolazi se do iskaza Hallovog teorema u smislu teorije grafova koji govori o nužnom i dovoljnom uvjetu za postojanje potpunog sparivanja u bipartitnom grafu. Na kraju drugog poglavlja iskazane su i neke posljedice iz područja kombinatorike, teorije skupova, teorije grupa i linearne algebre. Nadalje, u posljednjem poglavlju smo pokazali da je on samo jedan u nizu ekvivalentnih teorema čija primjena se širi na mnoga područja matematike pa je i motivacija koju smo vidjeli na početku rada raznolika. Medu ekvivalentne teoreme pripadaju osim Hallovog još Koenigov, Dilworthov, Mengerov, Birkhoff-Von Neumannov teorem i drugi. U radu su iskazani i dokazani navedeni teoremi i prikazana je njihova primjena. Vidjeli smo da objekti koji se promatraju u tim teoremima sežu od jednostavnih grafova (Mengerov i Koenigov teorem), binarnih matrica (Koenigov teorem) i dvostruko stohastičih matrica (Birkhoff-Von Neumannov teorem) do parcijalno uredenih skupova (Dilworthov teorem). Upoznali smo se i s latinskim pravokutnicima, još jednom specifičnom vrstom matrica, te smo pokazali da je Hallov teorem zaslužan za dokaz jednog od važnijih teorema o latinskim pravokutnicima. The aim of this thesis was to prove and present the applications of one of the central theorems in the field of combinatorics, Hall’s theorem. Through the first part of the thesis we have expressed its combinatorial formulation, equivalence in terms of graph theory and matching and provided four different proofs. The first statement of the theorem gives the necessary and suficient condition for the existence of a system of distinct representatives of given sets. A simple generalization leads to the statement of Hall’s theorem in terms of graph theory, which speaks of a necessary and suficient condition for the existence of complete matching in a bipartite graph. At the end of the second chapter, several consequences from the field of combinatorics, set theory, group theory and linear algebra are presented. Furthermore, in the last chapter we have shown that Hall’s theorem is one in a sequence of equivalent theorems whose application extends to many areas of mathematics, so the motivation we saw at the beginning of the paper is diverse. Among the equivalent theorems, in addition to Hall’s theorem, there are Koenig’s, Dilworth’s, Menger’s, Birkhoff-Von Neumann theorem, and others. These theorems are stated and proved in this thesis and their application is presented. We have seen that the objects observed in these theorems vary from simple graphs (Menger’s and Koenig’s theorem), binary matrices (Koenig’s theorem) and doubly stochastic matrices (Birkhoff-Von Neumann theorem) to partially ordered sets (Dilworth’s theorem). We were also introduced to Latin rectangles, another specific type of matrix, and have shown that Hall’s theorem is responsible for proving one of the most important theorems in Latin rectangles.

Hrvatske protestantske knjige otisnute u drugoj polovini 16. stoljeća u Urachu i Regensburgu važne su za razumijevanje povijesti hrvatskoga književnog jezika, napose u razdoblju velikoga previranja koje je nastupilo tijekom 16. stoljeća. Poznato je također kako se s vremenom njihova jezična koncepcija mijenjala i nadograđivala te da je donesene odluke o tome što je u tekstovima potrebno promijeniti provodio čitav tim stručnjaka (prevoditelja, prepisivača, urednika, korektora) – predvođen glagoljašima Stipanom Konzulom i Antonom (Antunom) Dalmatinom. Hrvatski su protestanti nastojali oblikovati novi hrvatski književni (liturgijski) jezik – utemeljen na hrvatskoj (slavenskoj) knjiškoj tradiciji, ponajprije crkvenoslavenskoj te čakavskoj pismenosti, a istodobno inkluzivan prema obilježjima hrvatskih govornih idioma. U radu se pristupa analizi dvojinskih oblika imenskih riječi i glagola u evanđeoskim tekstovima pet odabranih protestantskih knjiga (prvi dio glagoljičkoga Novoga testamenta 1562, ćirilički Novi testament 1563, glagoljička Postila 1562, ćirilička Postila 1563, latinička Postila 1568). Izdvajaju se oblici kojima se izriče značenje parnosti i dvojnosti, a potom se utvrđuje stupanj njihove sročnosti. Usustavljuju se uočene sličnosti i razlike među analiziranim predlošcima, a iz rezultata usporedne jezične analize izvode se zaključci o korektoru (korektorima) glagoljske i ćirilske Postile kao i o izmjeni načela jezične koncepcije hrvatskih reformatora koja je očigledno nastupila krajem 1562. i početkom 1563. godine., Croatian Protestant books printed in the second half of the 16th century in Urach and Regensburg are crucial to understanding history of Croatian literary language, especially in the period of great turmoil that occurred in the 16th century. Their language policy and practice has frequently changed and all of the changes were carried out by a team of experts (translators, editors, correctors), led by Glagolites Stipan Konzul and Anton (Antun) Dalmatian. Croatian Protestants aimed to form a new Croatian literary (liturgical) language – based on the Croatian (Slavic) literary tradition, primarily (Old) Croatian Church Slavonic and Chakavian literacy, and at the same time inclusive of the characteristics of Croatian spoken idioms. This paper presents an analysis of dual forms of nominal words and verbs found in the biblical texts, attested in the Glagolitic edition of the New Testament (first part, 1562), the Cyrillic one (1563), and in Postila lectionaries (Glagolitic from 1562 and Cyrillic from 1563), all of which were printed at the Urach press. The Latin version of Postila was published at the Regensburg press in 1568. The similarities and differences between the analyzed texts are pointed out. From these results certain conclusions can be drawn: concerning the corrector(s) of the Glagolitic and Cyrillic version of Postila, as well as the changes in language policy and practice that took place in late 1562 and early 1563.

U ovom radu obrađeni su Catalanovi, Motzkinovi i Narayanini brojevi, te Dyckovi putevi. Pokazali smo da su Catalanovi brojevi niz brojeva. Catalanove brojeve smo koristili kao uvod u Dyckove puteve, te Dyckov jezik. Pokazali smo kako je broj Dyckovih puteva duljine n jednak Catalanovom broju Cn. Uveli smo i Naranayine brojeve koji određuju broj Dyckovih puteva duljine n s k vrhova. Zadnji pojam bili su Motzkinovi brojevi koji određuju broj istoimenih puteva. Uz to, napravljen je i računalni program koji računa i iscrtava Motzkinove i Dyckove puteve. This paper is about Catalan, Motzkin and Narayana numbers, as well as Dyck paths. We have shown that Catalan numbers are a sequence. Catalan numbers have been used as an introduction to Dyck paths and the Dyck language. We have shown that the number of Dyck paths for a certain n is equal to the Catalan number Cn, and that the number of Dyck paths for a certain n with k peaks is equal to the Narayana number N(n, k). The last term we introduced was Motzkin numbers, which count the number of paths with the same name. In addition to that, a computer program has been written that counts and draws Motzkin and Dyck paths.

U ovom radu smo obradili temu povijesti teorije grafova. Začetkom teorije grafova smatra se Eulerovo rješavanje problema sedam Konigsbergških mostova 1736. godine. Graf definiran kao skupina vrhova i bridova koji spajaju te vrhove, daje ljudima moć da pojednostave probleme uklanjanjem nepotrebnih detalja. Stoga, grafovi omogućuju prikaz jasnih i jednostavnih dijagrama s mnogo robusnih aplikacija u fizici, elektrotehnici, računalnoj znanosti, kemiji, sociologiji i prometu. In this paper, we covered the subject of the history of the theory of graphs. Eulers solving of famous „Seven bridges of Konigsberg“ problem in 1736. is considered as inception of graph theory. Graph in this context is made up of vertices which are connected by edges, they provide people with power to simplify the problem by removing unnecessary details. Therefore, graphs allow you to view clear and simple diagrams with many robust applications in physics, electrical engineering, computer science, chemistry, sociology and transport.

Ovaj članak donosi nekoliko zanimljivosti te dvije vrste zadataka o LEGO kockicama. Zadaci su, iako istog tipa, vrlo različitih težina, tj., neki će biti laki i predškolskoj djeci dok će se za neke od zadataka morati pomučiti i profesionalni matematičari. Zadaci su uglavnom kombinacija geometrije i kombinatorike., This paper contains some interesting facts as well as two types of exercises referring to LEGO building bricks. Although being of the same type, the exercises are of different complexity, i.e. some of them are going to be easy even for pre-school children, whereas some of the exercises will be difficult even for professional mathematicians. They mostly represent a combination of geometry and combinatorics.