Analitičke metode modularnih formi

Ovaj je rad uvod u analitičku teoriju brojeva i teoriju modularnih formi. U prvom dijelu rada dokazujemo Dirichletov teorem o prostim brojevima u aritmetičkim nizovima. U drugom dijelu proučavamo modularne forme: konstruiramo osnovne primjere modularnih funkcija i modularnih formi (Eisensteinov red, modularnu diskriminantu i modularnu invarijantu) i istražujemo njihova svojstva; određujemo dimenzije i konstruiramo baze prostora modularnih formi; uvodimo Heckeove operatore kao korespondencije na skupu rešetki u $\mathbb{C}$ i istražujemo svojstva njihovih svojstvenih funkcija.This thesis is an introduction to analytic number theory and the theory of modular forms. In the first part of the thesis, we prove the Dirichlet’s theorem on primes in arithmetic progressions. In the second part, we study modular forms: we construct basic examples of modular functions and modular forms (the Eisenstein series, the modular discriminant and the modular invariant) and investigate their properties; we find the dimensions and construct bases of spaces of modular forms; we introduce the Hecke operators as correspondences on the set of lattices in $\mathbb{C}$ and investigate properties of their eigenfunctions

Analitičke metode modularnih formi

Ovaj je rad uvod u analitičku teoriju brojeva i teoriju modularnih formi. U prvom dijelu rada dokazujemo Dirichletov teorem o prostim brojevima u aritmetičkim nizovima. U drugom dijelu proučavamo modularne forme: konstruiramo osnovne primjere modularnih funkcija i modularnih formi (Eisensteinov red, modularnu diskriminantu i modularnu invarijantu) i istražujemo njihova svojstva; određujemo dimenzije i konstruiramo baze prostora modularnih formi; uvodimo Heckeove operatore kao korespondencije na skupu rešetki u $\mathbb{C}$ i istražujemo svojstva njihovih svojstvenih funkcija.This thesis is an introduction to analytic number theory and the theory of modular forms. In the first part of the thesis, we prove the Dirichlet’s theorem on primes in arithmetic progressions. In the second part, we study modular forms: we construct basic examples of modular functions and modular forms (the Eisenstein series, the modular discriminant and the modular invariant) and investigate their properties; we find the dimensions and construct bases of spaces of modular forms; we introduce the Hecke operators as correspondences on the set of lattices in $\mathbb{C}$ and investigate properties of their eigenfunctions

Analitičke metode modularnih formi

Ovaj je rad uvod u analitičku teoriju brojeva i teoriju modularnih formi. U prvom dijelu rada dokazujemo Dirichletov teorem o prostim brojevima u aritmetičkim nizovima. U drugom dijelu proučavamo modularne forme: konstruiramo osnovne primjere modularnih funkcija i modularnih formi (Eisensteinov red, modularnu diskriminantu i modularnu invarijantu) i istražujemo njihova svojstva; određujemo dimenzije i konstruiramo baze prostora modularnih formi; uvodimo Heckeove operatore kao korespondencije na skupu rešetki u $\mathbb{C}$ i istražujemo svojstva njihovih svojstvenih funkcija.This thesis is an introduction to analytic number theory and the theory of modular forms. In the first part of the thesis, we prove the Dirichlet’s theorem on primes in arithmetic progressions. In the second part, we study modular forms: we construct basic examples of modular functions and modular forms (the Eisenstein series, the modular discriminant and the modular invariant) and investigate their properties; we find the dimensions and construct bases of spaces of modular forms; we introduce the Hecke operators as correspondences on the set of lattices in $\mathbb{C}$ and investigate properties of their eigenfunctions

Vizualni identitet u kontekstu odabranog tržišta

Diplomski rad bavi se vizualnim identitetom proizvoda kroz identitet robne marke, ambalaže i promocije proizvoda u konačnici u kontekstu specifičnosti odabranog tržišta i ciljane skupine. Svrha mu je istraživanje osobnog projekta nastalog u sklopu kolegija Dizajn odabranog grafičkog proizvoda na Grafičkom fakultetu Sveučilišta u Zagrebu – dizajna robne marke premium proizvoda kućnih i uredskih žarulja Aha, njenih triju ambalaža i promocije vizualnog identiteta da bi se saznalo kako bi se Aha robna marka proizvoda ponašala na policama tržišta za koje nije dizajnirana u odnosu na tržište za koje jest dizajnirana, odnosno koji su potencijalni problemi dizajna kojima bi se valjalo posvetiti prije plasmana vizualnog identiteta na novo tržište. Istraživanje se vrši na hrvatskom tržištu i na odabranom španjolskom tržištu, na ciljanoj skupini visokoobrazovanih muškaraca od 25 do 50 godina. Teorijski su analizirana oba tržišta i njihov dizajn u društveno-gospodarskom kontekstu, a eksperimentalno je izvršeno opservacijsko istraživanje dizajna robnih marki i ambalaža u spomenutoj kategoriji žarulja na relevantnim svjetskim web portalima dizajna ambalaže i u njihovim distribucijskim kanalima na oba tržišta. Osim toga, izvršeno je anketiranje i intervjuiranje ciljane skupine na hrvatskom i španjolskom tržištu s ciljem da se otkrije kakvi su vizualni kodovi na pojedinom tržištu, kako ciljana skupina s pojedinog tržišta interpretira poruke i koje su im želje, potrebe, navike i vrijednosti

Non-vanishing of Poincaré series on the metaplectic group and applications

U ovoj se disertaciji proučava neponištavanje Poincaréovih redova na metaplektičkom natkrivaču, $SL_2(\mathbb{R})^{\sim}$, grupe $SL_2(\mathbb{R})$ i istražuju se primjene tih redova u teoriji modularnih formi polucijele težine. Pritom se koriste tri osnovna alata: Harish-Chandrin pristup teoriji reprezentacija povezanih poluprostih Liejevih grupa s konačnim centrom, klasična korespondencija između kusp-formi polucijele težine i kuspidalnih automorfnih formi na grupi $SL_2(\mathbb{R})^{\sim}$ i integralni kriterij neponištavanja za Poincaréove redove na lokalno kompaktnim Hausdorffovim grupama koji je dokazao G. Muić. U disertaciji su pomoću Poincaréovih redova $K$-konačnih matričnih koeficijenata integrabilnih reprezentacija grupe $SL_2(\mathbb{R})^{\sim}$ konstruirani sustavi izvodnica za prostore kusp--formi polucijele težine $m \geq \frac{5}{2}$. Dokazana je i formula za Peterssonov skalarni produkt spomenutih Poincaréovih redova s proizvoljnom kuspidalnom automorfnom formom na $SL_2(\mathbb{R})^{\sim}$ iste težine. Iz te formule i njena dokaza proizašao je niz rezultata o kusp-formama polucijele težine. Nadalje, dokazane su blago ojačane varijante Muićeva integralnog kriterija neponištavanja za Poincaréove redove na Liejevim grupama i za Poincaréove redove polucijele težine na gornjoj kompleksnoj poluravnini. Njihovom su primjenom dokazani rezultati o neponištavanju Poincaréovih redova pridruženih $K$-konačnim matričnim koeficijentima integrabilnih reprezentacija grupe $SL_2(\mathbb{R})^{\sim}$ i pripadnih kusp-formi polucijele težine, rezultati o neponištavanju nekih klasičnih Poincaréovih redova polucijele težine i rezultati o neponištavanju $L$-funkcija pridruženih kusp-formama polucijele težine $m \geq \frac{9}{2}$ u točkama pruge $\mathbb{C}_{\frac{m}{2} < \Re (s) < m-1}$.In this thesis, we study the non-vanishing of Poincaré series on the metaplectic cover, $SL_2(\mathbb{R})^{\sim}$, of $SL_2(\mathbb{R})$ and investigate their applications in the theory of modular forms of halfintegral weight. Our methods rely on the following three tools: Harish-Chandra’s approach to representation theory of connected semisimple Lie groups with finite center, the classical correspondence between cusp forms of half-integral weight and cuspidal automorphic forms on $SL_2(\mathbb{R})^{\sim}$, and the integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups that was proved by G. Muić. In the thesis, we use the Poincaré series of $K$-finite matrix coefficients of integrable representations of $SL_2(\mathbb{R})^{\sim}$ to construct spanning sets for spaces of cusp forms of halfintegral weight $m \geq \frac{5}{2}$. We prove a formula for the Petersson inner product of these Poincaré series with any cuspidal automorphic form on $SL_2(\mathbb{R})^{\sim}$ of the same weight. From this formula and its proof, we derive a series of results on cusp forms of half-integral weight. Next, we prove strengthened variants of Muić’s integral non-vanishing criterion for Poincaré series on Lie groups and for Poincaré series of half-integral weight on the upper half-plane. Using these criteria, we prove results on the non-vanishing of Poincaré series of $K$-finite matrix coefficients of integrable representations of $SL_2(\mathbb{R})^{\sim}$ and of corresponding cusp forms, results on the non-vanishing of some classical Poincaré series of half-integral weight, and results on the non-vanishing of $L$-functions associated to cusp forms of halfintegral weight $m \geq \frac{9}{2}$ in points of the strip $\mathbb{C}_{\frac{m}{2} < \Re (s) < m-1}$

Novel Approach in the Construction of Bioethanol-Producing Saccharomyces cerevisiae Hybrids

Za proizvodnju bioetanola iz lignoceluloznih hidrolizata potreban je proizvodni soj koji dobro podnosi prisutnost inhibitora rasta i fermentacije te veliku koncentraciju etanola. Stoga smo konstruirali hibridne diploide kvasca Saccharomyces cerevisiae međusobnim križanjem dvaju prirodnih izolata, Yllcl 7_E5 i UWOPS87-2421. Soj Yllcl 7_E5 izoliran je iz vina kao dobar proizvođač etanola, a soj UWOPS87-2421 izoliran je iz cvijeta kaktusa Opuntia megacantha i otporan je na inhibitore koji se mogu naći u lignoceluloznim hidrolizatima. Hibridni sojevi rasli su brže od ishodnih sojeva u odsutnosti i prisutnosti octene i levulinske kiseline te 2-furaldehida, koji se često nalaze kao inhibitori rasta u lignoceluloznim hidrolizatima, a pojačana ekspresija gena YAPI povećala je preživljenje testiranih sojeva. Također, pojedini su hibridni sojevi, iako potječu od dvaju istih ishodnih sojeva, pokazali različit fermentativni potencijal u testu proizvodnje C02, što upućuje na njihovu genetičku varijabilnost koja omogućava daljnju selekciju poželjnih svojstava. Iz naših se rezultata može zaključiti da se kombiniranjem konstrukcije hibridnih sojeva i metoda genetičkog inženjerstva mogu oplemeniti i razviti novi biotehnološki relevantni sojevi kvasca S. cerevisiae. Osim toga, utvrđeno je da je uspješnost ciljanja gena u prirodnim izolatima S. cerevisiae (Yllcl 7_E5a and UWOPS87-2421 a) daleko manja nego u laboratorijskim sojevima, a najčešći aberantni događaj bio je duplikacija ciljanog kromosoma.Bioethanol production from lignocellulosic hydrolysates requires a producer strain that tolerates both the presence of growth and fermentation inhibitors and high ethanol concentrations. Therefore, we constructed heterozygous intraspecies hybrid diploids of Saccharomyces cerevisiae by crossing two natural S. cerevisiae isolates, YIIc17_E5 and UWOPS87-2421, a good ethanol producer found in wine and a strain from the flower of the cactus Opuntia megacantha resistant to inhibitors found in lignocellulosic hydrolysates, respectively. Hybrids grew faster than parental strains in the absence and in the presence of acetic and levulinic acids and 2-furaldehyde, inhibitors frequently found in lignocellulosic hydrolysates, and the overexpression of YAP1 gene increased their survival. Furthermore, although originating from the same parental strains, hybrids displayed different fermentative potential in a CO2 production test, suggesting genetic variability that could be used for further selection of desirable traits. Therefore, our results suggest that the construction of intraspecies hybrids coupled with the use of genetic engineering techniques is a promising approach for improvement or development of new biotechnologically relevant strains of S. cerevisiae. Moreover, it was found that the success of gene targeting (gene targeting fidelity) in natural S. cerevisiae isolates (YIIc17_E5α and UWOPS87-2421α) was strikingly lower than in laboratory strains and the most frequent off-targeting event was targeted chromosome duplication

Transformation and genetic recombination In yeast Dekkera bruxellensis

U proizvodnji bioetanola kvasac Dekkera bruxellensis vrlo je zanimljiva alternativa kvascu Saccharomyces cerevisiae. Obje vrste brzo fermentiraju šećere i proizvode visoke koncentracije etanola, ali kvasac D. bruxellensis fermentira i disaharid celobiozu te kao izvor dušika koristi i nitrat. Međutim, biotehnološki potencijal ove vrste danas je neiskorišten jer do sada nije bilo moguće ciljano mijenjati njegov genom, tj. metodama genetičkog inženjerstva nije bilo moguće konstruirati sojeve željenih karakteristika. Stoga je cilj ovog rada bio razviti metodologiju za preciznu modifikaciju genoma ovog kvasca, sličnu onoj koja se svakodnevno provodi u modelnom kvascu S. cerevisiae. Razvijena je metoda genetičke transformacije, temeljena na elektroporaciji, kojom se u stanice kvasca D. bruxellensis unosi transformirajuća DNA. Nadalje, konstruirani su integrativni i replikativni plazmidi koji omogućavaju ekspresiju gena iz drugih organizama u ovom kvascu. U ovoj vrsti uspješnost uvođenja preciznih modifikacija u genom relativno je niska (5 – 10%) te ovisi o duljini homologne DNA na krajevima transformirajućeg fragmenta i mehanizmu njegove ugradnje. Međutim, ekspresija gena RAD54 iz kvasca S. cerevisiae, gena koji sudjeluje u procesu homologne genetičke rekombinacije, podigla je uspješnost genskog ciljanja u kvascu D. bruxellensis za oko 3 puta. S druge strane, ekspresija gena RAD52 iz kvasca S. cerevisiae, također uključenog u proces homologne rekombinacije, smanjila je već prethodno nisku uspješnost preciznih modifikacija genoma. Zahvaljujući razvijenoj metodologiji, u ovom radu konstruiran je i soj kvasca D. bruxellensis koji eksprimira transporter ksiloze, ksiloza reduktazu, ksilitol dehidrogenazu i ksilulokinazu pa stoga može rasti na podlozi na kojoj je ksiloza jedini izvor ugljika. Ovaj soj biotehnološki je značajan jer se u proizvodnji bioetanola druge generacije kao supstrat koriste lignocelulozni hidrolizati bogati ovim teško iskoristivim šećerom.BACKGROUND: In biotechnology, yeast Dekkera (Brettanomyces) bruxellensis plays a few opposing roles. It irreparably spoils wine by imparting on it the taste and smell of horse-sweat, urine, and burnt plastic. On the other hand, it is irreplaceable in the secondary fermentation of Lambic and English beers to which it bestows fruity aromas and aged character. Furthermore, its robust physiology, lifestyle, and capacity for vigorous fermentation advocate its use in the production of bioethanol. However, despite its versatility and importance, the biology of D. bruxellensis remains mostly unexplored. One of the obstacles to a better understanding of D. bruxellensis lies in the fact that until now was impossible to genetically modify this yeast, i.e., it was impossible to introduce and integrate foreign DNA into its genome, preferably at specific loci. This constraint also hindered construction of strains with improved properties, such as strains that produce higher alcohol content or that ferment xylose, a sugar that is abundant in raw materials used for the production of second-generation bioethanol. To overcome such limitations, this work aimed to develop a protocol for genetic transformation of D. bruxellensis and to determine how often the transforming fragment integrates at a homologous site in the genome. METHODS AND RESULTS: In this work, strain CBS 2499 of yeast D. bruxellensis was microbiologically characterized, genetically transformed, and used to investigate the efficiency of gene targeting. To ease and expedite the cultivation of D. bruxellensis and its transformants, different microbiological media were tested and, when required, new media were formulated. Bioinformatic analysis with newly developed software LIDD showed that almost all genes in strain CBS 2499 are present in two copies. This fact hindered but did not prevent isolation of auxotrophic mutants. Protocols used for routine transformation of model yeast Saccharomyces cerevisiae did not produce transformants of D. bruxellensis until they were reinforced with additional steps. Once fine-tuned, the protocols produced over 5000 transformants per microgram of circular plasmid DNA when they were combined with appropriate selective markers and newly developed set of centromeric plasmids. Analyses of transformants by Southern blot showed that in D. bruxellensis gene targeting is inefficient. The transforming fragment integrated at a specific locus only when it carried few kilobases of homologous DNA at its ends. Even then, gene targeting occurred in only 5-10% of the transformants. However, when D. bruxellensis expressed Rad54 protein from yeast S. cerevisiae, the frequency of ends-in targeting increased threefold. At the same time, the frequency of ends-out targeting remained unaffected. When D. bruxellensis expressed Rad52 protein from yeast S. cerevisiae, the efficiency of both ends-in and ends-out targeting decreased unexpectedly. Low efficiency of gene targeting suggests that illegitimate recombination is very efficient in D. bruxellensis. This high efficiency is also evident when multiple transforming fragments enter the cell as they are first ligated together and then illegitimately integrated into the genome. As such, they form tandem repeats which undergo frequent crossing-overs and are therefore unstable. By employing developed transformation protocol, a new strain of D. bruxellensis was constructed which encodes for xylose transporter, xylose reductase, xylitol dehydrogenase, and xylulokinase. Following UV mutagenesis and adaptive evolution, derivatives of this strain grew on a medium in which xylose was the only carbon source. SIGNIFICANCE AND IMPACT OF STUDY: Protocols described here for the first time allow the introduction of heterologous DNA into yeast D. bruxellensis. Subsequent measurements of gene targeting efficiency defined the length of the homologies at the end of the transforming fragment needed to introduce specific modifications into the genome. With this information, it is possible to construct biotechnologically interesting strains of D. bruxellensis which express new or lack existing genes. As such, these results represent a turning point for both fundamental and practical research of D. bruxellensis. As a proof-of-concept, a strain of D. bruxellensis was developed which can utilize xylose as a carbon source and as such is very interesting for the production of second-generation bioethanol from lignocellulose hydrolysates

Defect Detection in Ultrasonic Images Using Deep Neural Networks

Ultrazvučno testiranje jedna je od najzastupljenijih nedestruktivnih metoda ispitivanja materijala u industriji. Ultrazvučnom inspekcijom prikuplja se velika količina ultrazvučnih slika koju trenutno analiziraju ljudski inspektori svojim subjektivnim prosuđivanjem što rezultate čini osjetljivima na ljudsku pogrešku. Sve se češće koriste pristupi bazirani na strojnom učenju kako bi se automatizirao proces analiziranja takvih podataka. Korištenjem konvolucijskih neuronskih mreža moguće je otkriti oštećenja na materijalu, ali i odrediti njihove vrste i veličine. U okviru završnog rada implementirana su tri modela konvolucijskih neuronskih mreža te tumačenjem rezultata donesen je zaključak kako se takvi modeli mogu koristiti u izvedbenim rješenjima.For decades ultrasonic testing has been the most widely represented non-destructive method of materials testing in industry. Ultrasonic testing involves the acquisition of large volume of ultrasonic images that are currently analyzed by human inspectors in a subjective manner, making the results susceptible to human error. Recently, machine learning approaches have been increasingly used to automate the process of analyzing such data. Using convolutional neural networks, it is possible to detect damage to the material, but also to determine the type and size of the defect. Within the undergraduate thesis three models of convolutional neural networks were implemented and the interpretation of the results led to the conclusion how such models can be used in performance solutions