Ponceletov teorem

Ponceletov teorem, koji govori o postojanju mnogokuta istovremeno upisanog jednoj a opisanog drugoj konici, jedan je od najpoznatijih i najviše proučavanih teorema u projektivnoj geometriji. Kao što je izloženo u ovom radu, ako postoji mnogokut koji je upisan jednoj konici i opisan drugoj, tada postoji beskonačno mnogo mnogokuta s jednakim brojem strana. Upravo se zbog prirode postojanja rješenja ovaj teorem često naziva i Ponceletov porizam, gdje riječ porizam dolazi iz Stare Grčke iz koje dolaze brojni matematičari važni za razvoj matematike, posebno geometrije. U prvom poglavlju riječ je o zatvaranju u trokut gdje smo stekli prvi dojam o Ponceletovom teoremu, odnosno stekli smo dojam da mora postojati nekakav uvjet da bi do zatvaranja u trokut uopće došlo ako su dane dvije kružnice u ravnini, različitih polumjera, jedna unutar druge. Postojanje uvjeta za konstrukciju trokuta koji je upisan jednoj, a opisan drugoj kružnici navodi na dokaz Eulerovog teorema kojim se pokazuje da se takav trokut može konstruirati ako je zadovoljena Eulerova relacija. Eulerov teorem zapravo je temelj dokaza specijalnog slučaja Ponceletovog teorema za kružnice i trokut koji se nalazi u Poglavlju 3, te navodi na analogan uvjet za postojanje četverokuta koji je upisan vanjskoj, a opisan unutarnjoj kružnici. Najvažniji dio ovog rada zapravo je dokaz Ponceletovog teorema. Zbog kompleksnosti i dubine ovog teorema dokaz je vrlo zahtjevan, pogotovo zato što ne postoji elementarni dokaz. Teorem smo dokazali algebarsko-geometrijskim dokazom po uzoru na njemačkog matematičara Adolfa Hurwitza. U završnom dijelu rada izložena je mehanička interpretacija Ponceletovog teorema.Poncelet theorem is one of the most important and most explored theorems of projective geometry. This is a theorem about closing property for a sequence of polygons inscribed in one and circumscribed about the other given conic in the same plane. If described polygon exists, then there are infinitely many polygons with the same number of sides, inscribed in one and circumscribed about the other conic. The Poncelet theorem is usually called the Poncelet porism, where porism originates from Ancient Greece, home of many famous mathematicians important for the development of mathematics, especially geometry. First chapter is about special polygon, triangle. We have noticed that we need a special condition for its existence if there are two given circles, one inside the other, in the same plane. That condition is well known as ”Euler’s formula” which is specified in Euler’s theorem, proved in Chapter 2. Euler’s theorem is very important for a special case of Poncelet theorem, Poncelet theorem for triangle and two given circles in Chapter 3. Using analogy and special condition for quadrilaterals, there exists a quadrilateral inscribed in one, and circumscribed about the other given circle. The most important part is the proof of Poncelet theorem. Because of its complexity, a proof is very difficult especially because there is no elementary proof. We use an algebraic - geometric method to prove the theorem like the German mathematician Adolf Hurwitz. In the last chapter, we focus on the mechanical interpretation of Poncelet’s theorem

Ponceletov teorem

Ponceletov teorem, koji govori o postojanju mnogokuta istovremeno upisanog jednoj a opisanog drugoj konici, jedan je od najpoznatijih i najviše proučavanih teorema u projektivnoj geometriji. Kao što je izloženo u ovom radu, ako postoji mnogokut koji je upisan jednoj konici i opisan drugoj, tada postoji beskonačno mnogo mnogokuta s jednakim brojem strana. Upravo se zbog prirode postojanja rješenja ovaj teorem često naziva i Ponceletov porizam, gdje riječ porizam dolazi iz Stare Grčke iz koje dolaze brojni matematičari važni za razvoj matematike, posebno geometrije. U prvom poglavlju riječ je o zatvaranju u trokut gdje smo stekli prvi dojam o Ponceletovom teoremu, odnosno stekli smo dojam da mora postojati nekakav uvjet da bi do zatvaranja u trokut uopće došlo ako su dane dvije kružnice u ravnini, različitih polumjera, jedna unutar druge. Postojanje uvjeta za konstrukciju trokuta koji je upisan jednoj, a opisan drugoj kružnici navodi na dokaz Eulerovog teorema kojim se pokazuje da se takav trokut može konstruirati ako je zadovoljena Eulerova relacija. Eulerov teorem zapravo je temelj dokaza specijalnog slučaja Ponceletovog teorema za kružnice i trokut koji se nalazi u Poglavlju 3, te navodi na analogan uvjet za postojanje četverokuta koji je upisan vanjskoj, a opisan unutarnjoj kružnici. Najvažniji dio ovog rada zapravo je dokaz Ponceletovog teorema. Zbog kompleksnosti i dubine ovog teorema dokaz je vrlo zahtjevan, pogotovo zato što ne postoji elementarni dokaz. Teorem smo dokazali algebarsko-geometrijskim dokazom po uzoru na njemačkog matematičara Adolfa Hurwitza. U završnom dijelu rada izložena je mehanička interpretacija Ponceletovog teorema.Poncelet theorem is one of the most important and most explored theorems of projective geometry. This is a theorem about closing property for a sequence of polygons inscribed in one and circumscribed about the other given conic in the same plane. If described polygon exists, then there are infinitely many polygons with the same number of sides, inscribed in one and circumscribed about the other conic. The Poncelet theorem is usually called the Poncelet porism, where porism originates from Ancient Greece, home of many famous mathematicians important for the development of mathematics, especially geometry. First chapter is about special polygon, triangle. We have noticed that we need a special condition for its existence if there are two given circles, one inside the other, in the same plane. That condition is well known as ”Euler’s formula” which is specified in Euler’s theorem, proved in Chapter 2. Euler’s theorem is very important for a special case of Poncelet theorem, Poncelet theorem for triangle and two given circles in Chapter 3. Using analogy and special condition for quadrilaterals, there exists a quadrilateral inscribed in one, and circumscribed about the other given circle. The most important part is the proof of Poncelet theorem. Because of its complexity, a proof is very difficult especially because there is no elementary proof. We use an algebraic - geometric method to prove the theorem like the German mathematician Adolf Hurwitz. In the last chapter, we focus on the mechanical interpretation of Poncelet’s theorem

The Impact of the Organized Coach Tours on the Pollution in the Republic of Croatia

Tema diplomskog rada je „Utjecaj organiziranog autobusnog prijevoza na onečišćenje okoliša na području Republike Hrvatske“. Rad je temeljen na analizi sekundarnih podataka i izvora i na podacima prikupljenim primarnim istraživanjem. Onečišćenje okoliša definira se kao promjena stanja okoliša zbog nedopuštene emisije štetnih plinova ili zbog nekog drugog djelovanja koje bi moglo promijeniti kvalitetu okoliša. U tekstu se upućuje na probleme onečišćenja okoliša koji nastaje emisijama štetnih plinova iz motora autobusa. Također, upućuje se i na potrebu istraživanja u koje bi se mogli uključiti svi sudionici lokalne zajednice i turisti. Prvo je istraživanje provedeno 2002., u kojem je zaključeno da autobusi u turizmu ne zadovoljavaju ekološke standarde i da u razdobljima povećane potražnje autobusi ne zadovoljavaju turističke potrebe. Rezultati rada pokazuju da u Hrvatskoj prevladava još uvijek velik broj autobusa koji imaju zastarjelu tehnologiju. Također, dokumenti koje donosi zakonodavac sadržavaju implementaciju novih tehnologija i razvoja turizma prema načelima održivog razvoja. Zakonodavac regulira negativne efekte onečišćenja propisanom posebnom naknadom za onečišćenje i subvencijom za kupovinu novih vozila s čistom tehnologijom. U Hrvatskoj postoje preduvjeti za razvoj održivog turizma koji uključuje i novi trend koji se afirmira u Europskoj uniji a to je razvoj autobusnog turizma kao i uvođenje ekoloških zona u turističke destinacije. I na području Hrvatske bilježi se porast broja kružnih putovanja u organizaciji domaćih ili stranih turističkih agencija i organizatora putovanja. Međutim, unatoč brojnim zakonima, uredbama i strategijama o održivom turizmu, prometu ili razvoju društva u cjelini isti se ne provode u dovoljnoj mjeri u praksi. Ključni je problem još uvijek nedovoljno razvijena ideja i svijest o potrebi zaštite okoliša kao i nedovoljni izvori financiranja autobusnih prijevoznika za modernizaciju voznog parka.The subject of the thesis in this paper is "The Impact of the Organized Coach Tours on the Pollution in the Republic of Croatia." The pollution is defined as a change in the environment caused by the gas emissions from mobile sources into the environment or by some other impacts that are causing the changes of the environment. The thesis is written on the basis and analysis of the secondary research in which the papers and documents are included. Also, the second part of the thesis is the primary research conducted by the author. The first research in this field was carried out in 2002. The aim of the thesis is to analyse the problems of the air pollution caused by the gas emissions from the coaches and buses. Additionally, this paper is inspecting the possible solutions that could be conducted by government or individuals in the tourism and transport sector. The organized coach tours are the most growing trend in tourism industry which is based on the use of the ecologically equipped vehicles that are the dominant factor regarding the coach tours. This trend is developing in Croatia as well. The results of the research conducted in the thesis show the number of coaches that do not fulfill the technical and environmental requirements. The reasons are diverse and are co-related with the financial resources of the bus operators and the decreasing level of the environmental awareness. Despite the implementation of programs related to the sustainable development such as the subvention for buying new green vehicles there are problems that occur in communication between the policy and consumers. These problems are related to the information access and to the inefficency of the processing of legislative norms proscribed by government or NGO's. In conclusion, it is required to increase the environmental awarness by different campaigns in the tourism and the transport sector as well

Izraženost Wnt 4 u presađenim bubrezima [Expression of Wnt 4 in transplant kidneys]

With the gradual development of modern medicine the overall survival and quality of life of patients with chronic kidney disease significantly improved. Kidney transplantation significantly enhanced the treatment of end-stage renal disease patients. However, many transplanted kidneys are lost due to chronic graft changes caused by immunosupression therapy or occurence of late chronic immunologic response. The family of Wnt (wingless- type MMTV integration site family) genes are constituted of 19, till now, described members. Some of this proteins are present in kidney development but also in the process of kidney injury and reparation. Due to this published results it is assumed that the same molecules responsible for the regulation of nephrogenesis also participates in the patophysiology of tubulointerstitial kidney disease. It is still not clear which Wnt gene is responsible for protection and which for damage especially in transplanted kidneys. The aim of this study was to analyze whether Wnt 4 expression is increased in transplanted kidneys compared to healthy kdiney samples and is it dependable of different causes of graftectomies. In this study we have included 59 kidney transplanted patients from deceased donors and 24 nephrectomized patients due to localised kidney tumours as a control group. The obtained results did not confirmed our hypothesis. On the contrary, the Wnt 4 expression was significantly lower in the kidney transplanted patients group compared to healthy kidney tissue samples which was strongly correlated with age. Although the Wnt 4 expression was correlated with higher age, its most independent and significant correlation was with longer cold ischemia time. This are, at least to our knowledge, the first results which reported higher Wnt 4 expression and better kidney graft survival and its possible role as an factor in kidney graft regeneration especially in prolonged ischemia time and acute tubular necrosis. These results emphasize the importance on future studies of different Wnt receptors agonists as a novel therapy in treating tissue ichemia injuries

Efficiency of mechanised collection of soybean straw

Značaj soje proizlazi iz kakvoće njenog zrna (visok sadržaj bjelančevina i ulja), pa je jedna od najznačajnijih bjelančevinastih i uljnih kultura u svijetu. Sojina slama predstavlja važan izvor organske tvari te ima važan utjecaj na biološka, kemijska i fizikalna svojstva tla i ne treba ih smatrati otpadom. Biomasa poljoprivrednoga porijekla, pa tako i slama soje, vrlo je prihvatljivo gorivo s gledišta utjecaja na okoliš. Cilj ovog rada je utvrditi stvarne količine prikupljene slame nakon žetve soje korištenjem postojeće mehanizacije, prešom za valjkaste bale s pick-up uređajem. Poljski pokus proveden je na lokaciji u Velikoj Barni s tri sorte soje, Lucijom, Galinom i Ikom, zastupljenima u plodoredu kod domaćih poljoprivrednika. Učinkovitost baliranja razlikovala se između sorata i iznosila je od 33,3 % kod sorte Ika do 48,9 % kod sorte Lucija. Količina žetvenih ostataka koji će se moći prikupiti ovisi sadržaju vode u slami u trenutku baliranja, te konstrukciji kombajna i balirke.The importance of soybean comes from the quality of its grain (high protein and oil content), so it is one of the most important protein and oil crops in the world. Straw is an important source of organic matter and have important influence on the biological, chemical and physical properties of the soil and should not be considered as waste. Biomass of agricultural origin, like soy straw, is highly acceptable energy source from the environmental point of view. The aim of this paper is to determine the actual amounts of straw collected after harvesting soybean using existing mechanization, round baler with pick-up. The field experiment was set at location in Velika Barna with three soy varieties, Lucija, Galina and Ika, represented in the crop rotation by domestic farmers. Baling efficiency differed between soybean varieties and ranged from 33.3% for Ika to 48.9% for Lucija. The amount collected of post-harvest residues depended on the water content of the straw at the time of baling and the construction of combine and baler

Goat on the Istrian Coat of Arms Towards the Knowledge of the Historical Background of the Coat of Arms and the Istrian Goat-Breeding

U članku se analiziraju podaci o istarskom grbu, kako iz heraldičke literature, tako i iz ostalih raznorodnih izvora. Uglavnom se radi o oskudnim i današnjem čitatelju teško razumljivim prilozima, već i stoga što su bili namijenjeni ciljanoj, upućenoj publici. Posebnu poteškoću predstavljaju tekstovi, a takvih je većina, u kojima izneseni podaci o grbu nisu uopće ili nisu izravno dokumentirani. Stoga se, npr., pretpostavka o vremenu njegova nastanka, a podaci potvrđuju da ga je prvotno koristila Mletačka Republika, temelji na prikazu ovog grba s jedne strane na mletačkim, a s druge na zemljopisnim kartama ostalih europskih kartografa. Iznesena je i pretpostavka o tome kada ga je preuzela Austrija. Članak također razmatra popularno tumačenje, koje su sugerirali i neki autori, a neki ga smatraju tako očitim da mu argumentacija nije ni potrebna. Naime, da je ova domaća životinja imala iznimno važnu ulogu u istarskoj povijesti (kulturi, umjetnosti, religiji i gospodarstvu), što je dovelo do toga da postane obilježjem u grbu.Since 1990, Republic of Croatia has a new coat of arms. What is actually new is only its composition, while its constituent parts are historical Croatian coats of arms, including the Istrian one. Little is known about Istrian coat of arms, but a vague belief that it is an old feature of Istria is prevalent. This paper presents contributions on the coat of arms that the author has found so far, mostly descriptions of simple figures and tinctures. Discovered examples of the coat of arms show various deviations in detail of its execution. A change of a simple figure in l915 has also been documented, and from that point on, it was no longer a she-goat but a he-goat. The origin of this coat of arms is still an open issue, as well as the exact date of its origin and related details. The author explained her assumption that it didn\u27t exist prior to the 17th century. She was also interested whether the assumption that the goat was on the coat of arms because it was especially important for the existence of people in the past, and that it therefore left trace in religious life, language, monuments and various visual arts, could be proved. These traces probably wouldn\u27t exist if the goats weren\u27t so numerous. The data we presently have at our disposal affirm explicit importance and numerousness of goats in the history of Istria. They particularly left trace in the culture of Ćići (Istro-Romanians), specifically in their speech concerning the names of certain cattle-raising buildings. Croatian ethnologist A. Stojanović, who recognised the existence of particular goat-breeding layer in the Adriatic and parts of Mediterranean in regards to cattle raising in general, defined one technical detail as a criterion for belonging to that particular layer. Namely, the fact that sheep are not milked, but that goats are milked instead, as is the case on the island of Mljet, but that was not confirmed in Istria, and there was no confirmation that goat milk had been considerably represented in nourishment which is a regular feature of the goat-breeding layer in question. The author however feels that the role of the goat in the Istrian economy and tradition didn\u27t necessarily have to be so strong in order for it to become a trademark in the coat of arms. So far none of the official documents that would prove that choice are known. Data found have shown that this coat of arms had been officially used for Istria since the second part of the 17th century merely as a figure in primary Venetian and later Austrian official documents. The confirmation of using the coat of arms in the field work dates back only to the early 20th century. So far the least known aspect of this emblem, i. e. the goat as a simple figure, have been its national implications. Namely, in Stemmatographia by Vitezović from the early 18th century this Istrian coat of arms is represented among the Illyrian, therefore, according to Vitezović, Croatian coats of arms. In addition, it is left for the future to cast the light on the part of the history of this emblem, in which the goat became Italian-irredentist-fascist symbol late in the 19th and in the early 20th century

Electric vehicle charging infrastructure in Croatia – First-hand experiences and recommendations for future development

One of the most serious obstacles to massive deployment of electric vehicles is insufficient and cumbersome charging infrastructure. Both the number of charging points and their power capacity are often insufficient. On top of that, this new technology often suffers from many issues related to insufficient testing, immaturity and irregular handling. This paper summarizes the issues with the electric vehicle charging infrastructure and describes first-hand experiences with long-range electric vehicle trips originating from Zagreb, Croatia, during 2022. Issues with the charging points locations, power and availability are assessed. Based on these experiences, the paper provides some thoughts on the possible directions of the further development of the electric vehicle charging infrastructure in Croatia