Pappusov teorem o šesterovrhu

Pappusov teorem o šesterovrhu jedan je od najvažnijih teorema projektivne geometrije. On opisuje konfiguraciju od devet točaka i devet pravaca koja nastaje tako što se parovi nasuprotnih stranica šesterovrha, čiji vrhovi leže naizmjence na dva različita pravca, sijeku u trima kolinearnim točkama. U radu je izloženo osam različitih dokaza Papposovog teorema, s ciljem da se upoznaju i usporede različite metode pristupa. Posebno se razmatraju prednosti projektivno-geometrijskog pristupa u odnosu na osnovnu formulaciju teorema u okviru euklidske ravnine. Istaknuta je veza s Pascalovim teoremom i, preko Cayley-Bachrach-Chaslesovog teorema, dublja povezanost s algebarskom geometrijom. S druge strane, projektivnom interpretacijom euklidskog pojma kružnice pomoću para imaginarnih točaka na neizmjerno dalekom pravcu, dobiva se Miquelov teorem iz geometrije kružnica. Završno, naznačena je mogućnost formalnog algebarskog dokaza, preko homogenih koordinata, kojim se Pappusov teorem obuhvaća u svojoj punoj općenitosti.The hexagon theorem of Pappus is one of the most important theorems in projective geometry. It describes the configuration of nine lines and nine points which is generated by pairs of opposite sides of the hexagon, whose vertices lie alternatively on two different lines, intersect in three collinear points. In this paper we present eight proofs of Pappus’s theorem, with the aim of learning about different methods of approach to the problem and making a comparison between them. In particular, we expose the advantage of the projective geometry setting over the initial Euclidean formulation. We emphasize the connection to the Pascal’s theorem and, through the Cayley-Bachrach-Chasles theorem, the profound connection with Algebraic geometry. On the other hand, applying the projective interpretation of a circle, which is basically a Euclidean concept, by using a pair of imaginary points on the line at infinity, we derive Miquel’s theorem from the geometry of circles. Finally, we indicate the possibility of a formal algebraic proof using homogeneous coordinates, which encompasses Pappus’s theorem in its full generality

Ancient Roman marble sculpture of an empress from the Archaeological Museum Zadar, conservation works

Autor iznosi kratki pregled konzervatorsko-restauratorskih rado­va na antičkoj mramornoj skulpturi carice iz zbirke Arheološkog muzeja Zadar. Radovi su obuhvaćali demontažu skulpture iz po­stava, čišćenje površine mramora, istraživanje potencijalnih osta­taka polikromije, vakuumsku konsolidaciju mramora te montažu u antički postav Arheološkog muzeja Zadar. Zahvati su obrađeni u sklopu praktične nastave na Odsjeku za konzervaciju i restaura­ciju kamena Umjetničke akademije u Splitu.The author give a brief overview of the conservation-restoration work on the ancient marble sculpture of an empress from the collection of the Archaeological Museum Zadar. The process in­cluded dismantling the sculpture from the exhibition, cleaning the surface of the marble, exploring potential remains of poly­chromy, the vacuum consolidation of the marble, and installation in the ancient exhibition of the Archaeological Museum Zadar. The interventions were done as part of the practical course at the Department of conservation-restoration of stone at the Arts Academy in Split

Interpopulation genetic-ecological variation of Scots pine (Pinus sylvestris L.) in Serbia

The genetic-ecological variation of Scots pine (Pinus sylvestris L.) in Serbia was studied in the populations at five localities in western and south-western Serbia. Three groups of Scots pine (Pinus sylvestris L.) populations were differentiated based on genetic research (seed protein analysis) and plant community research. The first group consists of Scots pine populations on Šargan (FMU “Šargan“) and on Tara (FMU “Kaluderske Bare”), where the forests belong to the community of Scots pine and Austrian pine (Pinetum sylvestris-nigrae Pavlovic 1951). The second group covers the localities Stolovi (FMU “Radocelo-Crepuljnik“) and Zlatar (FMU “Zlatar I“), where the forests belong to the community of Scots pine and spruce (Piceo abietis-Pinetum sylvestris Stefanovic 1960). The third group comprises the Scots pine population on Pešter (FMU “Dubocica-Bare“) which belongs to the community of Scots pine with erica (Erico-Pinetum sylvestris Stefanovic 1963). Cluster analysis was performed on the basis of seed protein data and showed that there are three groups of Scots pine populations. The three populations coincide with plant communities. The community of Scots pine with erica (Erico-Pinetum sylvestris Stefanovic 1963) recorded on Pešter at the locality “Dubocica- Bare“ in the area of FE “Golija“ Ivanjica, is a special Scots pine population displayed at the greatest distance from all other populations in the cluster analysis dendrogram

Carboxyl Group as a Radical Scavenging Moiety: Thermodynamics of 2H+/2e– Processes of Phloretic Acid

Phloretic acid is one of the most abundant colon catabolites of various classes of polyphenols (e.g., polymeric proanthocyanidins, tea catechins, and ellagitannins). In this paper thermodynamics of 2H+/2e– radical scavenging mechanisms of phloretic acid was studied. For the first time the involvement of carboxyl group in double hydrogen atom transfer (dHAT), double electron transfer-proton transfer (dET-PT) and sequential double proton loss double electron transfer (SdPLdET) processes was investigated. Obtained results indicate that phloretic acid possesses potential for inactivating radicals of different characteristics (HO•, HOO•, CH3O•, CH3OO•, CH2=CH–O–O•, PhO•, Cl3COO• etc.) via dHAT and SdPLdET mechanisms. Because phloretic acid is usually better absorbed than its precursor molecules, it may contribute to health benefits associated with regular intake of polyphenol-rich diet by direct scavenging of radicals. This work is licensed under a Creative Commons Attribution 4.0 International License

Genetička karakterizacija genotipova populacija crnog bora (Pinus nigra Arnold) upotrebom proteinskih markera

The occurrence of Austrian pine over large areas attacked by erosion processes or on completely degraded and denuded areas on which it produces excellent results, classifies it among the most important economic species in forestry. The aim of present research is to study intra- and inter-population genetic variation in Austrian pine by the application of biochemical markers. A better knowledge of the genetic potential of Austrian pine populations will enhance the production of seed and planting material, and in this way also the success of a forestation and the establishment of Austrian pine specific-purpose plantations. The polymorphism of protein markers was determined based on the selected genotypes originating from six populations (seed stands), which actually included all Austrian pine populations in Serbia. Based on the derived electrophoregram, qualitative and quantitative differences (number and pattern) in protein fractions were identified and the seed protein profile was constructed for each tree, as well as for each of the six study populations. This showed the 'conservativeness' of some polypeptides, i.e. the presence in all analyzed genotypes. On the other hand, some protein fractions were variable at the population level, and some were variable depending on the population. Based on the electrophoregram, the coefficient of genetic similarity/distance was calculated by comparing the band numbers and patterns. Cluster analysis dendrograms were also constructed. The degree of genetic variation among the provenances was higher than the variation within the provenances. There was also a high genetic variation in seed proteins within the analyzed populations.Javljanje crnog bora na velikim površinama zahvaćenim erozionim procesima, ili potpuno degradiranim, ogoljenim površinama na kojima daje odlične rezultate, svrstavaju ga u najvažnije ekonomske vrste korišćene u šumarstvu. Cilj ovih istraživanja je bilo ispitivanje unutar i među populacijske genetičke varijabilnost crnog bora primenom biohemijskih markera. Bolje upoznavanje genetskog potencijala populacija crnog bora omogućiće poboljšanje proizvodnje semena i sadnog materijala, a time i uspeh prilikom pošumljavanja i podizanja namenskih kultura ovom vrstom. Polimorfizam proteinskih markera utvrđen je na osnovu odabranih genotipova poreklom iz šest populacija (semenskih sastojina) čime su obuhvaćene sve populacije crnog bora u Srbiji. Na osnovu dobijenih elektroforegrama utvrđene su kvalitativne i kvantitativne razlike (broj i raspored) proteinskih frakcija i određen je proteinski profil semena za svako stablo kao i za svaku od 6 ispitivanih populacija ponaosob. Tom prilikom je zapažena 'konzervativnost' pojedinih polipeptida, tj. prisustvo u svim analiziranim genotipovima. S druge strane utvrđene su proteinske frakcije koje su varijabilne na nivou populacija, kao i one čija varijabilnost varira u zavisnosti od populacija. Na osnovu dobijenih elektroforegrama je izračunat koeficijent genetičke sličnosti, odnosno udaljenosti, upoređivanjem broja i rasporeda traka i urađeni su dendogrami klaster analize. Utvrđen je veći stepen genetskog variranja između provenijencija nego unutar njih. Takođe je utvrđena velika genetska varijabilnost proteina semena unutar analiziranih populacija

Navika pušenja na radnom mjestu kod medicinskih sestara - usporedba između Opće bolnice i Doma zdravlja Dubrovnik

Danas kronične nezarazne bolesti uzrokuju 63 % ukupnog mortaliteta, a duhanski preparati jedan su od najvećih rizičnih čimbenika [1] . Pušenje je rizični čimbenik za nastanak velikog broja bolesti koje znatno smanjuju stupanj kvalitete života te uzrokuju prijevremenu smrt

Genetic divergence of Scots pine (Pinus sylvestris L.) populations in Serbia revealed by RAPD

The ability of random amplified polymorphic DNA (RAPD) to distinguish among Scots pine populations from Serbia was evaluated. Sixteen arbitrary 10-mer primers employed in the analysis produced 54 fragments of which 21 were polymorphic (38.89%). Certain rare and genotype-specific bands were identified which could be effectively used to distinguish between the populations. Polymorphism in RAPD markers among P. sylvestris populations was high and sufficient to distinguish each of the populations. The results obtained suggest that RAPD markers are valuable for the genetic divergence estimation in Pinus sylvestris and for the study of divergence among populations

Estimation of Random Accuracy and its Use in Validation of Predictive Quality of Classification Models within Predictive Challenges

Shortcomings of the correlation coefficient (Pearson's) as a measure for estimating and calculating the accuracy of predictive model properties are analysed. Here we discuss two such cases that can often occur in the application of the model in predicting properties of a new external set of compounds. The first problem in using the correlation coefficient is its insensitivity to the systemic error that must be expected in predicting properties of a novel external set of compounds, which is not a random sample selected from the training set. The second problem is that an external set can be arbitrarily large or small and have an arbitrary and uneven distribution of the measured value of the target variable, whose values are not known in advance. In these conditions, the correlation coefficient can be an overoptimistic measure of agreement of predicted values with the corresponding experimental values and can lead to a highly optimistic conclusion about the predictive ability of the model. Due to these shortcomings of the correlation coefficient, the use of standard error (root-mean-square-error) of prediction is suggested as a better quality measure of predictive capabilities of a model. In the case of classification models, the use of the difference between the real accuracy and the most probable random accuracy of the model shows very good characteristics in ranking different models according to predictive quality, having at the same time an obvious interpretation

Measuring and comparing the carbon footprints of different procurement models for primary school meals:Analysis of cases across five European countries

Sustainable Public Food Procurement (PFP) represents a key game changer for food systems transformation. It can influence both food consumption and food production patterns. It can deliver multiple social, economic and environmental benefits towards sustainable food systems for healthy diets. This publication aims to contribute to the improved understanding, dissemination and use of PFP as a development tool in particular in the case of school meals programmes. In Volume 1, researchers, policymakers and development partners can find evidence on how PFP can be used as a development tool and deliver multiple benefits for multiple beneficiaries. It argues that PFP can provide a market for local and smallholder farmers, promote the conservation and sustainable use of agrobiodiversity, and improve the nutrition and health of children and communities. Volume 2 of this publication, available at https://doi.org/10.4060/cb7969en, presents further analysis of the instruments, enablers and barriers for PFP implementation. It also provides case studies with local, regional and national experiences from Africa, Asia, Europe and North and South America