Powers of Hamilton cycles of high discrepancy are unavoidable

The P\'osa-Seymour conjecture asserts that every graph on $n$ vertices with minimum degree at least $(1 - 1/(r+1))n$ contains the $r^{th}$ power of a Hamilton cycle. Koml\'os, S\'ark\"ozy and Szemer\'edi famously proved the conjecture for large $n.$ The notion of discrepancy appears in many areas of mathematics, including graph theory. In this setting, a graph $G$ is given along with a $2$-coloring of its edges. One is then asked to find in $G$ a copy of a given subgraph with a large discrepancy, i.e., with many more edges in one of the colors. For $r \geq 2,$ we determine the minimum degree threshold needed to find the $r^{th}$ power of a Hamilton cycle of large discrepancy, answering a question posed by Balogh, Csaba, Pluh\'ar and Treglown. Notably, for $r \geq 3,$ this threshold approximately matches the minimum degree requirement of the P\'osa-Seymour conjecture

Globalna klasyfikacja krajów w kategorii zrównoważonych start-upów

In today's world, the pursuit of sustainable development has become a global priority. However, the integration of sustainability practices in entrepreneurial ventures, particularly in new startups, lags behind. This article aims to bridge the gap between sustainable development and entrepreneurship by examining the role of sustainability in the formation of sustainable startups. Through the use of cluster analysis, this study classifies countries based on their environmental and social objectives in the context of sustainable startups. The analysis reveals distinct clusters of countries with varying degrees of emphasis on environmental and social considerations. The results of the analysis provide valuable insights into the distribution of countries across different clusters, delineating the sustainable goals of startups in those countries. These findings align with existing research on sustainable entrepreneurship and contribute to the broader discourse in the field. The implications of these findings are significant for promoting sustainable entrepreneurship. While the study acknowledges several limitations, future research can expand upon these findings to develop a more comprehensive understanding of sustainable entrepreneurship on a global scale.W dzisiejszym świecie dążenie do zrównoważonego rozwoju stało się globalnym priorytetem. Jednak integracja  zrównoważonych praktyk w przedsiębiorczości, szczególnie w nowych start-upach, pozostaje w tyle. Artykuł  ma na celu wypełnienie luki pomiędzy zrównoważonym rozwojem a przedsiębiorczością poprzez zbadanie roli zrównoważonego rozwoju w tworzeniu zrównoważonych start-upów. Dzięki zastosowaniu analizy skupień w niniejszym badaniu dokonano klasyfikacji krajów na podstawie ich celów środowiskowych i społecznych w kontekście zrównoważonych start-upów. Analiza ukazuje wyraźne skupiska krajów o różnym stopniu spojrzenia na kwestie środowiskowe i społeczne. Wyniki analizy dostarczają cennych informacji na temat rozmieszczenia krajów w różnych klastrach, wyznaczają zrównoważone cele startupów w tych krajach. Ustalenia te pokrywają się z istniejącymi badaniami nad zrównoważoną przedsiębiorczością i przyczyniają się do szerszego dyskursu w tej dziedzinie. Konsekwencje tych ustaleń są istotne dla promowania zrównoważonej przedsiębiorczości. Chociaż w artykule wskazano kilka ograniczeń, przyszłe badania mogą rozszerzyć te ustalenia, aby uzyskać pełniejsze zrozumienie zrównoważonej przedsiębiorczości w skali globalnej

Tam multa in tectis crepitans salit horrida grando (Virgil, Georgics, 1.449)

Najveći neprijatelji ratara i vinogradara u antici bili su uvijek isti: suša, tuča, oluje i skakavci. Jedina obrana bili su im magijske riječi i postupci. Antički izvori prenose cijeli niz različitih magijskih formula za obranu od tuče, sačuvanih u književnim izvorima i na filakterijima, kao i cijeli niz postupaka koje je trebalo obaviti da bi se vinograd ili polje obranili od nepogode. Svi se ti postupci mogu okarakterizirati kao magijski, zato jer nisu imali kauzalne povezanosti s nastankom i odvraćanjem tuče. S druge strane, prirodni filozofi ostavili su rasprave o tome što tuča jest i gdje nastaje i uglavnom se slažu s tim da tuča ima veze s mjesecom i da vjerojatno ondje nastaje.So thick the horrid hail bounds rattling on the roofs. It is summer, it is hot, the hail has ravaged the vineyards. Virgil’s line in the title of this paper not only vividly conveys the terrible feeling that the rattling of hailstones on the roof tiles excites among people – he calls it horrid grando – but is also brilliant Latin onomatopoeia. As long as there have been farmers and viticulturists, their enemies have always been the same; drought, hail, storm and locusts. Since antiquity these four woes have constantly been repeated in agricultural texts, in prayers, in magical formulae, all having a single aim: save my vineyard (olive grove, field, holding), whatever god, daimon or angel there might be. The peasant or landowner could fight against hail in two ways. The first was with words. We can find words against hail written in the form of magical formulae and prayers on various objects; there are inscriptions on stone, on terracotta plaques, little sheets of metal and on papyri. Secondly, with actions. These are various magical procedures for protection against hail. Help was sought from the gods, and from gifted individuals, like Pythagoras, Empedocles, Epimenides of Crete or Abaris the Hyperborean, of whom it was said that they could ward off bad weather. Scientific minds like Seneca or Pliny the Elder could not come to terms with this kind of magic and had no faith in the effectiveness of such formulae. Seneca mocked the customs in Cleona where there were special anti-hail guardians (khalazophylakes) who watched when hail-bearing clouds approached and when the peasants though that they would gain favour with the clouds with the blood of victims. The setting of philosopher and intellectual, of educated and well-read people, is one world, and that of peasant and grape-grower another very different one. People in the fields and vineyards were ready to do everything, pay anyone who could halt the catastrophe. Witnesses to such rural magic are the many phylacteries (white magic spells for protection against all evils) which frequently referred precisely to hail. A whole series of phylacteries – inscribed either in lead or in stone – have been preserved throughout the Roman Empire, and tell of this magic of the viticulturists. They invoked various celestial beings, archangels and angels (there are most of them in Late Antiquity, and most of them are Christians) to protect vineyards and farms from the activities of evil beings that bring the hail, such as for example the demon Tartaruc(h)us on a phylactery from Trogir (tabella plumbea Traguriensis). And in literary sources we have recommendations for procedures that should protect the vineyards, most of them preserved in the Byzantine Geoponica , compiled in the 10th century, in an Arabic text known as Nabataean Agriculture, and in earlier agricultural writers like Columella and Palladius. In these procedures, the sovereign substances are the skins of seals, hyenas, hippopotami, hedgehogs, live turtles, live venomous snakes, and objects such as pictures of snakes, of bunches of grapes, wooden figures of bulls or wooden crosses placed in the vineyards, iron mirrors and iron keys. In pagantimes the blood of sacrificed animals also worked, which was in Christian times replaced with symbols of blood like red rags; menstrual blood was also held to be sovereign. Hail could also be stopped by a lyre being played. There were various theories about what hail was and where it came from; the most frequent suggestions were that it was linked with the moon; in the satirist Lucian we find that on the Moon the vineyards bore bunches of ice-berries, and when the wind shook these vines, hail fell here below. Baron Münchhausen, drawing on Lucian, went to the Moon and repeated the story of the lunar ice vineyards, recommending that it would be useful, the next time the hail fell, to make lunar wine of this ice

Exploring Dark Matter Properties from the Smallest to the Largest Scales

One of the main problems in cosmology is to understand the formation and evolution of galaxies, galaxy clusters, and large-scale structure. Whereas the basics of the current Cold Dark Matter (CDM) paradigm for structure formation are widely accepted, some controversial issues for CDM (e.g. cuspiness of dark-matter halos, substructure crisis) still remain. In order to test CDM predictions, one needs to investigate bound objects from the smallest (dwarf galaxies) to the largest (galaxy clusters) scales at different redshifts. In particular, their matter-content, luminous and dark, has to be studied in detail. We address these issues by using gravitational lensing. Specifically, we explore the properties of mass-substructure and the mass-profiles of galaxies, as well as the mass-profiles of galaxy clusters, using strong and weak gravitational lensing. In the beginning we give a short preface to the subject, intended for a non-expert reader. Following the general introduction to gravitational lensing we address the lens properties of quadruply imaged systems, lensed by numerically simulated galaxies. We focus on the substructure in the lens galaxies. We compare signatures of substructure on gravitational lensing properties and compare these with the observed lensed systems. The second part of this work deals with clusters of galaxies. We describe a method that improves the mass estimates of clusters considerably and develop a new cluster mass reconstruction technique which combines strong and weak lensing. We test our method using simulations and find that it can very successfully reconstruct both the enclosed mass and the mass distribution of a cluster. Finally we apply this technique to an observed cluster RX J1347-1145, the most X-ray luminous cluster

Institutional determinants and entrepreneurial action

The paper examines the effect of specific institutional factors on entrepreneurial activity. In the course of the examination, we encounter various viewpoints regarding entrepreneurship and different needs of national politics. The research objective is to determine whether institutional factors influence early-stage entrepreneurial activity. There is a broad array of opinions on appropriate set of factors that influence the entrepreneurship processes, on ways of their influence and on differentiating between the developed and less developed countries. Therefore, we examined the defined research hypothesis in the light of 24 countries (Argentina, Belgium, Brazil, Chile, Columbia, Croatia, Denmark, Finland, France, Greece, Hungary, Island, Italy, Japan, Latvia, Netherlands, Norway, Peru, Russia, Slovenia, Spain, United Kingdom, Uruguay and USA), in the period between the years 2006 and 2010 (24 countries * 5 years = 120 observations). The data was obtained from Global Entrepreneurship Monitor (GEM) database and complemented with data from other international sources such as Heritage Foundation, among others. With econometrics business methods, we determined that greater economic freedom in the institutional context of a country affects the extension of productive entrepreneurship, while the individual\u27s decision for the entrepreneurship is conditioned significantly by the prevailing cultural and social norms

Focusing Cosmic Telescopes: Exploring Redshift z~5-6 Galaxies with the Bullet Cluster 1E0657-56

The gravitational potential of clusters of galaxies acts as a cosmic telescope allowing us to find and study galaxies at fainter limits than otherwise possible and thus probe closer to the epoch of formation of the first galaxies. We use the Bullet Cluster 1E0657-56 (z = 0.296) as a case study, because its high mass and merging configuration makes it one of the most efficient cosmic telescopes we know. We develop a new algorithm to reconstruct the gravitational potential of the Bullet Cluster, based on a non-uniform adaptive grid, combining strong and weak gravitational lensing data derived from deep HST/ACS F606W-F775W-F850LP and ground-based imaging. We exploit this improved mass map to study z~5-6 Lyman Break Galaxies (LBGs), which we detect as dropouts. One of the LBGs is multiply imaged, providing a geometric confirmation of its high redshift, and is used to further improve our mass model. We quantify the uncertainties in the magnification map reconstruction in the intrinsic source luminosity, and in the volume surveyed, and show that they are negligible compared to sample variance when determining the luminosity function of high-redshift galaxies. With shallower and comparable magnitude limits to HUDF and GOODS, the Bullet cluster observations, after correcting for magnification, probe deeper into the luminosity function of the high redshift galaxies than GOODS and only slightly shallower than HUDF. We conclude that accurately focused cosmic telescopes are the most efficient way to sample the bright end of the luminosity function of high redshift galaxies and - in case they are multiply imaged - confirm their redshifts.Comment: 12 pages, Accepted for publication in Ap

Weighted Tur\'an theorems with applications to Ramsey-Tur\'an type of problems

We study extensions of Tur\'an Theorem in edge-weighted settings. A particular case of interest is when constraints on the weight of an edge come from the order of the largest clique containing it. These problems are motivated by Ramsey-Tur\'an type problems. Some of our proofs are based on the method of graph Lagrangians, while the other proofs use flag algebras. Using these results, we prove several new upper bounds on the Ramsey-Tur\'an density of cliques. Other applications of our results are in a recent paper of Balogh, Chen, McCourt and Murley.Comment: 19 pages, 10 figure