Percolation on nonunimodular transitive graphs

We extend some of the fundamental results about percolation on unimodular nonamenable graphs to nonunimodular graphs. We show that they cannot have infinitely many infinite clusters at critical Bernoulli percolation. In the case of heavy clusters, this result has already been established, but it also follows from one of our results. We give a general necessary condition for nonunimodular graphs to have a phase with infinitely many heavy clusters. We present an invariant spanning tree with $p_c=1$ on some nonunimodular graph. Such trees cannot exist for nonamenable unimodular graphs. We show a new way of constructing nonunimodular graphs that have properties more peculiar than the ones previously known.Comment: Published at http://dx.doi.org/10.1214/009117906000000494 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Socioeconomic Background of the Divergence of Belarusian and Ukrainian Political Systems

The aim of this paper is to analyze the political, social and economic background of the divergence of Belarusian and Ukrainian transitions. We focus on Belarus in order to find explanation for questions such as why could Lukashenko remain the authoritarian leader of Belarus, while in Ukraine the position of the political elite had proved less stable and collapsed in 2004. On the theoretical framework of elite-sociology, we seek to determine whether the internal factors (as macroeconomic conditions, standard of living, the oppressive nature of the political system and the structure of the political elite) play a significant role in the operation of the domino effect. This article emphasises the determining role of immanent internal factors, thus the political stability in Belarus can be explained by the role of the suppressing political regime, the hindrance of democratic rights and the relatively good living conditions that followed the transformational recession. Whilst in Ukraine, the markedly different circumstances brought forth the success of the Orange Revolution

Enumeration of diagonally colored Young diagrams

In this note we give a new proof of a closed formula for the multivariable generating series of diagonally colored Young diagrams. This series also describes the Euler characteristics of certain Nakajima quiver varieties. Our proof is a direct combinatorial argument, based on Andrews' work on generalized Frobenius partitions. We also obtain representations of these series in some particular cases as infinite products.Comment: Final version, 12 pages. To appear in Monatshefte f\"ur Mathemati

The analysis and evaluation of the relation between road transportation and climate change

The target of this article is to analyse and evaluate the relation between road transportation and climate change, through the long time series of average CO2 concentration in the atmosphere and global average temperature of Earth. This article is built on data from the age % of which has information before the human impact on Earth. It can be clearly seen from the research that the human impact on air quality has different tendency than it had before. The trend of time-series was nearly independent from time: it was constant. It can be identified from the data that the increase of temperature was usually faster than the decrease in decreasing periods. There is a strong correlation between the average CO2 concentration in air and the average temperature of the Earth. The CO2 emitted into the environment increases the global temperature of the Earth. A huge part of the CO2 emitted by mankind into the atmosphere comes from transportation, mainly from the sector of road transportation

Ends in free minimal spanning forests

We show that for a transitive unimodular graph, the number of ends is the same for every tree of the free minimal spanning forest. This answers a question of Lyons, Peres and Schramm.Comment: Published at http://dx.doi.org/10.1214/009117906000000025 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Byzantine Missions among the Magyars in the Later 10th Century?

Byzantine missions among the Magyars during the later 10th century? For many 10th century Christian observers, as they frequently noted, the arrival of the conquering Hungarians at the end of the 9th century meant the beginning of the Apocalypse. Therefore it is hardly surprising, that in the eyes of Christian authors the newly arrived People of Gog and Magog appeared as the par excellence pagans of their age. This view is clearly attested by all extant historical writings of the time, whether Byzantine Greek, Western European Latin or Eastern European Slavic. On the other hand, archaeological excavations conducted over the last one and a half century in the Carpathian Basin, produced a number of cross finds, datable to 10th and 11th centuries that continue to provoke a lively debate among historians and archaeologists, most of whom have been speculating how these crosses are to be interpreted. Some leading experts of early Hungarian history were in favour of and others were against the presence and spread of Christianity in the Carpathian Basin before the time of the state-enforced conversion under Saint Stephen. The present paper aims to revisit the main arguments established by the debating parties and introduce new ones in order to better understand the background against which Saint Stephen’s efforts in Christianizing his kingdom are to be contextualized. My object is to question the usefulness of applying strict theological/canonical criteria when hints of an early evangelizing activity in the burials of the given period are searched for. On the other hand, by reviewing the known ecclesiastical regulations I argue that in the first century of official Christianization of the Árpádian Age, the Church left the question of burial up to the family of the deceased; a fact which, in my judgement, helps to explain why it is nearly impossible to find a criterion or a set of criteria for determining the burial of a Christian or a partly Christianized individual before the use of churchyard cemeteries