On equal values of power sums of arithmetic progressions

In this paper we consider the Diophantine equation \begin{align*}b^k +\left(a+b\right)^k &+ \cdots + \left(a\left(x-1\right) + b\right)^k=\\ &=d^l + \left(c+d\right)^l + \cdots + \left(c\left(y-1\right) + d\right)^l, \end{align*} where $a,b,c,d,k,l$ are given integers. We prove that, under some reasonable assumptions, this equation has only finitely many integer solutions.Comment: This version differs slightly from the published version in its expositio

The existence of an inverse limit of inverse system of measure spaces - a purely measurable case

The existence of an inverse limit of an inverse system of (probability) measure spaces has been investigated since the very beginning of the birth of the modern probability theory. Results from Kolmogorov [10], Bochner [2], Choksi [5], Metivier [14], Bourbaki [3] among others have paved the way of the deep understanding of the problem under consideration. All the above results, however, call for some topological concepts, or at least ones which are closely related topological ones. In this paper we investigate purely measurable inverse systems of (probability) measure spaces, and give a sucient condition for the existence of a unique inverse limit. An example for the considered purely measurable inverse systems of (probability) measure spaces is also given

Methane activation and exchange by titanium-carbon multiple bonds

We demonstrate that a titanium-carbon multiple bond, specifically an alkylidyne ligand in the transient complex, (PNP)Ti≡C^(t)Bu (A) (PNP^− = N[2-P(CHMe_2)_(2)-4-methylphenyl]_2), can cleanly activate methane at room temperature with moderately elevated pressures to form (PNP)Ti=CHtBu(CH_3). Isotopic labeling and theoretical studies suggest that the alkylidene and methyl hydrogens exchange, either via tautomerization invoking a methylidene complex, (PNP)Ti=CH_(2)(CH_(2)^(t)Bu), or by forming the methane adduct (PNP)Ti≡C^(t)Bu(CH_4). The thermal, fluxional and chemical behavior of (PNP)Ti=CH^(t)Bu(CH_3) is also presented in this study

Population and age structure in Hungary: a residential preference and age dependency approach to disaggregate census data

We present a simple model to disaggregate age structured population census data to a 1-km grid for Hungary. A dasymetric approach was used to predict the spatial distribution of population in different age groups by distinguishing residential preferences (in relation to accessible social, economic and green amenities) for working age groups (15–29, 30–49 and 50–64) and population dependencies for children and the elderly (aged 0–14 and 65+). By using open-access land cover data and fine-level population census data as inputs, the model predicts the likely spatial distribution of population and age structure for Hungary in 2011. The resulting map and gridded data provide information to support spatial planning of residential development and urban infrastructure. The model is less data-demanding than most existing approaches, but provides greater power for describing population patterns. It can also be used to create scenarios of future demographic change

Modelling regional cropping patterns under scenarios of climate and socio-economic change in Hungary

Impacts of socio-economic, political and climatic change on agricultural land systems are inherently uncertain. The role of regional and local-level actors is critical in developing effective policy responses that accommodate such uncertainty in a flexible and informed way across governance levels. This study identified potential regional challenges in arable land use systems, which may arise from climate and socio-economic change for two counties in western Hungary: Veszprém and Tolna. An empirically-grounded, agent-based model was developed from an extensive farmer household survey about local land use practices. The model was used to project future patterns of arable land use under four localised, stakeholder-driven scenarios of plausible future socio-economic and climate change. The results show strong differences in farmers' behaviour and current agricultural land use patterns between the two regions, highlighting the need to implement focused policy at the regional level. For instance, policy that encourages local food security may need to support improvements in the capacity of farmers to adapt to physical constraints in Veszprém and farmer access to social capital and environmental awareness in Tolna. It is further suggested that the two regions will experience different challenges to adaptation under possible future conditions (up to 2100). For example, Veszprém was projected to have increased fallow land under a scenario with high inequality, ineffective institutions and higher-end climate change, implying risks of land abandonment. By contrast, Tolna was projected to have a considerable decline in major cereals under a scenario assuming a de-globalising future with moderate climate change, inferring challenges to local food self-sufficiency. The study provides insight into how socio-economic and physical factors influence the selection of crop rotation plans by farmers in western Hungary and how farmer behaviour may affect future risks to agricultural land systems under environmental change