Results on singular products of the distributions $x_{\pm }^{-p}$ and $x^{-p}$ for natural $p$ are derived, when the products are balanced so that their sum exists in the distribution space. These results follow the pattern of a known distributional product published by Jan Mikusiński in 1966. The results are obtained in the Colombeau algebra of generalized functions, which is the most relevant algebraic construction for tackling nonlinear problems of Schwartz distributions.
Let $P_k$ denote a path with $k$ edges and $łK_{n,n}$ denote the $ł$-fold complete bipartite graph with both parts of size $n$. In this paper, we obtain the necessary and sufficient conditions for $łK_{n,n}$ to have a balanced $P_k$-decomposition. We also obtain the directed version of this result.
The present-day sea-level rise is a major indicator of climate change. The sea level in European seas has risen at a rate of 2.5 to 4 millimeters per year (von Schuckmann et al., 2018). The aim of this paper is to present the sea level variability in the Baltic Sea, as based on satellite altimetry. For this purpose, the paper presents a methodology of the investigation of the Baltic Sea level changes based on the Optimum Dataset (OptD) method. The OptD method was used to identify characteristic points from the analyzed data set. For detailed theoretical and empirical tests, the sea level anomaly was used. The time series were created from the data set after introducing the OptD method, in the period from January 1993 to December 2017. The time series are then used to characterize sea level trends, and inter-annual and semi-annual variability in the Baltic Sea region. The results prove that the linear change is higher at points which are located in the northern part of the Baltic Sea, while it is lower at points located in the western part of the Baltic Sea. The average trend is 4.1±0.2 mm/yr. However, the annual cycles in the sea level variations measured by altimetry reach maximum values in approximately the same months (November/December) in the whole Baltic Sea area. We find that there occur substantial regional deviations in sea level depending on the latitude and longitude. Our results confirm the need for research into the sea level variability in the Baltic Sea region.