Integral operators in the spaces of functions of bounded Schramm variation

In this paper we provide a full characterization of linear integral operators acting from the space of functions of bounded Jordan variation to the space of functions of bounded Schramm variation in terms of their generating kernels.Comment: This paper grew out of the unpublished part of the paper by the same authors, Compactness in normed spaces: a unified approach through semi-norms, arXiv:2111.10547v

Compactness in normed spaces: a unified approach through semi-norms

In this paper we prove two new abstract compactness criteria in normed spaces. To this end we first introduce the notion of an equinormed set using a suitable family of semi-norms on the given normed space satisfying some natural conditions. Those conditions, roughly speaking, state that the norm can be approximated (on the equinormed sets even uniformly) by the elements of this family. As we are given some freedom of choice of the underlying semi-normed structure that is used to define equinormed sets, our approach opens a new perspective for building compactness criteria in specific normed spaces. As an example we show that natural selections of families of semi-norms in spaces $C(X,\mathbb{R})$ and $l^p$ for $p\in[1,+\infty)$ lead to the well-known compactness criteria (including the Arzel\`a-Ascoli theorem). In the second part of the paper, applying the abstract theorems, we construct a simple compactness criterion in the space of functions of bounded Schramm variation and provide a full characterization of linear integral operators acting from the space of functions of bounded Jordan variation to the space of functions of bounded Schramm variation in therms of their generating kernels.Comment: Th version accepted for publication in Topological Methods in Nonlinear Analysi

Energy security of Polish consumers in 2004-2021

Theoretical background: Energy security is one of the most important components of economic security. It is influenced not only by access to energy sources and the economic situation of the state and the individual consumer. The authors adopted the definition of energy security of an individual consumer as included in the UN report. It states that the energy security of an individual consumer is guaranteed when the average consumer has access to energy at all times, in various forms, in sufficient quantity and at a reasonable price, with the simultaneous development of support mechanisms for socially and economically sensitive consumers, and after implementation of independent mechanisms for resolving disputes with the energy company, which eliminate or at least alleviate the economic advantage of the enterprises.Purpose of the article: The main aim of the paper is to present the impact of macroeconomic indicators of the Polish economy on the energy security of Polish consumers with a minimum and average monthly income. Detailed objectives include determining the possibility of purchasing by consumers with such income, diesel oil, Pb95 gasoline, LPG, electricity, heat and present changes in the energy security of individual consumers in the analyzed period (i.e. 2004–2021).Methods: The study covered Poland, country which joined the European Union in 2004. The period of the analysis described the years from 2004 to 2021. The source of the collected information was a review of the literature on the subject and statistical data. Descriptive, tabular and graphical methods, constant dynamic indicators were used for the analysis and presentation of the results.Main findings: The last two decades of the 21st century have brought Poland a number of positive socio-economic changes. The Polish economy noted that inflation remained at the level of National Bank of Poland’s expectations, increased trade, a marked decline in unemployment and rising wages, and thus a general increase in the quality of life as well as energy security of the individual consumer. The authors assumed that in the years of 2004–2021 the consumer had good access to energy, heat and other fuels. The supplies of energy resources were ensured by long-term contracts, including the Yamal contract, and the demand for heating coal was satisfied with the raw material largely mined in Poland. Heat, mainly in large clusters of people, was generated by coal-fired power plants. This raw material was also used by power plants. For heating their houses, individual consumers – mainly in the countryside – used gas or oil.Theoretical background: Energy security is one of the most important components of economic security. It is influenced not only by access to energy sources and the economic situation of the state and the individual consumer. The authors adopted the definition of energy security of an individual consumer as included in the UN report. It states that the energy security of an individual consumer is guaranteed when the average consumer has access to energy at all times, in various forms, in sufficient quantity and at a reasonable price, with the simultaneous development of support mechanisms for socially and economically sensitive consumers, and after implementation independent mechanisms for resolving disputes with the energy company, which eliminate or at least alleviate the economic advantage of the enterprises.Purpose of the article: The main aim of the paper is to present the impact of macroeconomic indicators of the Polish economy on the energy security of Polish consumers with a minimum and average monthly income. Detailed objectives include determining the possibility of purchasing by consumers with such income, diesel oil, EU 95 gasoline, LPG, electricity, heat and present changes in the energy security of individual consumers in the analyzed period (i.e. 2004-2021).Methods: The study covered Poland, country which joined the European Union in 2004. The period of the analysis described the years from 2004 to 2021. The source of the collected information was a review of the literature on the subject and statistical data. Descriptive, tabular and graphical methods, constant dynamic indicators were used for the analysis and presentation of the results.Main Findings: The last two decades of the 21st century have brought Poland with a number of positive socio-economic changes. The Polish economy noted that inflation remained at the level of National Bank of Poland’s expectations, increased trade, a marked decline in unemployment and rising wages, and thus a general increase in the quality of life as well as energy security of the individual consumer (UN Report, 2000). The authors assumed that in the years of 2004-2021 the consumer had good access to energy, heat and other fuels. The supplies of energy resources were ensured by long-term contracts, including the Yamal contract, and the demand for heating coal was satisfied with the raw material largely mined in Poland (Musiałek, 2020), (Rogus, Mazanek, Maczuga,  2019).  Heat, mainly in large clusters of people, was generated by coal-fired power plants (Urban, Dzikuć, 2013). This raw material was also used by power plants (Olkulski, 2014). For heating their houses, individual consumers mainly in the countryside used gas or oil