On Factorization of Functional Operators with Reflection on the Real Axis

Problems of factorization of matrix functions are closely connected with the solution of matrix Riemann boundary value problems and with the solution of vector singular integral equations. In this article, we study functional operators with orientation-reversing shift reflection on the real axes. We introduce the concept of multiplicative representation of functional operators with shift and its partial indices. Based on the classical notion of matrix factorization, the correctness of the definitions is shown. A theorem on the relationship between factorization of functional operators with reflection and factorization of the corresponding matrix functions is proven.

Relationship between the Geotourism Potential and Function in the Polish Part of the Roztocze Transboundary Biosphere Reserve

The Polish part of the Roztocze Transboundary Biosphere Reserve area is characterized by diversified geotourism resources with relatively high value. However, their potential seems not to be fully used in the current product offer. The aim of the study was therefore to assess the spatial variability of the geotourism potential and function and to determine their interrelations in view of further development of geotourism in the Roztocze TBR and the perspective of creation of the “Kamienny Las na Roztoczu” geopark. The study was carried out with the use of the taxonomic method of multidimensional comparative analysis consisting of calculation and analysis of general, total, and partial indices of the geotourism potential and function in 22 communes. The results showed the highest total indicator of geotourism potential in two communes, i.e. Józefów and Krasnobród, and the highest value of the total geotourism function index in Krasnobród. The results of the analysis of the relationships between the geotourism potential and function indicate that the geotourism resources and products are fully used in terms of the development of the function only in Krasnobród commune. In turn, the value of the total geotourism function index in the Zwierzyniec commune exceeds the geotourism potential indicator, which implies that this area is overloaded by tourist movement. The total indicators of geotourism potential in the other communes, especially Józefów, Krasnobród, Lubycza Królewska, and Susiec, indicate the possibility of more intensive exploitation of geotourism resources in preparation of interesting products in compliance with the principles of sustainable development and, consequently, the development of the geotourism function.

On explicit Wiener–Hopf factorization of 2 × 2 matrices in a vicinity of a given matrix

As it is known, the existence of the Wiener–Hopf factorization for a given matrix is a well-studied problem. Severe difficulties arise, however, when one needs to compute the factors approximately and obtain the partial indices. This problem is very important in various engineering applications and, therefore, remains to be subject of intensive investigations. In the present paper, we approximate a given matrix function and then explicitly factorize the approximation regardless of whether it has stable partial indices. For this reason, a technique developed in the Janashia–Lagvilava matrix spectral factorization method is applied. Numerical simulations illustrate our ideas in simple situations that demonstrate the potential of the method.

On effective criterion of stability of partial indices for matrix polynomials

In the work, we obtain an effective criterion of the stability of the partial indices for matrix polynomials under an arbitrary sufficiently small perturbation. Verification of the stability is reduced to calculation of the ranks for two explicitly defined Toeplitz matrices. Furthermore, we define a notion of the stability of the partial indices in the given class of matrix functions. This means that we will consider an allowable small perturbation such that a perturbed matrix function belong to the same class as the original one. We prove that in the class of matrix polynomials the Gohberg–Krein–Bojarsky criterion is preserved, i.e. new stability cases do not arise. Our proof of the stability criterion in this class does not use the Gohberg–Krein–Bojarsky theorem.

Exact conditions for preservation of the partial indices of a perturbed triangular 2 × 2 matrix function

The possible instability of partial indices is one of the important constraints in the creation of approximate methods for the factorization of matrix functions. This paper is devoted to a study of a specific class of triangular matrix functions given on the unit circle with a stable and unstable set of partial indices. Exact conditions are derived that guarantee a preservation of the unstable set of partial indices during a perturbation of a matrix within the class. Thus, even in this probably simplest of cases, when the factorization technique is well developed, the structure of the parametric space (guiding the types of matrix perturbations) is non-trivial.

Towards Sustainable Development and Preventing Exclusions—Determining Road Accessibility at the Sub-Regional and Local Level in Rural Areas of Poland

The improvement in a regions’ accessibility that accounts for various means of inter-regional transport and inter-regional communication is one of the main determinants of sustainable regional development. This study focuses on road accessibility in rural areas where an insufficient number and scope of international and domestic investments can lead to an imbalance in the transport infrastructure. Therefore, the aim of this study was to evaluate road accessibility at the level of the NUTS4 units in view of their sustainable development and their divergence from rural units that are less distant from the main transport routes. The studied area—a province—is situated in the northeast of Poland. Data concerning the travel time, condition, and density of the road network were analysed to develop and compare the measures of road accessibility in individual units and to perform an overall assessment of the NUTS4 units. Partial indices were calculated with the support of the QGIS and ArcGIS software. Peripheral regions in post-socialist countries appear to be particularly underinvested in terms of the transport, including at the local and regional level. The applied methodology supported the presentation of problem peripheral areas that are underinvested and threatened with exclusion.

REGIONAL CONCENTRATION OF CHARCOAL PRODUCTION IN THE STATE OF PARAÍBA, BRAZIL (1994 - 2016)

ABSTRACT Charcoal has an economic, social and environmental importance, because in addition to being a source of energy, it generates employment and income in the rural environment. Therefore, knowing your market is fundamental for the decision-making of those segments that depend on this raw material. This work analyzed the regional concentration of charcoal production in the state of Paraíba, Brazil, from 1994 to 2016. The data used to measure the regional production concentration (in tons) of native Paraíba charcoal were obtained from the Brazilian Institute of Geography and Statistics (IBGE) from 1994 to 2016. The indicators used were the Concentration Ratio [CR(k)], the Herfindahl-Hirschman Index (HHI), Theil's entropy index (E) and the Gini Index (G). The main results show that the mesorregions of Borborema and Sertão Paraibano present concentrations in charcoal production from Paraíba. The CR(k) of the municipalities had a low to moderately low concentration and a moderately high to a high concentration for the microregions; the HHI and E showed deconcentration tendencies from competitive markets; the G showed strong to very strong inequality for the municipalities and microregions on average. It is concluded that the concentration of charcoal production at regional levels is not concentrated, even though it presents a moderate concentration in the partial indices [CR(k)] for the Paraíba microregions.

Use of IT platform in determination of efficiency of mining machines

Determination of effective use of mining devices has very significant meaning for mining enterprises. High costs of their purchase and tenancy cause that these enterprises tend to the best use of possessed technical potential. However, specifics of mining production causes that this process not always proceeds without interferences. Practical experiences show that determination of objective measure of utilization of machine in mining enterprise is not simple. In the paper a proposition for solution of this problem is presented. For this purpose an IT platform and overall efficiency model OEE were used. This model enables to evaluate the machine in a range of its availability performance and quality of product, and constitutes a quantitative tool of TPM strategy. Adapted to the specificity of mining branch the OEE model together with acquired data from industrial automatic system enabled to determine the partial indicators and overall efficiency of tested machines. Studies were performed for a set of machines directly use in coal exploitation process. They were: longwall-shearer and armoured face conveyor, and beam stage loader. Obtained results clearly indicate that degree of use of machines by mining enterprises are unsatisfactory. Use of IT platforms will significantly facilitate the process of registration, archiving and analytical processing of the acquired data. In the paper there is presented methodology of determination of partial indices and total OEE together with a practical example of its application for investigated machines set. Also IT platform was characterized for its construction, function and application.

Regular approximate factorization of a class of matrix-function with an unstable set of partial indices

From the classic work of Gohberg & Krein (1958 Uspekhi Mat. Nauk. XIII , 3–72. (Russian).), it is well known that the set of partial indices of a non-singular matrix function may change depending on the properties of the original matrix. More precisely, it was shown that if the difference between the largest and the smallest partial indices is larger than unity then, in any neighbourhood of the original matrix function, there exists another matrix function possessing a different set of partial indices. As a result, the factorization of matrix functions, being an extremely difficult process itself even in the case of the canonical factorization, remains unresolvable or even questionable in the case of a non-stable set of partial indices. Such a situation, in turn, has became an unavoidable obstacle to the application of the factorization technique. This paper sets out to answer a less ambitious question than that of effective factorizing matrix functions with non-stable sets of partial indices, and instead focuses on determining the conditions which, when having known factorization of the limiting matrix function, allow to construct another family of matrix functions with the same origin that preserves the non-stable partial indices and is close to the original set of the matrix functions.