Dynamics of tax revenues in the budget of Ukraine and their forecast during the crisis period

It is extremely important for the budget process to obtain accurate forecasts of potential tax revenues, especially in periods of disruption and crisis. The paper is devoted to the study of dynamics of tax revenues’ volumes in the budget of Ukraine and the forecast of their values during the crisis.The dynamics of tax revenues in the Consolidated Budget of Ukraine, studied by using randomized R|S-analysis, fractal and probabilistic analyses as well as entropy calculation based on the data on monthly tax revenues for the period 2011–2021, is anti-persistent, fractal-like and unpredictable based on parametric dependencies, simple and complex trends. The topological dimension of the lines of dynamics for tax revenues of all types of taxes is much higher than 1, and the Hirst index indicates either fractal similarity of dynamics or its chaos. The map of dissipation periods of tax revenues in Ukraine, determined on the basis of entropy calculation and periods of negative entropy production according to the dynamics of tax revenues, coincided with the periods of maximum reduction in their volumes. The most crisis periods in the formation of tax revenues are 2019–2020, for certain types of taxes – 2016–2020, but the dissipation of tax revenues is projected for 2021–2022.The comparison of the level of fractal similarity in dynamics of the volume of tax revenues and peculiarities of the dynamics of entropy and entropy production, allowed to substantiate the division of taxes into nine types, of which five were found in Ukraine.

CONDITION (K) FOR BOOLEAN DYNAMICAL SYSTEMS

We generalize Condition (K) from directed graphs to Boolean dynamical systems and show that a locally finite Boolean dynamical system $({{\mathcal {B}}},{{\mathcal {L}}},\theta )$ with countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ satisfies Condition (K) if and only if every ideal of its $C^*$ -algebra is gauge-invariant, if and only if its $C^*$ -algebra has the (weak) ideal property, and if and only if its $C^*$ -algebra has topological dimension zero. As a corollary we prove that if the $C^*$ -algebra of a locally finite Boolean dynamical system with ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ countable either has real rank zero or is purely infinite, then $({{\mathcal {B}}}, {{\mathcal {L}}}, \theta )$ satisfies Condition (K). We also generalize the notion of maximal tails from directed graph to Boolean dynamical systems and use this to give a complete description of the primitive ideal space of the $C^*$ -algebra of a locally finite Boolean dynamical system that satisfies Condition (K) and has countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ .

𝐶*-algebras and their nuclear dimension

We review the notion of nuclear dimension for C ∗ \mathrm {C}^* -algebras introduced by Winter and Zacharias. We explain why it is a non-commutative version of topological dimension. After presenting several examples, we give a brief overview of the state of the art.

Political topology of Europe

The objective scope of the analysis performed in the text encompasses selected aspects of policy in its topological dimension. The space of policy is understood as both a theoretical construct (a policy field) and relations between the characteristics of political actors and their special kind of geographical co-existence. The following have been recognised as essential characteristics of policymaking: (1) electoral process and pluralism, (2) functioning of government, (3) political participation, (4) political culture and (5) civil liberties. These features can become an object of analysis in the assessment of democratic and authoritarian tendencies in selected countries. The text uses two statistical methods of multidimensional comparative analysis (Ward’s method and k-means method), apart from which use has been made of basic descriptive statistics and a comparative analysis of the values of the parameters of political characteristics. A selection of 40 European countries (EU-28 and 12 other countries) have been subjected to a statistical analysis according to the 2018 data. The main goal of the analysis is to connect facts and characteristics attributed to policy with a specific geographical area. In order to elaborate the objective scope of the research problem, the following research questions have been presented in the text: (1) Which of the characteristics of policy will determine the division of state entities according to a special type of clusters?, (2) Will political characteristics determine the division of particular state entities according to a special type of geographical division? The addressed research questions have been related to the hypotheses subjected to verification in the text.

On groups whose actions on finite-dimensional CAT(0) spaces have global fixed points

We give a criterion for group elements to have fixed points with respect to a semi-simple action on a complete CAT(0) space of finite topological dimension. As an application, we show that Thompson’s group T and various generalizations of Thompson’s group V have global fixed points when they act semi-simply on finite-dimensional complete CAT(0) spaces, while it is known that T and V act properly on infinite-dimensional CAT(0) cube complexes.

Marked length rigidity for one-dimensional spaces

In a compact geodesic metric space of topological dimension one, the minimal length of a loop in a free homotopy class is well-defined, and provides a function [Formula: see text] (the value [Formula: see text] being assigned to loops which are not freely homotopic to any rectifiable loops). This function is the marked length spectrum. We introduce a subset [Formula: see text], which is the union of all non-constant minimal loops of finite length. We show that if [Formula: see text] is a compact, non-contractible, geodesic space of topological dimension one, then [Formula: see text] deformation retracts to [Formula: see text]. Moreover, [Formula: see text] can be characterized as the minimal subset of [Formula: see text] to which [Formula: see text] deformation retracts. Let [Formula: see text] be a pair of compact, non-contractible, geodesic metric spaces of topological dimension one, and set [Formula: see text]. We prove that any isomorphism [Formula: see text] satisfying [Formula: see text], forces the existence of an isometry [Formula: see text] which induces the map [Formula: see text] on the level of fundamental groups. Thus, for compact, non-contractible, geodesic spaces of topological dimension one, the marked length spectrum completely determines the subset [Formula: see text] up to isometry.