The paper is devoted to the study of conditions for the Hausdorff-Besicovitch faithfulness of the family of cylinders generated by Cantor series expansions. We show that there exist subgeometric Cantor series expansions for which the corresponding families of cylinders are not faithful for the Hausdorff-Besicovitch dimension on the unit interval. On the other hand we found a rather wide subfamily of subgeometric Cantor series expansions generating faithful families of cylinders. We also study conditions for the Hausdorff-Besicovitch dimension preservation on [0;1] by probability distribution functions of random variables with independent symbols of arithmetic Cantor series expansions.

We study properties of Bernoulli convolutions generated by the second Ostrogradsky series, i.e., probability distributions of random variables ξ = ∑_k=1^∞(-1)^k+1ξ_k/q_k, where q_k is a sequence of positive integers with q_k+1≥ q_k(q_k+1), and {ξ_k} are independent random variables taking the values 0 and 1 with probabilities p_0k and p_1k respectively. We prove that ξ has an anomalously fractal Cantor type singular distribution (_H (S_ξ)=0) whose Fourier-Stieltjes transform does not tend to zero at infinity. We also develop different approaches how to estimate a level of "irregularity" of probability distributions whose spectra are of zero Hausdorff dimension. Using generalizations of the Hausdorff measures and dimensions, fine fractal properties of the probability measure μ_ξ are studied in details. Conditions for the Hausdorff–Billingsley dimension preservation on the spectrum by its probability distribution function are also obtained.

The paper is devoted to the study of fractal properties of distributions of random variables with independent symbols of factorial expansion, as well as to the applications of the obtained results to the metric and dimensional number theory. In particular, we prove that for almost all (in the sense of Lebesgue measure) real numbers from the unit interval the frequency nu i0 (x) of an arbitrary digit i0 is equal to zero: nu(i0) (x) = 0, for all i(0) is an element of {0,1, 2, ..., k, ...} =: N-0. On the other hand, we show that the set of essentially non-normal numbers (i.e., those real numbers which have no frequency of any symbol i0 is an element of N0) for the factorial expansion is a superfractal set (i.e., it is a set of zero Lebesgue measure and of maximal (1) Hausdorff-Besicovitch dimension).

The paper is devoted to the study of conditions for the preservation of mutual singularity resp. absolute continuity, and discreteness of probability measures under measurable mappings of probability spaces. Under very general assumptions we have found such conditions for the preservations. At the same time a series of important counterexamples are presented. The results obtained can simplified essentially the study the Lebesgue structure (i.e., finding necessary and sufficient conditions for the singular continuity, absolute continuity and discreteness of a wide spectra of probability measures with independent digits of symbolic expansions of real numbers and their multidimensional generalizations.

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Being popular world-wide, virtual laboratories enter into different fields of education and research and practitioners have to be responsible for choosing the most suitable and then adapt them to particular field. The aim of the present work was to assess the effectivity of the implementation of Praxilab, Labster, and LabXchange virtual laboratories as the powerful digital tool into teaching protocols of “Clinical and laboratory diagnostics” discipline for physical therapists and rehabilitologist. We have carried out the online survey for 45 students enrolled in physical rehabilitation degree program. About 70\% surveyed students reported that implementation of virtual laboratories in “Clinical and laboratory diagnostics” discipline met individual learning needs of students, helped acquired digital skills (25\%), and supported them to stay ahead of the curve. The virtual lab applications, not only assisted harness students fair against lack of practical skills, but also brought about a new dimension to the classes and helped overcome digital alienation and gain their digital skills and abilities. Indeed, a virtual lab can’t completely replace the experimental work and teacher’s explanation, but it might support teaching activities of a modern mentor and learning activities of a modern student. Almost all of surveyed students (82\%) expected that in near future the virtual laboratories would take the dominant place in the education market due to possibility of students’ pre-train the key points of practical activities before real experiments in lab and better understand their theoretical backgrounds. Thus, this study is intended to contribute to utilization of virtual labs by students enrolled in study physical therapy/physical rehabilitation with expected efficiency.

We study families $\Phi$ of coverings which are faithful for the Hausdorff dimension calculation on a given set $E$ (i. e., special relatively narrow families of coverings leading to the classical Hausdorff dimension of an arbitrary subset of $E$) and which are natural generalizations of comparable net-coverings. They are shown to be very useful for the determination or estimation of the Hausdorff dimension of sets and probability measures. We give general necessary and sufficient conditions for a covering family to be faithful and new techniques for proving faithfulness/non-faithfulness for the family of cylinders generated by expansions of real numbers. Motivated by applications in the multifractal analysis of infinite Bernoulli convolutions, we study in details the Cantor series expansion and prove necessary and sufficient conditions for the corresponding net-coverings to be faithful. To the best of our knowledge this is the first known sharp condition of the faithfulness for a class of covering families containing both faithful and non-faithful ones. Applying our results, we characterize fine fractal properties of probability measures with independent digits of the Cantor series expansion and show that a class of faithful net-coverings essentially wider that the class of comparable ones. We construct, in particular, rather simple examples of faithful families $\mathcal{A}$ of net-coverings which are extremely non-comparable to the Hausdorff measure.

Distance education has become the mandatory component of higher education establishments all over the world including Ukraine regarding COVID-19 lockdown and intentions of Universities to render valuable knowledge and provide safe educational experience for students. The present study aimed to explore the student’s and academic staff’s attitude towards e-learning and the most complicated challenges regarding online learning and distance education. Our findings disclosed that the online learning using Zoom, Moodle, Google Meet, BigBlueButton and Cisco has become quite popular among the students and academic staff in Ukraine in time of the lockdown period and beyond. Based on the Principal Component Analysis data processing we can conclude that students’ satisfaction and positive e-learning perception are in a good correlation with quality of e-learning resources and set of apps which are used while e-learning and distance education. Also, education style, methods, and manner predict willingness of students to self-study. The self-motivation, time-management, lack of practice, digital alienation, positive attitude towards ICT, and instruction strategy belong to the most important challenges of COVID-19 lockdown based on the students and academic staff interviews. Online learning on daily purpose should be used in the favor of strengthening of classical higher education rather than replacing the former. Blended education is the best alternative to face-to-face education, because the communication with mentor in a live environmental even virtual should have ushered the learners to complete online learning and improve its results.

According to the Development Concept of the Digital Economy and Society in Ukraine, the priority of this area is to develop a substantial national policy on digitalization of education, as this is the key part of the education reform in Ukraine. For this reason, universities should firstly take into account the particularities of teaching the current generation of students and the needs of the digital society as a whole. This paper considers the process of transition from informatization to digitalization in society, implementation of digital support for the educational process in the university, development of the digital educational environment for the training university teachers, and proposes the digital tools for such an environment. The authors propose several ways to improve the development level of digitalization of the educational environment in the university. This is to take into account the needs of the digital society and the modern generation of students, provide a high level of the digital literacy formation of university graduates and support the development of a new digital security system of the modern university. Aiming to design the digital educational environment for increasing the of educators’ digital literacy level, the authors propose to develop and implement the following computer, multimedia and computer-based learning tools and equipment, which includes blended and distance learning classes, cloud technologies, tools of virtual and augmented reality, tools for gamification of the educational process, educational robotics, tools for learning 3D technologies, MOOCs.

We establish several new fractal and number theoretical phenomena connected with expansions which are generated by infinite linear iterated function systems. We show that the systems of cylinders of generalized Lüroth expansions are, generally speaking, not faithful for the Hausdorff dimension calculation. Using Yuval Peres' approach, we prove sufficient conditions for the non‐faithfulness of such families of cylinders. On the other hand, rather general sufficient conditions for the faithfulness of such covering systems are also found. As a corollary, we obtain the non‐faithfullness of the family of cylinders generated by the classical Lüroth expansion.

A new method is developed to construct the probabilistic and dimensional theories for families of representations of real numbers based on studies of special mappings that preserve the Lebesgue measure and Hausdorff–Besicovitch dimension. These mappings are characterized by the property that a preimage and image have the same symbols for two representations of the same family (the set of points of discontinuity of such mappings can be everywhere dense). These mappings are said to be $G$-mappings ($G$-isomorphisms of representations). The probabilistic, metric, and dimensional theories of $G$-isomorphic representations are identical. We establish a rather deep connection between the faithfulness of systems of coverings generated by different representations and the property of preservation of the Hausdorff-Besicovitch dimension of sets by the above-mentioned mappings. General sufficient conditions on faithfulness are found to evaluate the Hausdorff–Besicovitch dimension of families of cylinder sets generated by $F$ and $I$-$F$-representations of real numbers.

The paper is devoted to restricted Oppenheim expansion of real numbers (ROE),which includes as partial cases already known Engel, Silvester and Lüroth expansions. We find conditions under which for almost all (with respect to Lebesgue measure) real numbers from the unit interval their ROE-expansion contain arbitrary digit i only finitely many times. Main results of the paper states the singularity (w.r.t. Lebesgue measure) of the distribution of random variable with i.i.d increments of symbols of restricted Oppenheim expansion. General non-i.i.d. case are also studied and sufficient conditions for the singularity of the corresponding probability distributions are found.

The article is devoted to finding conditions for the packing dimension preservation by distribution functions of random variables with independent Q̃-digits. The notion of “faithfulness of fine packing systems for packing dimension calculation” is introduced, and connections between this notion and packing dimension preservation are found.

In the present paper we study the dependence of fractal and metric properties of numbers which are non-normal resp. essentially non-normal w.r.t. a chosen system of numeration. In particular, we solve open problems mentioned in [1] and prove that there exist expansions (the Q* -expansions or Q* -representations) for real numbers such that the corresponding sets of essentially non-normal numbers and even the whole set of non-normal numbers are of zero Hausdorff dimension. On the other hand, we show that in the same model of Q* -expansions it is possible to choose the matrix Q* in such a way that the corresponding set of essentially non-normal numbers is of full Lebesgue measure. Sufficient conditions forfull dimensionality resp. zero dimensionality of the set of essentially non-normal numbers are also presented. (C) 2016 Published by Elsevier Masson SAS.

We develop a new technique to prove the faithfulness of the Hausdorff--Besicovitch dimension calculation of the family $\varPhi(Q^*)$ of cylinders generated by $Q^*$-expansion of real numbers. All known sufficient conditions for the family $\varPhi(Q^*)$ to be faithful for the Hausdorff--Besicovitch dimension calculation use different restrictions on entries $q_{0k}$ and $q_{(s-1)k}$. We show that these restrictions are of purely technical nature and can be removed. Based on these new results, we study fine fractal properties of random variables with independent $Q^*$-digits.

This paper is devoted to the development of a probabilistic approach to transformations preserving the Hausdorff-Besicovitch dimension. New relations between fractal faithfulness of fine covering systems and DP-properties of related probability distribution functions are found. Necessary and sufficient conditions for the probability distribution functions of random variables with independent $Q^*$-symbols to be DP-functions are obtained.

We develop a new method for the construction of metric, probabilistic and dimensional theories for families of representations of real numbers via studies of special mappings, under which symbols of a given representation are mapped into the same symbols of other representation from the same family, and they preserve the Lebesgue measure and the Hausdorff-Besicovitch dimension (for such mappings the set of points of discontinuity can be everywhere dense). These mappings are said to be G-mappings (G-isomorphisms of representations). Probabilistic, metric and dimensional theories of G-isomorphic representations are identical. We show a rather deep connection between the faithfulness of systems of coverings, generated by different representations, and the preservation of the Hausdorff-Besicovitch dimension of sets by the above mentioned mappings.