7 research outputs found
Topological and fractal properties of real numbers which are not normal
No full textAbstractThe set L of essentially non-normal numbers of the unit interval (i.e., the set of real numbers having no asymptotic frequencies of all digits in their nonterminating s-adic expansion) is studied in details. It is proven that the set L is generic in the topological sense (it is of the second Baire category) as well as in the sense of fractal geometry (L is a superfractal set, i.e., the Hausdorff–Besicovitch dimension of the set L is equal 1). These results are substantial generalizations of the previous results of the two latter authors [M. Pratsiovytyi, G. Torbin, Ukrainian Math. J. 47 (7) (1995) 971–975].The Q∗-representation of real numbers (which is a generalization of the s-adic expansion) is also studied. This representation is determined by the stochastic matrix Q∗. We prove the existence of such a Q∗-representation that almost all (in the sense of Lebesgue measure) real numbers have no asymptotic frequency of all digits. In the case where the matrix Q∗ has additional asymptotic properties, the Hausdorff–Besicovitch dimension of the set of numbers with prescribed asymptotic properties of their digits is determined (this is a generalization of the Eggleston–Besicovitch theorem). The connections between the notions of “normality of numbers” respectively of “asymptotic frequencies” of their digits is also studied
On fine fractal properties of generalized infinite Bernoulli convolutions
No full textAbstractThe paper is devoted to the investigation of generalized infinite Bernoulli convolutions, i.e., the distributions μξ of the following random variables:ξ=∑k=1∞ξkak, where ak are terms of a given positive convergent series; ξk are independent random variables taking values 0 and 1 with probabilities p0k and p1k correspondingly.We give (without any restriction on {an}) necessary and sufficient conditions for the topological support of ξ to be a nowhere dense set. Fractal properties of the topological support of ξ and fine fractal properties of the corresponding probability measure μξ itself are studied in details for the case where ak⩾rk:=ak+1+ak+2+⋯ (i.e., rk−1⩾2rk) for all sufficiently large k. The family of minimal dimensional (in the sense of the Hausdorff–Besicovitch dimension) supports of μξ for the above mentioned case is also studied in details. We describe a series of sets (with additional structural properties) which play the role of minimal dimensional supports of generalized Bernoulli convolutions. We also show how a generalization of M. Cooper's dimensional results on symmetric Bernoulli convolutions can easily be derived from our results
Digitalization of the educational process for the training of the pre-service teachers
No full textAccording 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
Strengthening of e-learning at the leading Ukrainian pedagogical universities in the time of COVID-19 pandemic
No full textDistance 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
Enhancing digital and professional competences via implementation of virtual laboratories for future physical therapists and rehabilitologist
No full textBeing 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
Author Correction: A MHz-repetition-rate hard X-ray free-electron laser driven by a superconducting linear accelerator
No full textAnalysis of the effects of a Constructivist-Based Mathematics Problem Solving Instructional Program on the achievement of Grade Five Students in Belize, Central America.
No full textThis thesis examined whether social constructivist activities can improve the mathematical competency of grade five students in Belize, Central America. The sample included 342 students and eight teachers from two rural and urban schools. A switching replication design was employed enabling students in the experimental groups to be taught using social constructivist activities for 12 weeks and the controls exposed to similar instructional practices from weeks 7 to 12. Students‘ performance was assessed using Pre-test, Post test 1 and 2 with an internal consistency of 0.89, 0.90 and 0.93 respectively. As revealed by the repeated measures ANOVA within subject analysis, there were significant differences among the pre-test and post test 1 and 2 results. That is, students in the control groups, who were instructed using a procedural approach from weeks 1 to 6, demonstrated higher gains than the experimental groups who were immersed in social constructivist activities. Furthermore, when the control groups became immersed in similar activities from weeks 7 to 12, they continued to outperform the experimental groups who were exposed to social constructivist activities alone. Hence, due to this unexpected result, the aim of this thesis became to explain why these results came about and what implications for teaching were highlighted by the consideration.
Besides the quantitative results highlighted above, qualitative data was also obtained as part of the study. For example, students were videoed within constructivist math groups and their performance analyzed using Pirie and Kieren‘s (1994) Model of Growth for Mathematical Understanding. The data from the video recording revealed that use of one step math problems did not enabled students to restructure their thinking to solve innovative problems. Data from semi-structured interviews also revealed that some students lacked basic math skills and were not exposed or guided to solve complex problems. Besides the need for careful examination of social constructivist activities on performance, this thesis underscores the importance of relevant teaching and learning activities, the important role of teachers during social constructivist activities and the need to identify suitable forms of assessment to measure performance
