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Ratio statistics and distributions play a crucial role in various fields, including linear regression, metrology, nuclear physics, operations research, econometrics, biostatistics, genetics, and engineering. In this work, we examine the statistical properties and probability calculations of the Hake normalized gain as a measure of effect size and educational effectiveness in physics education. Leveraging existing knowledge about the Hake ratio as a ratio of normal variables and utilizing open data science tools, we developed two novel computational approaches for computing ratio distributions. Our pilot numerical study demonstrates the speed, accuracy, and reliability of calculating ratio distributions through (1) DE quadrature with/without barycentric interpolation, a very quick and efficient quadrature method, and (2) a 2D vectorized numerical inversion of characteristic functions, which offers broader applicability by not requiring knowledge of PDFs or the independence of ratio constituents. These numerical explorations not only deepen the understanding of the Hake ratio's distribution but also showcase the efficiency, precision, and versatility of our proposed methods, making them highly suitable for fast data analysis based on exact probability ratio distributions. This capability has potential applications in multidimensional statistics and uncertainty analysis in metrology, where precise and reliable data handling is essential., Comment: 23 pages, 5 figures, 1 table

High school physics competitions serve as a platform for talented students to showcase their skills, engage in challenging problems, and foster a passion for science. This paper explores innovative approaches to enhance these competitions by harnessing the power of open science and artificial intelligence (AI) tools. Particularly we delve into the capabilities of state-of-the-art AI chatbots, i.e. ChatGPT, Bard, Claude, related to problem solving in physics. Together with open science tools like SageMath and Jupyter AI, they have the potential to serve as intelligent, powerful co-pilots, tutors, and assistants in understanding and applying physics, as well as knowledge from connected STEM fields. Furthermore, these innovative approaches can revolutionize high school physics competitions, providing students and their tutors with powerful resources to excel in their scientific pursuits., Comment: 10 pages, 4 figures, preprint DIDFYZ2023 conference

The COVID-19 pandemic has underscored the importance of blended learning in contemporary physics and, more generally, STEM education. In this contribution, we summarize current pedagogical models of blended learning, such as rotational and flexible non-rotational models, and customizable configurations of physical and virtual learning spaces. With the inevitable integration of digital technology as one of the pillars of blended learning, teachers find themselves in an unprecedented position to not only obtain data more frequently but also analyze it and adjust instruction accordingly. Consequently, we discuss a crucial element of blended learning effectiveness: data management and usage. In this context, data literacy for teaching emerges as an essential skill for effective blended learning, encompassing the ability to transform various data types into actionable instructional knowledge and practices. In other words, current research in physics education shows that a data-literate science teacher is a more prosperous and effective teacher., Comment: 9 pages, 4 figures, preprint conference DIDFYZ2023

Nowadays interactive digital scientific environments have become an integral part of scientific computing in solving various scientific tasks in research, but also STEM education. We introduce SageMath or shortly Sage -- a free open Python-based alternative to the well-known commercial software -- in the frame of our course Methods of Physical Problems Solving for future scientists and science teachers. Particularly, in the 1st illustrative example from the Physics Olympiad, we present Sage as a scientific open data source, symbolic, numerical, and visualization tool. The 2nd example from the Young Physicists' Tournament shows Sage as a multimedia, modeling, and programming tool. By employing SageMath as an open digital environment for scientific computing in the education of all STEM disciplines, teachers and students are empowered not only with a universal educational tool, but a real research tool, enabling them to engage in interactive visualization, modeling, programming, and solving of authentic, complex interdisciplinary problems, thus naturally enhancing their motivation to pursue science in alignment with the core mission of STEM education., Comment: 9 pages, 3 figures, 1 table, conference DIDSCI2022

The contribution provides a detailed exploration of the online platform Perusall as an advanced social annotation technology in teaching and learning STEM disciplines. This exploration is based on the authors' insights and experiences from three years of implementing Perusall at P.J. \v{S}af\'arik University in Ko\v{s}ice, Slovakia. While the concept of social annotation technology and its educational applications are not novel, Perusall's advanced features, including AI and data science reports, enable its use in both synchronous and asynchronous blended and flipped learning environments. In this context, Perusall serves as a digital tool for collecting formative data, monitoring student progress, and identifying areas of difficulty. This assessment data can be effectively utilized in preparing and personalizing subsequent face-to-face group interactions, thereby enhancing and improving the learning experience. From a pedagogical viewpoint, Perusall's role was particularly significant during the Covid-19 pandemic, enabling effective, continuous, and engaging learning amidst social distancing and physical restrictions. Today, Perusall has become a key tool in blended learning, facilitating higher-order cognitive processes during the educational process and, with its multifaceted applications, serves as a modern catalyst in redefining educational experiences and outcomes., Comment: 8 pages, 4 Figures, conference DIDSCI2022 preprint

A streamflow time series encompasses a large amount of hidden information and reliable prediction of its behavior in the future remains a challenge. It seems that the use of information measures can significantly contribute to determining the time horizon of rivers and improving predictability. Using the Kolmogorov complexity (KC) and its derivatives (KC spectrum and its highest value), and Lyapunov exponent (LE), it has previously been shown that the degree of streamflow predictability depends on human activities, environmental factors, and natural characteristics. This paper applied the KC and LE measures to investigate the randomness and chaotic behavior of monthly streamflow of 1879 rivers from the United States for a period from 1950 to 2015 and evaluated their time horizons via the Lyapunov and Kolmogorov time (LT and KT, respectively)., Comment: 27 pages, 10 figures

At present, there is still no officially accepted and extensively verified implementation of computing the gamma difference distribution allowing unequal shape parameters. We explore four computational ways of the gamma difference distribution with the different shape parameters resulting from time series kriging, a forecasting approach based on the best linear unbiased prediction, and linear mixed models. The results of our numerical study, with emphasis on using open data science tools, demonstrate that our open tool implemented in high-performance Python(with Numba) is exponentially fast, highly accurate, and very reliable. It combines numerical inversion of the characteristic function and the trapezoidal rule with the double exponential oscillatory transformation (DE quadrature). At the double 53-bit precision, our tool outperformed the speed of the analytical computation based on Tricomi's $U(a, b, z)$ function in CAS software (commercial Mathematica, open SageMath) by 1.5-2 orders. At the precision of scientific numerical computational tools, it exceeded open SciPy, NumPy, and commercial MATLAB 5-10 times. The potential future application of our tool for a mixture of characteristic functions could open new possibilities for fast data analysis based on exact probability distributions in areas like multidimensional statistics, measurement uncertainty analysis in metrology as well as in financial mathematics and risk analysis., Comment: 26 pages, 4 figures, 2 tables, 1 box, corrected typos in text and tables, more precise runtimes for MMA and MTB, added references, correct links for online materials

During the last ten years advances in open-source digital technology, used especially by data science, led to very accessible ways how to obtain, store, process, analyze or share data in almost every human activity. Data science tools bring not only transparency, accessibility, and reproducibility in open science, but also give benefits in open education as learning tools for improving effectiveness of instruction. Together with our pedagogical introduction and review of Jupyter as an interactive multimedia learning tool, we present our three-years long research in the framework of a complex mixed-methods approach which examines physics teachers' perception of Jupyter technology in three groups: Ph.D. candidates in physics education research (PER) ($N = 9$), pre-service physics teachers ($N = 33$) and in-service physics teachers ($N = 40$). Despite the fact that open-source Jupyter notebooks are natural and easy as email or web, the results suggest that in-service teachers are not prepared for Jupyter technology and open analysis, but positively accept open education data presented via another open-source data science tool, R Shiny interactive web application, as an important form of immediate feedback and learning about the quality of their instruction. Simultaneously our instruction results in the frame of the Flipped Learning also indicate that young beginning PER researchers and pre-service physics teachers can master key digital skills to work with Jupyter technology appreciating its big impact on their learning, data and statistical literacy or professional development. All results support the ongoing worldwide effort to implement Jupyter in traditional education as a promising free open-source interactive learning tool to foster learning process, especially for the upcoming young generation., Comment: 32 pages, 10 figures, 5 tables

We propose a two-stage estimation method of variance components in time series models known as FDSLRMs, whose observations can be described by a linear mixed model (LMM). We based estimating variances, fundamental quantities in a time series forecasting approach called kriging, on the empirical (plug-in) best linear unbiased predictions of unobservable random components in FDSLRM. The method, providing invariant non-negative quadratic estimators, can be used for any absolutely continuous probability distribution of time series data. As a result of applying the convex optimization and the LMM methodology, we resolved two problems $-$ theoretical existence and equivalence between least squares estimators, non-negative (M)DOOLSE, and maximum likelihood estimators, (RE)MLE, as possible starting points of our method and a practical lack of computational implementation for FDSLRM. As for computing (RE)MLE in the case of $ n $ observed time series values, we also discovered a new algorithm of order $\mathcal{O}(n)$, which at the default precision is $10^7$ times more accurate and $n^2$ times faster than the best current Python(or R)-based computational packages, namely CVXPY, CVXR, nlme, sommer and mixed. We illustrate our results on three real data sets $-$ electricity consumption, tourism and cyber security $-$ which are easily available, reproducible, sharable and modifiable in the form of interactive Jupyter notebooks., Comment: 29 pages, 1 figure, 5 tables