Functions represented into fractional Taylor series

Fractional Taylor series are studied. Then solutions of fractional linear ordinary differential equations (FODE), with respect to Caputo derivative, are approximated by fractional Taylor series. The Cauchy-Kowalevski theorem is proved to show the existence and uniqueness of local solutions for FODE with Cauchy initial data. Sufficient conditions for the global existence of the solution and the estimate of error are given for the method using fractional Taylor series. Two illustrative numerical examples are given to demonstrate the validity and applicability of this method

Full Hermite Interpolation and Approximation in Topological Fields

By using generalized divided differences, we study the simultaneous interpolation of an m times continuously differentiable function and its derivatives up to a fixed order in a topological field K. If K is a valued field, then simultaneous Hermite interpolation and approximation are considered. Newton interpolating series are used in the case of an infinite number of conditions of interpolation. Applications to the numerical approximation of variational problems, the solution of a functional equation and, in the case of p-adic fields, the representation of solutions of a boundary value problem for an equation of the Fuchsian type illustrate the efficiency of the theoretical results

Full Hermite Interpolation and Approximation in Topological Fields

By using generalized divided differences, we study the simultaneous interpolation of an m times continuously differentiable function and its derivatives up to a fixed order in a topological field K. If K is a valued field, then simultaneous Hermite interpolation and approximation are considered. Newton interpolating series are used in the case of an infinite number of conditions of interpolation. Applications to the numerical approximation of variational problems, the solution of a functional equation and, in the case of p-adic fields, the representation of solutions of a boundary value problem for an equation of the Fuchsian type illustrate the efficiency of the theoretical results

On a Surface Associated with Pascal’s Triangle

An open problem in reliability theory is that of finding all the coefficients of the reliability polynomial associated with particular networks. Because reliability polynomials can be expressed in Bernstein form (hence linked to binomial coefficients), it is clear that an extension of the classical discrete Pascal’s triangle (comprising all the binomial coefficients) to a continuous version (exhibiting infinitely many values in between the binomial coefficients) might be geometrically helpful and revealing. That is why we have decided to investigate the geometric properties of a continuous extension of Pascal’s triangle including: Gauss curvatures, mean curvatures, geodesics, and level curves, as well as their symmetries

On a Surface Associated with Pascal’s Triangle

An open problem in reliability theory is that of finding all the coefficients of the reliability polynomial associated with particular networks. Because reliability polynomials can be expressed in Bernstein form (hence linked to binomial coefficients), it is clear that an extension of the classical discrete Pascal’s triangle (comprising all the binomial coefficients) to a continuous version (exhibiting infinitely many values in between the binomial coefficients) might be geometrically helpful and revealing. That is why we have decided to investigate the geometric properties of a continuous extension of Pascal’s triangle including: Gauss curvatures, mean curvatures, geodesics, and level curves, as well as their symmetries

On a Surface Associated to the Catalan Triangle

We define a surface that interpolates the ballot numbers in the Catalan triangle corresponding to every pair of nonnegative integers (except for the origin). We study the geometric properties of this surface and prove that it contains exactly five half-lines. The mean curvature and the Gauss curvature of the surface are also calculated

Characterization of Rectifying Curves by Their Involutes and Evolutes

A rectifying curve is a twisted curve with the property that all of its rectifying planes pass through a fixed point. If this point is the origin of the Cartesian coordinate system, then the position vector of the rectifying curve always lies in the rectifying plane. A remarkable property of these curves is that the ratio between torsion and curvature is a nonconstant linear function of the arc-length parameter. In this paper, we give a new characterization of rectifying curves, namely, we prove that a curve is a rectifying curve if and only if it has a spherical involute. Consequently, rectifying curves can be constructed as evolutes of spherical twisted curves; we present an illustrative example of a rectifying curve obtained as the evolute of a spherical helix. We also express the curvature and the torsion of a rectifying spherical curve and give necessary and sufficient conditions for a curve and its involute to be both rectifying curves

The Impact of Agile Management and Technology in Teaching and Practicing Physical Education and Sports

The context of the COVID-19 pandemic has caused educational institutions and sports clubs to change their management strategy. Due to the modernization of computer technology, physical education and sports (PES) teachers decided to include this technology in their teaching activity to help students and athletes to acquire PES-specific transversal skills and digital skills, and also to increase the attractiveness of the lessons. The present research aims at assessing the challenges and opportunities of technology and adopting an Agile Management style to improve the teaching, learning, and practice of PES. Therefore, a survey was conducted on PES teachers and trainers, as they have a clear perspective on the field and their views are therefore very important and relevant to our study, even if they do not have solutions for all the challenges facing them. They were asked to share their professional opinions regarding the implementation of digital methods and applications on the sportive results of performance sportsmen, athletes, and students. The survey, conducted on 144 respondents, contained mostly multiple-choice questions rated on a Likert scale and open-ended questions allowing respondents to offer solutions and express their opinion freely. This article demonstrates the positive influence of Agile Management in the choice and implementation of technology dedicated to PES

The Role of Physical Education and Sports in Modern Society Supported by IoT—A Student Perspective

The COVID-19 pandemic in recent years and the massive presence of information technology generate one of the biggest challenges facing humanity, namely the technological challenge. In this context, educational technologies have a positive impact on the correct and effective teaching and learning of physical education and sports (PES), with a great positive impact on future sustainable higher education (HE). Thus, various innovative techniques could be of interest, such as the use of social networks and fitness sites, e-learning platforms, computer games, and telephone applications involving video analysis and age-specific images of students and the skills taught. This study aims to establish the main means used by technology, through which it can improve the teaching, learning, and practice of PES. This paper demonstrates the positive effects of technology on the PES field in modern society through a regression model, applied to data collected from 260 students from 2 Romanian PES Universities. The pedagogical and educational elements of our model also highlight the role of technology as a facilitator of knowledge, functioning as a tool that comes to the aid of specialists in the PES field