A Note on Horadam Hybrinomials

The article is devoted to studying innovative technologies for teaching physical and mathematical disciplines in higher education institutions of Ukraine in the conditions of war. The study aims to consider and develop innovative approaches to teaching physical and mathematical disciplines in higher education institutions, taking into account martial law. The study’s relevance is due to the limitations in the educational process due to the war, which requires the use of not only distance technologies but also a comprehensive solution to the problem of providing quality education in Ukraine. It is essential when teaching physical and mathematical disciplines that can become the basis for the further development of the defense industry of Ukraine. For the purpose of the study, the methods of general scientific knowledge were used, which include the analysis of scientific literature, synthesis of information, and grouping of data, which eventually allowed to identify the problems of teaching physical and mathematical disciplines and to propose directions for their solution using the method of induction and deduction. The results of the study show that the main innovative approaches to teaching physical and mathematical disciplines are distance education, the organization of a single knowledge base, and the use of a problem-based method of teaching physical and mathematical disciplines. Today in Ukraine, there is no single platform that higher education institutions can use for effective student learning. Therefore educational institutions use existing international and domestic platforms to organize and ensure communication processes between teachers and students, a platform for assessing knowledge, and a platform for storing lectures and other teaching materials. It is recommended to improve the approach to teaching physical and mathematical disciplines, taking into account the relevance of military topics, which would allow to educate the population, to form qualified labor resources in various sectors of the economy that would contribute to the development of the defense sector. The practical significance of the study lies in the possibility of its use in improving the methods of teaching physical and mathematicaldisciplines by higher education institutions during the war and in the period of post-war reconstruction.

In this paper, bicomplex $k$-Fibonacci quaternions are defined. Also, some algebraic properties of bicomplex $k$-Fibonacci quaternions are investigated. For example, the summation formula, generating functions, Binet's formula, the Honsberger identity, the d'Ocagne's identity, Cassini's identity, Catalan's identity for these quaternions are given. In the last part, a different way to find $n-th$ term of the bicomplex $k$-Fibonacci quaternion sequence was given using the determinant of a tridiagonal matrix.

In this paper, we introduced the extended generalized Fibonacci polynomial sequence \{$Y_{2,n}$\} and extended generalized Tribonacci polynomial \{$Y_{3,n}$\} with arbitrary initial values and established a recursive matrix and then presented some properties of these. Further, we investigated some well-known identities like Binet's formula, Catalan's identity, Cassini's identity, d'Ocagne's identity, generating function, explicit sum formula, sum of first $n$ terms for the extended generalized Fibonacci polynomial sequence and extended generalized Tribonacci polynomial sequence.

In this paper, bicomplex Pell and bicomplex Pell-Lucas numbers are defined. Also, negabicomplex Pell and negabicomplex Pell-Lucas numbers are given. Some algebraic properties of bicomplex Pell and bicomplex Pell-Lucas numbers which are connected between bicomplex numbers and Pell and Pell-Lucas numbers are investigated. Furthermore, d'Ocagne's identity, Binet's formula, Cassini's identity and Catalan's identity for these numbers are given.

The aim of the study is to obtain new binomial transforms for the $k-$ Narayana sequence. The first of these is the binomial transform, which is its normal form, and in the first step, after finding the recurrence relation of this new binomial transform, the generating function and Binet formula were obtained. Finally, Pascal's triangle was calculated. In the rest of the article, $k-$binomial transform was performed for the $k-$ Narayana sequence and the recurrence relation, generating function, Binet formula and Pascal's triangle were examined for the new sequence obtained. Then, by performing the falling binomial transform and the rising binomial transform, the features listed above were found again for these sequences.

This research introduces a novel category of hybrid numbers, with components represented by unrestricted Lichtenberg numbers. We present some recurrence relations, summation formulas, the Binet formula, and the generating function associated with these numbers. Additionally, a comprehensive Vajda's identity is derived, which encompasses Catalan's, Cassini's and d'Ocagne's identities as specific cases.

The main object of the present paper is to consider the binomial transforms for Horadam quaternion sequences. We gave new formulas for recurrence relation, generating function, Binet formula and some basic identities for the binomial sequence of Horadam quaternions. Working with Horadam quaternions, we have found the most general formula that includes all binomial transforms with recurrence relation from the second order. In the last part, we determined the recurrence relation for this new type of quaternion by working with the iterated binomial transform, which is a different type of binomial transform.

1R40. Microcontinuum Field Theories II: Fluent Media. - AC Eringen (15 Red Tail Dr, Littleton CO 80126-5001). Springer-Verlag, New York. 2001. 340 pp. ISBN 0-387-98969-2. $149.00.Reviewed by ME Ryan (Dept of Chem Eng, SUNY, 505 Furnas Hall, Buffalo NY 14260).The book, Microcontinuum Field Theories II. Fluent Media, is a mathematically-based exposition of the continuum mechanics of structured fluid media, such as anisotropic fluids, liquid crystals, suspensions, etc. Volume II builds upon and extends the theoretical foundation of the subject developed in the first volume, Microcontinuum Field Theories I. Foundations and Solids. The development in Volume II extends the classical field theories of elasticity, fluid dynamics, and electromagnetism to continua constituted of material points or internal structures having directors that may be deformable as well as orientable. The book is largely based on the author’s contribution to the subject over the past few decades. The book is a reference source that assumes the reader has sufficient background in the subject to follow the derivation and application of the field equations to a variety of different situations. The book is intended for research scientists and graduate students in the disciplines of mathematics, physics, engineering, and related fields. This volume comprises nine chapters (Chs 9-17) which can be divided into four parts: I) Theory of micropolar fluids (Chs 9-11), II) Liquid crystals (Chs 12-14), III) Microstretch fluids (Chs 15-16), and IV) Micromorphic fluids (Ch 17). Electromag-netic interactions are also incorporated into the first three parts. Chapter 9 introduces basic measures of strain and rotation, constitutive equations, and the basic field equations. Analyses of pipe flow, flow in an orthogonal rheometer, lubrication in a journal bearing, Stokes flow around a sphere, stagnation flow, thermal instability, and boundary layer flow are provided. Mixed convection in vertical flow and a discussion of turbulence are also provided within the context of micropolar fluid dynamics. Chapter 10 develops the constitutive and field equations of magneto-hydrodynamics. Analyses of channel flow, shear flow, and thermal stability are presented. Chapter 11 addresses anisotropic fluids and suspensions. The basic theory is developed and applied to channel and couette flows.Chapter 12 gives a detailed presentation of the kinematics, balance laws, constitutive equations, and field equations of liquid crystals. The electrodynamics of liquid crystals is presented in Chapter 13. A variety of problems involving liquid crystals are analyzed in Chapter 14 including shear flow and heat conduction in a channel. The kinematics, constitutive equations, and field equations relating to microstretch fluids are provided in Chapter 15. Applications to the theory of acoustic waves in bubbly fluids and blood flow in small arteries are presented. Chapter 16 presents a continuum theory of the electrodynamics of polymeric liquid crystals. The last chapter addresses the subject of micromorphic fluids with applications of the theory to channel flow, the rheological behavior of blood in arteries, and the behavior of deformable suspensions. A few problems are briefly stated at the end of each chapter. A list of references, a subject index, and a list of errata for Volume I are provided at the end of the book. The treatment provided in this monograph will serve as a useful reference for the academician or research scientist interested in the fundamental theoretical development of this subject.

The theory of dynamical systems is a very complex subject that has produced several surprises in the recent past in connection with the theory of chaos and fractals. The application of the tools of dynamical systems in cosmological settings is less known, in spite of the number of published scientific papers on this subject. In this paper, a mostly pedagogical introduction to the cosmological application of the basic tools of dynamical systems theory is presented. It is shown that, in spite of their amazing simplicity, these tools allow us to extract essential information on the asymptotic dynamics of a wide variety of cosmological models. The power of these tools is illustrated within the context of the so-called Λ-cold dark matter (ΛCDM) and scalar field models of dark energy. This paper is suitable for teachers, undergraduate students, and postgraduate students in the disciplines of physics and mathematics.

Modern developments in the ceramic industries have called for more highly trained personnel for management, and for research and development. This article describes a sandwich course for a Council for National Academic Awards (C.N.A.A.) degree in ceramic technology, embracing as well the disciplines of physics, chemistry and mathematics.

PurposeThe aims of this article are to describe the current status, usage, and acceptance of domestic preprint servers in mainland China by investigating three integrated preprint servers in mainland China. These are the Qiji e‐print archive (Qiji), the Chinese Preprint Server (CPS), Chinese Science Papers Online (CSPO).Design/methodology/approachThis research gives a quantitative analysis of the submission numbers to three preprint servers, the subject distribution and citations. An investigation of university websites and their policies was also carried out.FindingsPreprint submissions have a strong correlation with official promotion and policies. The preprints communication pattern in mainland China concentrates on the disciplines of physics, mathematics and computer science which reflects international preprint communication behaviour. As yet the domestic preprint servers in mainland China attract little attention from excellent scholars and have a low impact on scientific communication.Research limitations/implicationsNo author survey was conducted.Originality/valueThis is the first paper to characterize the usage and acceptance of domestic preprint servers in mainland China using a quantitative method.

A. F. Horadam defined the complex Fibonacci numbers and Fibonacci quaternions in the middle of the 20th century. Half a century later, S. Hal{\i}c{\i} introduced the complex Fibonacci quaternions by inspiring from these definitions and discussed some properties of them. Recently, the elliptic biquaternions, which are generalized form of the complex and real quaternions, have been presented. In this study, we introduce the set of Fibonacci elliptic biquaternions that includes the set of complex Fibonacci quaternions as a special case and investigate some properties of Fibonacci elliptic biquaternions. Furthermore, we give the Binet formula and Cassini's identity in terms of Fibonacci elliptic biquaternions. Finally, we give elliptic and real matrix representations of Fibonacci elliptic biquaternions.

This paper introduces generalised incomplete beta functions defined by the generalised beta function. Firstly, we provide some of the generalised beta function's basic properties, such as integral representations, summation formulas, Mellin transform, and beta distribution. We then present several fundamental properties, such as integral representations, summation formulas, and recurrence relations with the help of the generalised incomplete beta functions.

International scientific conference «Research activities and achievements in pedagogy and psychology» : conference proceedings (April 5–6, 2023. Częstochowa, the Republic of Poland). Riga, Latvia : «Baltija Publishing», 2023. 188 pages.

The relevance of the research: the development of high-end manufacturing leads to an increase in the negative impact of man on the environment. So, today the issues of developing environmental consciousness in college students are relevant. The problem statement: the negative impact of a person on the environment is increasing. However, the development ofthe ecological consciousness of students, awareness ofthe consequences of environmental impact is not at the proper level that is necessary in the transition of society to the sixth technological mode. The purpose of the research is to prove that the formation of the cognitive activity of college students in the ecological aspect is a necessary condition for the development of the ecological consciousness of college students. To solve the problem, the article considers such research methods as the analysis of pedagogical literature, observation, synthesis, generalization, considers the issues of practical application of methods on the example of classes in physical and mathematical disciplines and classes in Ecology. The conclusions: The use of mechanisms for the formation of students' ability to show initiative in problem-dialogical interaction in matters relating to environmental problems contributes to the development of students' environmental consciousness, the growth of their responsibility for the effectiveness of their professional activities, and the impact on the ecosystem.