In this paper, we study new ideas in the generalization of additive and multiplicative pulsating Fibonacci sequences. Then, we construct two types of pulsating Fibonacci sequences of the
m
th order. Moreover, the closed forms of the two sequences are derived by basic linear algebra.
In this work, we explore edge direction, transitivity, and connectedness of Cayley graphs of gyrogroups. More specifically, we find conditions for a Cayley graph of a gyrogroup to be undirected, transitive, and connected. We also show a relationship between the cosets of a certain type of subgyrogroups and the connected components of Cayley graphs. Some examples and applications regarding these findings are provided.
To learn a programming language, the students have to understand the logical flow of the commands as well as the syntax. The logical flow might be more difficult to understand when compared with a syntax which can detect easily. The primary flow of commands or the control structures includes the sequence, condition or selection, and iteration. The students construct the program flowchart by using these control structure. They also have to understand the result of each command execution, step by step. In this research, we propose the technique for developing the learning tool (AR flowchart) to simulate the result of the commands in program flowchart by using augmented reality (AR), so the learners can visualize the result. With this tool, the students can construct a program flowchart as a series of commands by using AR markers. The result of the execution of these commands can be displayed so the students can see whether the logic of the program is correct or not. The design of this tool aims at increasing student engagement and helping them to understand program logic better. The evaluation of the concept results by the group of university students supports our propose.
Let H n (F) be the space of all n × n upper Hessenberg matrices over a field F where n is a positive integer greater than 2. In this paper, linear maps preserving rank-1 on H n (F) are characterized, i.e. T is a linear rank-1 preserver on H n (F) if and only if (i) im T is an n-dimensional rank-1 subspace or (ii) there exist nonsingular upper Hessenberg matrices P and Q such that
In this study, we investigate Hamiltonian cycles in the right-Cayley graphs of gyrogroups. More specifically, we give a gyrogroup version of the factor group lemma and show that some right-Cayley graphs of certain gyrogroups are Hamiltonian.
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