In this study, we investigate the norm of difference operator on some sequence spaces such as Hilbert and Cesaro matrix domains. Therefore the present study is a complement for those results obtained in [1].

This paper deals with the solution behavior and periodic nature of the solutions of the difference equation $$ s_{n+1}=\alpha s_{n}+\dfrac{\beta s_{n}s_{n4}}{\gamma s_{n4}+\delta s_{n5} },\;\;\;n=0,1,... $$ {\Large \noindent }where the initial conditions $s_{5},\ s_{4},\ s_{3},\ s_{2},\ s_{1},\ s_{0}$ are arbitrary positive real numbers and $\alpha ,\ \beta ,\ \gamma ,\ \delta \ $are positive constants. Also we obtain the closed form of the solutions of some special cases of this equation.

In this paper, we give a parametrization of algebraic points of degree at most $4$ over $\mathbb{Q}$ on the schaeffer curve $\mathcal{C}$ of affine equation : $ y^{2}=x^{5}+1 $. The result extends our previous result which describes in [5] ( Afr. Mat 29:11511157, 2018) the set of algebraic points of degree at most $3$ over $\mathbb{Q}$ on this curve.

Since people existed, they have prioritized confidentiality in information sharing and communication. Although there are independent studies on encryption and music in literature, no study is seen on encryption methods that are created by using the properties of mathematical number strings and can be expressed with musical instruments. The purpose in this research is to develop ideas for an effective encryption method and to create a time and location variable encryption method considering this deficiency in the literature by getting advantage of the additive feature in Fibonacci and Lucas number sequences and moving from here to develop new perspectives on encryption science. In the research letters in alphabet, numbers and 10 of the most used symbols were selected and ASCII codes were determined. The objects to be encrypted are divided into 6 main groups (uppercase vowel, uppercase consonant, lowercase vowel, lowercase consonant letters, numbers, and symbols). ASCII codes are written with the additive property of the Fibonacci and Lucas numbers (Zeckendorf's Theorem) and matched with the corresponding notes. In addition to the first method in the study, the encryption system is encrypted by shifting depending on time. In addition to this method, the encryption system was encrypted by shifting depending on the location. In the last method, the text to be encrypted was encrypted by shifting depending on both location and time. The software of the first stage of the encryption system has been created. The encryption method we have created can be transmitted in both audio and text. Since encryption can be applied with various instruments, it offers variety in terms of data privacy. In the encryption system, people who have a musical ear can audibly decipher the password regardless of the written source. In the research, the same text differs as time and location change. This method allows multiple transformations of a character in a text. With these features, it differs from the encryption methods made until now.

The theory of curves has a very long history. Moving frames defined on curves are important parts of this theory. They have never lost their importance. A point particle of constant mass moving along a trajectory in space may be seen as a point of the trajectory. Therefore, there is a very close relationship between the differential geometry of the trajectory and the kinematics of the particle moving on it. One of the most important elements of the particle kinematics is the jerk vector of the moving particle. Recently, a new resolution of the jerk vector, along the tangential direction and two special radial directions, has been presented by \"Ozen et al. (JTAM 57(2)(2019)). By means of these two special radial directions, we introduce a new moving frame for the trajectory of a moving particle with non vanishing angular momentum in this study. Then, according to this frame, some characterizations for the trajectory to be a rectifying curve, an osculating curve, a normal curve, a planar curve and a general helix are given. Also, slant helical trajectories are defined with respect to this frame. Afterwards, the necessary and sufficient conditions for the trajectory to be a slant helical trajectory (according to this frame) are obtained and some special cases of these trajectories are investigated. Moreover, we provide an illustrative numerical example to explain how this frame is constructed. This frame is a new contribution to the field and it may be useful in some specific applications of differential geometry, kinematics and robotics in the future.

This paper evaluates the impact of an effective preventive vaccine on the control of some infectious diseases by using a new deterministic mathematical model. The model is based on the fact that the immunity acquired by a fully effective vaccination is permanent. Threshold $\mathcal{R}_{0}$, defined as the basic reproduction number, is critical indicator in the extinction or spread of any disease in any population, and so it has a very important role for the course of the disease that caused to an epidemic. In epidemic models, it is expected that the disease becomes extinct in the population if $\mathcal{R}_{0}<1.$ In addition, when $\mathcal{R}_{0}<1$ it is expected that the diseasefree equilibrium point of the model, and so the model, is stable in the sense of local and global. In this context, the threshold value $\mathcal{R}_{0}$ regarding the model is obtained. The local asymptotic stability of the diseasefree equilibrium is examined with analyzing the corresponding characteristic equation. Then, by proved the global attractivity of diseasefree equilibrium, it is shown that this equilibria is globally asymptotically stable.

This paper investigates the impact of the various parameters of the mathematical model for Hepatitis B virusHepatitis D virus (HBVHDV) coinfection with controls (awareness, vaccine and therapy). It establishes that the model is biologically meaningful and epidemiologically well posed. Furthermore, simulations are carried out on the equations of the model using MATLAB and the results indicate that; when $c_1$(awareness) increase from $0.08$ to $0.70$, then the number of exposed HB individuals in the population will also increase. Conversely, we notice a drastic decrease in the number of exposed HBD individuals in the population when $c_1$(awareness) increase from $0.08$ to $0.70$. Again, we observe a decrease in the number of exposed treated individuals in the population when $c$(therapy) increase from $0.08$ to $0.50$. Similarly, we notice an increase in the number of recovered HBD individuals in the population upon the increase of $c$(therapy) from $0.08$ to $0.50$. We therefore conclude that awareness, vaccine and therapy are good measure which can be used to effectively control HBVHDV coinfection in a population. However, awareness and vaccine are better control strategies than therapy. Hence, these simulation results provide the best framework for the control of the disease; Hepatitis B virusHepatitis D virus (HBVHDV) coinfection in a population.

In this paper, we obtain some new results on the equiboundedness of solutions and asymptotic stability for a class of nonlinear difference systems with variable delay of the form x(n+1)=ax(n)+B(n)F(x(n−m(n))), n=0,1,2,...x(n+1)=ax(n)+B(n)F(x(n−m(n))),\ \ \ \ \ \ n=0,1,2,... where FF is the real valued vector function, m:Z→Z+,m:Z→Z+, which is bounded function and maximum value of mm is kk and is a k×kk×k variable coefficient matrix. We carry out the proof of our results by using the Banach fixed point theorem and we use these results to determine the asymptotic stability conditions of an example.

This paper presents kinematics form of pronation and supination movement. The algorithm of Stewart platform motion can be used to create a new motion of supination (or pronation) motion. Pronation motion can be taken as Stewart motion which has not any rotation on xaxis and yaxis. In this case, pronation motion has only one parameter. Supination movement creates a helix curve. Additionally, the correlation between rotation angle and extension is 1. This allows us to use artificial intelligence in pronation motion. In this article, the algorithm and Matlab applications of pronation motion are given in the concepts of artificial intelligence approach. This is a new and important approach.

In this paper, we study dynamics and bifurcation of the third order rational difference equation \begin{eqnarray*} x_{n+1}=\frac{\alpha+\beta x_{n2}}{A+Bx_{n}+Cx_{n2}}, ~~n=0, 1, 2, \ldots \end{eqnarray*} with positive parameters $\alpha, \beta, A, B, C$ and nonnegative initial conditions $\{x_{k}, x_{k+1}, \ldots, x_{0}\}$. We study the dynamic behavior, the sufficient conditions for the existence of the NeimarkSacker bifurcation, and the direction of the NeimarkSacker bifurcation. Then, we give numerical examples with figures to support our results.
