These days different e-learning architecture provide different kinds of e-learning experiences due to-one size fits for all‖ concept. This is no way better than the traditional learning and does not exploit the technological advances. Thus the e-learning system began to evolve to adaptable e-learning systems which adapts or personalizes the learning experience of the learners. Systems infer the characteristics of the learners and identify the preferences of the learners and automatically generate personalized learning path and customize learning contents to the individuals needs. This process is known as adaptation and systems which adapt are known are adaptive systems. So the main objective of this research was to provide an adaptive e-learning system framework which personalizes the learning experience in an efficient way. In this paper a framework for adaptive e-learning system using user response theory is proposed to meet the research objectives identified in section 1.D.

The main objective of this work is to develop the fuzzy fractional mathematical model that will be used to examine the dynamics of monkeypox viral transmission. The proposed dynamical model consists of human and rodents individuals and this monkeypox infection model is mathematically formulated by fuzzy fractional differential equation defined in Caputo’s sense. We provide results that demonstrate the existence and uniqueness of the considered model’s solution. We observe that our results are accurate, and that our method is applicable to the fuzzy system of fractional ordinary differential equations (ODEs). Furthermore, this monkeypox virus model has been identified as a generalization of SEIQR and SEI models. The results show that keeping diseased rodents apart from the human population reduces the spread of disease. Finally, we present brief discussions and numerical simulations to illustrate our findings.

The major goal of this study is to create an optimal technique for managing COVID‐19 spread by transforming the SEIQR model into a dynamic (multistage) programming problem with continuous and discrete time‐varying transmission rates as optimizing variables. We have developed an optimal control problem for a discrete‐time, deterministic susceptible class (S), exposed class (E), infected class (I), quarantined class (Q), and recovered class (R) epidemic with a finite time horizon. The problem involves finding the minimum objective function of a controlled process subject to the constraints of limited resources. For our model, we present a new technique based on dynamic programming problem solutions that can be used to minimize infection rate and maximize recovery rate. We developed suitable conditions for obtaining monotonic solutions and proposed a dynamic programming model to obtain optimal transmission rate sequences. We explored the positivity and unique solvability nature of these implicit and explicit time‐discrete models. According to our findings, isolating the affected humans can limit the danger of COVID‐19 spreading in the future.

The main focus of this paper is to develop certain types of fundamental theorems using q, $q(\alpha )$, and h difference operators. For several higher order difference equations, we get two forms of solutions: one is closed form and another is summation form. However, most authors concentrate only on the summation part. This motivates us to develop closed-form solutions, and we succeed. The key benefit of this research is finding the closed-form solutions for getting better results when compared to the summation form. The symmetric difference operator is the combination of forward and backward difference symmetric operators. Using this concept, we employ the closed and summation form for q, $q(\alpha )$, and h difference symmetric operators on polynomials, polynomial factorials, logarithmic functions, and products of two functions that act as a solution for symmetric difference equations. The higher order fundamental theorems of q and $q(\alpha )$ are difficult to find when the order becomes high. Hence, by inducing the h difference symmetric operator in q and $q(\alpha )$ symmetric operators, we find the solution easily and quickly. Suitable examples are given to validate our findings. In addition, we plot the figures to examine the value stability of q and $q(\alpha )$ difference equations.

In this paper, we present and analyze the upper and lower bounds on the Hyper Zagreb index $\chi^2(G)$ of graph $G$ in terms of the number of vertices $(n)$, number of edges $(m)$, maximum degree $(\Delta)$, minimum degree $(\delta)$ and the inverse degree $(ID(G))$. In addition, we give a counter example on the upper bound of the second Zagreb index for Theorems 2.2 and 2.4 from \cite{ranjini}. Finally, we present lower and upper bounds on $\chi^2(G)+\chi^2(\overline G)$, where $\overline G$ denotes the complement of $G$.