The theory of fuzzy sets, since its foundation, has advanced in a wide range of means and in many fields. One of the areas to which fuzzy set theory has been applied extensively is mathematical programming. Nevertheless, the applications of fuzzy theory can be found in e.g. logic, decision theory, artificial intelligence, computer science, control engineering, expert systems, management science, operations research, robotics, and others. Theoretical improvements have been made in many directions. Nowadays it has a lot of applications also on possibility theory, actuarial credibility theory, fuzzy logic and approximate reasoning, fuzzy control, fuzzy data analysis, fuzzy set models in operations research, etc. The aim of this paper is to investigate some topological properties of a set X when the topology defined on it is the collection of all the α-level subsets of a fuzzy subset A of X.We have been able to establish some results regarding fuzzy cluster level subsets, convergence of level subsets and quasicompactness among others.

In this paper, we study the De et al.'s approach for the application of intuitionistic fuzzy relation (i.e., max-min-max composite relation), and improve the approach for better output. The validity of the improved intuitionistic fuzzy composite relation is carried out in comparison to De et al.'s approach using numerical experiments. It is shown that the improved intuitionistic fuzzy composite relation yields a better output. Finally, an application of the improved approach to medical diagnostic process is carried out using a hypothetical medical database. This improved intuitionistic fuzzy composite relation could be used as a sustainable approach in applying intuitionistic fuzzy sets to other real-life decision-making problems.

In this paper, on the basis of the three-dimensional Chen system, a smooth continuous nonlinear flux-controlled memristor model is used as the positive feedback term of this system, a hyper-chaotic circuit system is successfully constructed, and a simulated equivalent circuit is built for simulation using Multisim software, which agrees with the numerical simulation results by comparison. Meanwhile, a new impulsive control mode called the three-stage-impulse is put forward. It is a cyclic system with three components: continuous inputs are exerted in the first and third parts of the cycle while giving no input in the second part of the cycle, an impulse is exerted at the end of each continuous subsystem, the controller is simple in structure and effective in stabilizing most existing nonlinear systems. The Chen hyper-chaotic system will be controlled based on the three-stage-impulse control method combined with the Lyapunov stability principle. At the end of this paper, we have employed and simulated a numerical example; the experimental results show that the controller is effective for controlling and stabilizing the newly designed hyper-chaotic system.

So many real life problems ranging from medicine, agriculture, biology and finance are modelled by nonlinear systems. In this case, a chaotic nonlinear system is considered and, as opposed to solving Linear Matrix Inequality (LMI), which is the usual approach but cumbersome, a completely different approach was used. In some other cases, the computation of singular value of matrix was used but the method in this study needs not such. In addition, most models, if not all, concentrate on finding a control matrix J under some sufficient conditions. The problem is that only one such matrix J is provided. In reality, the actual control quantity may have a little deviation from the theoretical J. Hence, the study in this paper provides a set of infinite uncertain matrices Jα which are able to adapt to control the system under uncertain conditions. It turns out that this new method controls the system in shorter time with less computational complexities.

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