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In many physical models, internal energy will run out without external energy sources. Therefore, finding optimal energy sources and studying their behavior are essential issues. In this article, we study the following variational problem: G ϵ ∗ = sup ∫ Ω G (u) ϵ q (x) d x : ∥ ∇ u ∥ L p (.) ≤ ϵ , u = 0   on   ∂ Ω , with the help of Γ -convergence, where G : R → R is upper semicontinuous, non zero in the L 1 sense, 0 ≤ G (u) ≤ c | u | q (.) , Ω is a bounded open subset of R n , n ≥ 3 , 1 < p (.) < n , and p (.) ≤ q (.) ≤ p ∗ (.) . For special choices of G, we can study Bernoulli's free-boundary and plasma problems in variable exponent Lebesgue spaces. [ABSTRACT FROM AUTHOR]

Conflicts occur naturally in the real world at all levels of society, individually, in groups or society as a whole. Almost all the existing conflict resolution models are unilateral in their decision‐making process. They do not consider the actions of the involved parties simultaneously. Therefore, in this paper, we aim to design a novel conflict resolution model based on game‐theoretic rough sets by constructing a game between all the concerned parties (players), computing the payoff of different strategies and classifying them following equilibrium rules. The proposed model yields more realistic and accurate results as it explores all possibilities and is flexible in determining different threshold values relative to the complexities of real‐life problems. Three real‐life conflict situations are solved with the proposed model, and a comprehensive analysis is done to validate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Type-2 intuitionistic fuzzy sets are proposed as functions from non empty set U to T T where T = { (μ , ν) : μ + ν ≤ 1 , μ ≥ 0 , ν ≥ 0 } and T T is the set of all mappings from T to T . The members of T T are called intuitionistic fuzzy values (IFV). In this paper, we develop a mathematical framework for IFVs by defining a set of generalized operations on T T and proved it to be an algebra. The other important properties like convexity, normality of IFVs and many important subalgebras are also explored and studied. Furthermore, two partial orders based on generalized operations are defined, which enable us to study the lattices in T T . [ABSTRACT FROM AUTHOR]

In this paper, an in depth study is done on topological properties of intuitionistic fuzzy rough sets in light of different conditions like serial, strongly serial, left continuity, transitivity on intuitionistic fuzzy relations, t-norms, implicators by adopting a axiomatic approach with the ingredients of intuitionistic fuzzy logic. Numerous intuitionistic fuzzy topologies based on many different kinds of intuitionistic fuzzy relations are explored. Also, a special class of intuitionistic fuzzy relations known as T-similarity class has been studied algebraically and found interesting lattices to model real life problems for better applications of intuitionistic fuzzy rough sets. [ABSTRACT FROM AUTHOR]

In this paper, the characterization of Γ-convergence for the first countable topological spaces, characterization of convergence in supremum metric in general setting and some mutual relation between these convergences are discussed. The Γ-convergence is defined as the Kuratowaski-Painlevé convergence of the endographs of the intuitionistic fuzzy sets. The supremum metric is the supremum of Hausdroff distance among the η-cuts of the intuitionistic fuzzy sets. To study these convergences is an important part of the theoretical fundamentals for intuitionistic fuzzy set theory. Some results are given as an application to variational analysis. [ABSTRACT FROM AUTHOR]

Making decisions are very common in the modern socio-economic environments. However, with the increasing complexity of the social, today's decision makers (DMs) face such problems in which they hesitate and irresolute to provide their views. To cope with these uncertainties, many generalizations of fuzzy sets are designed, among them dual hesitant fuzzy set (DHFS) is quite resourceful and efficient in solving problems of a more vague nature. In this article, a novel concept called proportional dual hesitant fuzzy set (PDHFS) is proposed to further improve DHFS. The PDHFS is a flexible tool composed of some possible membership values and some possible non-membership values along with their associated proportions. In the theme of PDHFS, the proportions of membership values and non-membership values are considered to be independent. Some basic operations, properties, distance measure and comparison method are studied for the proposed set. Thereafter, a novel approach based on PDHFSs is developed to solve problems for multi-attribute group decision-making (MAGDM) in a fuzzy situation. It is totally different from the traditional approach. Finally, a practical example is given in order to elaborate the proposed method for the selection of the best alternative and detailed comparative analysis is given in order to validate the practicality. [ABSTRACT FROM AUTHOR]

Preference relations are very useful to express decision makers' preferences over alternatives in the process of group decision-making. The multiple self confidence levels also exist in preference relations due to different knowledge levels, expertise, cultural and educational backgrounds of decision makers. In this study, the hesitant fuzzy preference relation with self confidence and hesitant multiplicative preference relation with self confidence are defined by taking multiple self confidence levels into consideration in the form of a linguistic term set. Latter on, priority weight vectors are derived from linear programming models, and then, to avoid heterogenous hesitant preference relation effects, a collective priority weight vector is derived by using ordered weighted average operator for group decision making. Finally, some practical case studies are done to illustrate the proposed model, and a comparative analysis is carried out to show how self confidence levels influence the group decision-making results. [ABSTRACT FROM AUTHOR]

Preference of an alternative over another alternative is a useful way to express the opinion of decision maker. In the process of group decision making, preference relations are used in preference modelling of the alternatives under given criteria. The probability is an important tool to deal with uncertainty; in many scenarios of decision making probabilities of different events affect the decision making process directly. In order to deal with this issue, in this paper, hesitant probabilistic fuzzy preference relation (HPFPR) is defined. Furthermore, consistency of HPFPR and consensus among decision makers are studied in the hesitant probabilistic fuzzy environment. In this respect, many novel algorithms are developed to achieve consistency of HPFPRs and reasonable consensus between decision makers and a final algorithm is proposed comprehending all other algorithms, presenting a complete decision support model for group decision making. Lastly, we present a case study with complete illustration of the proposed model and discussed the effects of probabilities on decision making validating the importance of the introduction of probability in hesitant fuzzy preference relation. [ABSTRACT FROM AUTHOR]

The preference of one alternative over another is a useful way to express the opinion of the decision-maker. In the process of group decision-making, preference relations are used in preference modeling of the alternatives under given criteria. The probability is an important tool to deal with uncertainty and, in many scenarios of decision-making problems, the probabilities of different events affect the decision-making process directly. In order to deal with this issue, the hesitant probabilistic multiplicative preference relation (HPMPR) is defined in this paper. Furthermore, consistency of the HPMPR and consensus among decision makers are studied here. In this respect, many algorithms are developed to achieve consistency of HPMPRs, reasonable consensus between decision-makers and a final algorithm is proposed comprehending all other algorithms, presenting a complete decision support model for group decision-making. Lastly, we present a case study with complete illustration of the proposed model and discuss the effects of probabilities on decision-making validating the importance of the introduction of probability in hesitant multiplicative preference relations. [ABSTRACT FROM AUTHOR]

In variational analysis, Γ-convergence proved its importance because almost all other convergences can be expressed as Γ-convergence. The discussion about maximization and minimization is also fundamental in variational analysis. In this paper, we define three topologies τM-,τM+,τM on the collection of intuitionistic fuzzy sets based on maxima operator. Furthermore, convergences in these topologies are proved to be compatible with Γ-, Γ+, Γ convergences when intuitionistic fuzzy sets are upper semicontinuous with locally compact Hausdroff domain. [ABSTRACT FROM AUTHOR]

Nowadays, in the modern digital era, the use of computer technologies such as smartphones, tablets and the Internet, as well as the enormous quantity of confidential information being converted into digital form have resulted in raised security issues. This, in turn, has led to rapid developments in cryptography, due to the imminent need for system security. Low-dimensional chaotic systems have low complexity and key space, yet they achieve high encryption speed. An image encryption scheme is proposed that, without compromising the security, uses reasonable resources. We introduced a chaotic dynamic state variables selection procedure (CDSVSP) to use all state variables of a hyper-chaotic four-dimensional dynamical system. As a result, less iterations of the dynamical system are required, and resources are saved, thus making the algorithm fast and suitable for practical use. The simulation results of security and other miscellaneous tests demonstrate that the suggested algorithm excels at robustness, security and high speed encryption. [ABSTRACT FROM AUTHOR]

The classical matrix theory is deficient to express the vagueness of the real life. The fuzzy set theory has been successfully applied to bridge this gap. Much work has already been done on a two-person zero sum matrix game with fuzzy goals. In continuation, this paper is dedicated to define and study a multi-criteria two-person zero sum game with intuitionistic fuzzy goals. It is shown that solving such games is equivalent to solving two crisp multi object linear programming problems. Our work generalizes the previous study on a multi-criteria game with fuzzy goals by adopting the approach of linear programming with intuitionistic fuzzy sets. Finally, an illustrative numerical example is provided to elaborate the proposed approach. [ABSTRACT FROM AUTHOR]

This paper aims to study fuzzy order bounded linear operators between two fuzzy Riesz spaces. Two lattice operations are defined to make the set of all bounded linear operators as a fuzzy Riesz space when the codomain is fuzzy Dedekind complete. As a special case, separation property in fuzzy order dual is studied. Furthermore, we studied fuzzy norms compatible with fuzzy ordering (fuzzy norm Riesz space) and discussed the relation between the fuzzy order dual and topological dual of a locally convex solid fuzzy Riesz space. [ABSTRACT FROM AUTHOR]

In this paper, we prove two generalized concentration-compactness principles for variable exponent Lebesgue spaces and as an application study the asymptotic behaviour of low energy extremals. [ABSTRACT FROM AUTHOR]

The fuzzy order convergence in fuzzy Riesz spaces is defined only for fuzzy order bounded nets. The aim of this paper is to define and study unbounded fuzzy order convergence and some of its applications. Furthermore, some theoretical concepts like the fuzzy weak order unit and fuzzy ideals are studied in relation to unbounded fuzzy order convergence. [ABSTRACT FROM AUTHOR]

Decision making is the key component of people's daily life, from choosing a mobile phone to engaging in a war. To model the real world more accurately, probabilistic linguistic term sets (PLTSs) were proposed to manage a situation in which several possible linguistic terms along their corresponding probabilities are considered at the same time. Previously, in linguistic term sets, the probabilities of all linguistic term sets are considered to be equal which is unrealistic. In the process of decision making, due to the vagueness and complexity of real life, an expert usually hesitates and unable to express its opinion in a single term, thus making it difficult to reach a final agreement. To handle real life scenarios of a more complex nature, only membership linguistic decision making is unfruitful; thus, some mechanism is needed to express non-membership linguistic term set to deal with imprecise and uncertain information in more efficient manner. In this article, a novel notion called probabilistic hesitant intuitionistic linguistic term set (PHILTS) is designed, which is composed of membership PLTSs and non-membership PLTSs describing the opinions of decision makers (DMs). In the theme of PHILTS, the probabilities of membership linguistic terms and non-membership linguistic terms are considered to be independent. Then, basic operations, some governing operational laws, the aggregation operators, normalization process and comparison method are studied for PHILTSs. Thereafter, two practical decision making models: aggregation based model and the extended TOPSIS model for PHILTS are designed to classify the alternatives from the best to worst, as an application of PHILTS to multi-attribute group decision making. In the end, a practical problem of real life about the selection of the best alternative is solved to illustrate the applicability and effectiveness of our proposed set and models. [ABSTRACT FROM AUTHOR]