A NOVEL FUZZY ENTROPY MEASURE AND ITS APPLICATION IN COVID-19 WITH FUZZY TOPSIS

Fuzzy sets (FSs) are an important tool to model uncertainty and vagueness. Entropy is being used to measure the fuzziness within a fuzzy set (FS). These entropies are used to find multicriteria decision-making. For measuring uncertainty with TOPSIS techniques an axiomatic definition of entropy measure for fuzzy sets is also given in this paper. The proposed entropy is provided to satisfy all the axioms. Several numerical examples are presented to compare the proposed entropy measure with existing entropies. The corresponding results show that the newly proposed entropy can be computed easily and give reliable results. Finally, the decision-making algorithm TOPSIS (Techniques of ordered preference similarity to ideal solution) is utilized to solve multicriteria decision-making problems (MCDM) related to daily life. In the current situation, COVID-19 has no proper medical treatment. We use TOPSIS technique to suggest an effective medicine for this pandemic. Numerical results and practical examples show the effectiveness and practical applicability of the proposed entropy.

The Mathematical Structure of Integrated Information Theory

Integrated Information Theory is one of the leading models of consciousness. It aims to describe both the quality and quantity of the conscious experience of a physical system, such as the brain, in a particular state. In this contribution, we propound the mathematical structure of the theory, separating the essentials from auxiliary formal tools. We provide a definition of a generalized IIT which has IIT 3.0 of Tononi et al., as well as the Quantum IIT introduced by Zanardi et al. as special cases. This provides an axiomatic definition of the theory which may serve as the starting point for future formal investigations and as an introduction suitable for researchers with a formal background.

Entropy in Thermodynamics: from Foliation to Categorization

We overview the notion of entropy in thermodynamics. We start from the smooth case using differential forms on the manifold, which is the natural language for thermodynamics. Then the axiomatic definition of entropy as ordering on a set that is induced by adiabatic processes will be outlined. Finally, the viewpoint of category theory is provided, which reinterprets the ordering structure as a category of pre-ordered sets.

Austrian behavioral economics

This paper explores the potential for gains from trade between Austrian and behavioral economics, with a focus on how the two schools of thoughts can constructively critique each other. Among other things, the Austrian critique of behavioral economics would urge it to jettison its restrictive and axiomatic definition of rationality, and to treat humans as active agents rather than passive recipients of environmental and cognitive influences. Meanwhile, the behavioral critique of Austrian economics would push it to take more seriously the fundamental question of how individuals arrive at choices and to analyze how such choices can interact with ‘micro-institutional’ choice environments.

Cladistic hypotheses as degree of equivalence relational structures: implications for three-item statements

Three-item statements, as minimal informative rooted binary phylogenetic trees on three items, are the minimal units of cladistic information. Their importance for phylogenetic reconstruction, consensus and supertree methods relies on both (i) the fact that any cladistic tree can always be decomposed into a set of three-item statements, and (ii) the possibility, at least under some conditions, to build a new cladistic tree by combining all or part of the three-item statements deduced from several prior cladistic trees. In order to formalise such procedures, several k-adic rules of inference, i.e., rules that allow us to deduce at least one new three-item statement from exactly k other ones, have been identified. However, no axiomatic background has been proposed, and it remains unknown if a particular k-adic rule of inference can be reduced to more basic rules. In order to solve this problem, we propose here to define three-item statements in terms of degree of equivalence relations. Given both the axiomatic definition of the latter and their strong connection to hierarchical classifications, we establish a list of the most basic properties for three-item statements. With such an approach, we show that it is possible to combine five three-item statements from basic rules although they are not combinable only from dyadic rules. Such a result suggests that all higher k-adic rules are well reducible to a finite set of simpler rules.

Representations of a Comparison Measure between Two Fuzzy Sets

This paper analyzes the representation behaviors of a comparison measure between two compared fuzzy sets. Three types of restrictions on two fuzzy sets are considered in this paper: two disjoint union fuzzy sets, two disjoint fuzzy sets and two general fuzzy sets. Differences exist among the numbers of possible representations of a comparison measure for the three types of fuzzy sets restrictions. The value of comparison measure is constant for two disjoint union fuzzy sets. There are 49 candidate representations of a comparison measure for two disjoint fuzzy sets, of which 13 candidate representations with one or two terms are obtained. For each candidate representation, a variant of the general axiomatic definition for a comparison measure is presented. Choosing the right candidate representation for a given application, we can easily and efficiently calculate and compare a comparison measure.

Quasi-Perfect Stackelberg Equilibrium

Equilibrium refinements are important in extensive-form (i.e., tree-form) games, where they amend weaknesses of the Nash equilibrium concept by requiring sequential rationality and other beneficial properties. One of the most attractive refinement concepts is quasi-perfect equilibrium. While quasiperfection has been studied in extensive-form games, it is poorly understood in Stackelberg settings—that is, settings where a leader can commit to a strategy—which are important for modeling, for example, security games. In this paper, we introduce the axiomatic definition of quasi-perfect Stackelberg equilibrium. We develop a broad class of game perturbation schemes that lead to them in the limit. Our class of perturbation schemes strictly generalizes prior perturbation schemes introduced for the computation of (non-Stackelberg) quasi-perfect equilibria. Based on our perturbation schemes, we develop a branch-and-bound algorithm for computing a quasi-perfect Stackelberg equilibrium. It leverages a perturbed variant of the linear program for computing a Stackelberg extensive-form correlated equilibrium. Experiments show that our algorithm can be used to find an approximate quasi-perfect Stackelberg equilibrium in games with thousands of nodes.

A New Axiomatics for Masures

Masures are generalizations of Bruhat–Tits buildings. They were introduced by Gaussent and Rousseau to study Kac–Moody groups over ultrametric fields that generalize reductive groups. Rousseau gave an axiomatic definition of these spaces. We propose an equivalent axiomatic definition, which is shorter, more practical, and closer to the axiom of Bruhat–Tits buildings. Our main tool to prove the equivalence of the axioms is the study of the convexity properties in masures.

DEFORMATION CONDITIONS FOR PSEUDOREPRESENTATIONS

Given a property of representations satisfying a basic stability condition, Ramakrishna developed a variant of Mazur’s Galois deformation theory for representations with that property. We introduce an axiomatic definition of pseudorepresentations with such a property. Among other things, we show that pseudorepresentations with a property enjoy a good deformation theory, generalizing Ramakrishna’s theory to pseudorepresentations.