A New Model of Fuzzy Logic: Monadic Monoidal T-Norm Based Logic

In this article, we introduce the variety of monadic MTL-algebras as MTL-algebras equipped with two monadic operators. After a study of the basic properties of this variety, we define and investigate monadic filters in monadic MTL-algebras. By using the notion of monadic filters, we prove the subdirect representation theorem of monadic MTL-algebras and characterize simple and subdirectly irreducible monadic MTL-algebras. Moreover, present monadic monoidal t-norm based logic (MMT L), a system of many valued logic capturing the tautologies of monadic MTL-algebras and prove a completeness theorem.AMS Classification: 08A72, 03G25, 03B50, 03C05.

Weak Convergence of Distributions

This chapter introduces the fundamentals of weak convergence for real sequences. Definitions and examples are given. The Skorokhod representation theorem is proved and the chapter then considers the preservation of weak convergence under transformations. Next, the role of moments and characteristic functions is considered. In the leading case of random sums, the criteria for weak convergence and the concept of a stable distribution are studied.

Weak Convergence

This chapter reviews the theory of weak convergence in metric spaces. Topics include Skorokhod’s representation theorem, the metrization of spaces of measures, and the concept of tightness of probability measures. The key relation is shown between weak convergence and uniform tightness. Considering the space C of continuous functions in particular, the functional central limit theorem is proved for martingales, together with extensions to the multivariate case.

FUZZY LIMITS OF FUZZY FUNCTIONS

In this paper, we study the theory of fuzzy limit of fuzzy function depending on the Altai’s principle and using the representation theorem (resolution principle) to run the fuzzy arithmetic.The novelty underlying this theory is that we can provethe convergence of afuzzy function to its fuzzy limit through proving the convergence of its 𝛼-cuts’boundaries to their limits for the membership degree 0<𝛼𝑜<𝛼1≤𝛼≤1.

Inner Product Groups and Riesz Representation Theorem

In this paper, we introduce an inner product on abelian groups and, after investigating the basic properties of the inner product, we first show that each inner product group is a torsion-free abelian normed group. We give examples of such groups and describe the norms induced by such inner products. Among other results, Hilbert groups, midconvex and orthogonal subgroups are presented, and a Riesz representation theorem on divisible Hilbert groups is proved.

Certain Class of Analytic Functions with respect to Symmetric Points Defined by Q-Calculus

In this study, we familiarise a novel class of Janowski-type star-like functions of complex order with regard to j , k -symmetric points based on quantum calculus by subordinating with pedal-shaped regions. We found integral representation theorem and conditions for starlikeness. Furthermore, with regard to j , k -symmetric points, we successfully obtained the coefficient bounds for functions in the newly specified class. We also quantified few applications as special cases which are new (or known).

Banach-Steinhaus Theorem for the Space P of All Primitives of Henstock-Kurzweil Integrable Functions

In this paper, it is shown how the Banach-Steinhaus theorem for the space P of all primitives of Henstock-Kurzweil integrable functions on a closed bounded interval, equipped with the uniform norm, can follow from the Banach-Steinhaus theorem for the Denjoy space by applying the classical Hahn-Banach theorem and Riesz representation theorem.