A note on ranking generalized fuzzy numbers

Ranking fuzzy numbers plays an important role in decision making under uncertain environment. Recently, Chen and Sanguansat (2011) [Chen, S. M. & Sanguansat, K. (2011). Analyzing fuzzy risk based on a new fuzzy ranking method between generalized fuzzy numbers. Expert Systems with Applications, 38(3), (pp. 2163-2171)] proposed a method for ranking generalized fuzzy numbers. It considers the areas on the positive side, the areas on the negative side and the heights of the generalized fuzzy numbers to evaluate ranking scores of the generalized fuzzy numbers. Chen and Sanguansat’s method (2011) can overcome the drawbacks of some existing methods for ranking generalized fuzzy numbers. However, in the situation when the score is zero, the results of the Chen and Sanguansat’s ranking method (2011) ranking method are unreasonable. The aim of this short note is to give a modification on Chen and Sanguansat’s method (2011) to make the method more reasonable.     

A note on the sendograph metric of fuzzy numbers

In this paper, by studying the attainable properties of sendograph metric in fuzzy number space, we generalize Monotone convergence theorem and Nested theorem of intervals from the space of real numbers to the space of fuzzy numbers.     

\lambda -Statistical convergence of fuzzy numbers and fuzzy functions of order \theta

In this paper, we introduce the concept of $\lambda $ -statistical convergence of order $\theta $ and strong $\lambda $ -summability of order $\theta $ for the sequence of fuzzy numbers. Further the same concept is extended to the sequence of fuzzy functions and introduce the spaces like $S_\lambda ^\theta (\hat{f})$ and $\omega _{\lambda p} ^\theta (\hat{f})$ . Some inclusion relations in those spaces and also the underlying relation between these two spaces are also obtained.     

A theorem on Liapunov functions using $$\ddot V$$

Theory of Computing Systems -     

Construction of Perfect q-ary Codes by Sequential Switchings of $$\widetilde\alpha $$ -Components

We suggest a construction of perfect q-ary codes using sequential switchings of special-type components of the Hamming code. The construction yields a lower bound on the number of different q-ary codes.     

Levelwise statistical cluster and limit points of a sequence of fuzzy numbers

In the current work we introduce the notions of levelwise statistical limit point and levelwise statistical cluster point of a sequence of fuzzy numbers and discuss the relations between the sets of ordinary limit points, levelwise limit points, levelwise statistical limit points, statistical cluster points and levelwise statistical cluster points of a sequence of fuzzy numbers. Finally, we show that the Bolzano–Weierstrass Theorem is not valid for levelwise statistical convergence.     

A new fuzzy MCDM approach based on centroid of fuzzy numbers

This paper proposes a new fuzzy MCDM (FMCDM) approach based on centroid of fuzzy numbers for ranking of alternatives. The FMCDM approach allows decision makers (DMs) to evaluate alternatives using linguistic terms such as very high, high, slightly high, medium, slightly low, low, very low or none rather than precise numerical values, allows them to express their opinions independently, and also provides an algorithm to aggregate the assessments of alternatives. Three numerical examples are investigated using the FMCDM approach to illustrate its applications. It is shown that the FMCDM approach offers a flexible, practical and effective way of group decision making.     

A note on probabilities of fuzzy events

If X is a set of random elements and A is a fuzzy subset of X, a new definition for the probability of A is presented. This definition is compared with Zadeh's definition of using the expected value of A as the probability of A. Various properties of this new definition are presented. Furthermore it is shown that the distinction between these two definitions is based upon the choice of definition for the cardinality of a fuzzy subset.     

Symmetric fuzzy numbers and additive equivalence of fuzzy numbers

To obtain the group properties of fuzzy quantities, Mare? introduced an equivalence relation between fuzzy quantities. However, the Mare?’s method used to prove his main theorems demands to limit the investigation to fuzzy quantities with finite support. In this paper, we discuss the properties of symmetric fuzzy numbers, show an equivalent characterization of convex fuzzy sets, and present a way to construct a symmetric convex fuzzy set with a convex fuzzy set. Based on these results, we restrict ourselves to fuzzy numbers and prove Mare?’s results without the limitation to fuzzy numbers with finite support using the refined equivalence relation due to Hong and Do. Our results prove one of Mare?’s open questions in the literature.     

Counting information: A note on physicalized numbers

Existing work on the ultimate limits of computation has urged that the apparatus of real numbers should be eschewed as an investigative tool and replaced by discrete mathematics. The present paper argues for a radical extension of this viewpoint: not only the continuum but all infinitary constructs including the rationals and the potential infinite sequence of whole numbers need to be eliminated if a self-consistent investigative framework is to be achieved.     

A note on generalized Euler numbers and polynomials

In this paper we construct new generalized Euler polynomials and generalized Euler numbers attached to χ. We investigate some of the properties that are related to generalized Euler polynomials. We also derive the existence of a specific interpolation function that interpolates generalized Euler polynomials at negative integers. Finally, we investigate computationally the roots of the generalized Euler polynomials E _{n, χ}(x) for values of the index n.     

Square roots of $${3\times 3}$$ matrices

We find square roots of a complex-valued matrix \(A_{3 \times 3}\) using equation \(B^{2}=A\). The proposed method is faster than Higham’s method and provides up to 8 square roots with less relative residual and error.     

A note on a fuzzy measure of typicality

The concept of typicality is studied and the relationship between a typical value and the mode is explored. It is suggested that a typical value should be a fuzzy subset. A measure of typicality associated with any fuzzy subset is introduced. We provide an algorithm for finding the best typical value which is of the form of an interval. © 1997 John Wiley & Sons, Inc.     

A semantically complete extension sequence of the system $$\mathcal{L}^* $$

In this paper, the method of well-combined semantics and syntax proposed by Pavelka is applied to the research of the propositional calculus formal system . The partial constant values are taken as formulas, formulas are fuzzified in two manners of semantics and syntax, and inferring processes are fuzzified. A sequence of new extensions { } of the system is proposed, and the completeness of is proved.