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Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy the canonical anti-commutation relation (CAR) in equal time. This paper aimed to discuss the classical reduction and ergodicity of quantum exclusion semigroups constructed by QBN. We first study the classical reduction of the quantum semigroups to an Abelian algebra of diagonal elements and the space of off-diagonal elements. We then provide an explicit representation formula by separating the action on off-diagonal and diagonal operators, on which they are reduced to the semigroups of classical Markov chains. Finally, we prove that the asymptotic behavior of the quantum semigroups is equivalent to one of its associated Markov chains, and that the semigroups restricted to the off diagonal space of operators have a zero limit.

Up to now, there have been many research results about approximations of fuzzy numbers by specific fuzzy numbers under preservation of some operators. However, there are few investigations about approximations of n-dimensional fuzzy numbers. The main reason is that the structure of n-dimensional fuzzy numbers are more complex. This paper devotes to seek an approximation operator that produces a special kind of 2-dimensional fuzzy number which is the nearest one to the given 2-dimensional fuzzy number along the direction k. The concept of four edges approximation for 2-dimensional fuzzy numbers is defined, which is a special kind of 2-dimensional fuzzy numbers. At the same time, the concept of fuzzy 2-cell prismoid numbers is introduced, and the nearest fuzzy 2-cell prismoid numbers of 2-dimensional fuzzy numbers which preserve the centroid of the core for the 2-dimensional fuzzy numbers with respect to the weighted pseudometric are discussed. Finally, with respect to the criteria: scale invariance, identity criterion and nearness criterion, some properties of the approximation operator are presented.

In this paper, we present a characteristic theorem for the generalized difference concept of n-dimensional fuzzy-numbers by means of support functions. The concepts of differentiability for n-dimensional fuzzy-number-valued functions are presented and discussed using the new generalized difference. The relationships of g-differentiability, s-differentiability, and l-differentiability are characterized. Furthermore, optimization problems with s-convex and s-differentiable fuzzy-number-valued functions are described.

The aim of this study is to generalize convexity from vector-valued maps to n-dimensional fuzzy number-valued functions and to investigate some relations among the convexity and generalized convexity of n-dimensional fuzzy number-valued functions, which are based on a new ordering of n-dimensional fuzzy numbers proposed in this study. Furthermore, some criteria are obtained for the convexity and generalized convexity under the conditions of lower and upper semicontinuity, respectively. Finally, we study the local-global minimum properties of the convex fuzzy number-valued functions.

Taking into account that interval-valued fuzzy numbers can provide more flexibility to represent the imprecise information and interval-valued trapezoidal fuzzy numbers are widely used in practice, this paper devotes to seek an approximation operator that produces an interval-valued trapezoidal fuzzy number which is the nearest one to the given interval-valued fuzzy number, and the approximation operator preserves the core of the original interval-valued fuzzy number with respect to the weighted distance. As an application, we use the interval-valued trapezoidal approximation to handle fuzzy risk analysis problems, which overcome the drawback of existing fuzzy risk analysis methods.