GATEAUX and FRÉCHET DERIVATIVE IN INTUITIONISTIC FUZZY NORMED LINEAR SPACES

In this paper, we introduce intuitionistic fuzzy derivative, intuitionistic fuzzy Gateaux derivative and intuitionistic fuzzy Fréchet derivative and some of their properties are studied. The relations between intuitionistic fuzzy Gateaux derivative and intuitionistic fuzzy Fréchet derivative are studied.

The Gâteaux derivative and orthogonality in C∞

The general problem in this paper is minimizing the C∞− norm of suitable affine mappings from B(H) to C∞, using convex and differential analysis (Gateaux derivative) as well as input from operator theory. The mappings considered generalize the so-called elementary operators and in particular the generalized derivations, which are of great interest by themselves. The main results obtained characterize global minima in terms of (Banach space) orthogonality.

GENERAL FORMULATION OF QUANTUM ANALYSIS

A general formulation of noncommutative or quantum derivatives for operators in a Banach space is given on the basis of the Leibniz rule, irrespective of their explicit representations such as the Gâteaux derivative or commutators. This yields a unified formulation of quantum analysis, namely the invariance of quantum derivatives, which are expressed by multiple integrals of ordinary higher derivatives with hyperoperator variables. Multivariate quantum analysis is also formulated in the present unified scheme by introducing a partial inner derivation and a rearrangement formula. Operator Taylor expansion formulas are also given by introducing the two hyperoperators [Formula: see text] and [Formula: see text] with the inner derivation δA:Q↦[A,Q]≡AQ-QA. Physically the present noncommutative derivatives express quantum fluctuations and responses.