“Fuzzy” calculus: The link between quantum mechanics and discrete fractional operators

2020 ◽  
Vol 23 (3) ◽  
pp. 764-786
Author(s):  
Raoul R. Nigmatullin ◽  
Paolo Lino ◽  
Guido Maione
Keyword(s):  
Fractional Integral ◽  
Continuous Spin ◽  
First Order ◽  
Discrete Signals ◽  
Resonance Frequencies ◽  
Fuzzy Calculus ◽  
New Form ◽  
Hidden States ◽  
Extended Interpretation ◽  
Continuous Range

AbstractIn this paper, based on the “fuzzy” calculus covering the continuous range of operations between two couples of arithmetic operations (+, –) and (×, :), a new form of the fractional integral is proposed occupying an intermediate position between the integral and derivative of the first order. This new form of the fractional integral satisfies the C1 criterion according to the Ross classification. The new calculus is tightly related to the continuous values of the continuous spin S = 1 and can generalize the expression for the fractional values of the shifting discrete index. This calculus can be interpreted as the appearance of the hidden states corresponding to unobservable values of S = 1. Many well-known formulas can be generalized and receive a new extended interpretation. In particular, one can factorize any rectangle matrix and receive the “perfect” filtering formula that allows transforming any (deterministic or random) function to another arbitrary function and vice versa. This transformation can find unexpected applications in data transmission, cryptography and calibration of different gadgets and devices. One can also receive the hybrid (”centaur”) formula for the Fourier (F-) transformation unifying both expressions for the direct and inverse F-transformations in one mathematical unit. The generalized Dirichlet formula, which is obtained in the frame of the new calculus to allow selecting the desired resonance frequencies, will be useful in discrete signals processing, too. The basic formulas are tested numerically on mimic data.

Fuzzy Calculus

2015 ◽  
pp. 41-67
Author(s):  
Luciana Takata Gomes ◽  
Laécio Carvalho de Barros ◽  
Barnabas Bede
Keyword(s):  
Fuzzy Calculus

scholarly journals Foundation of Interval-Valued Intuitionistic Fuzzy Limit and Differential Theory and an Application to Continuous Data

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Zhenghai Ai ◽  
Xiaoqin Shu ◽  
Zeshui Xu
Keyword(s):  
Fuzzy Numbers ◽  
Primary Task ◽  
Continuous Data ◽  
Intuitionistic Fuzzy Numbers ◽  
Intuitionistic Fuzzy ◽  
Operational Laws ◽  
Fuzzy Calculus ◽  
Interval Valued

The intuitionistic fuzzy calculus (IFC), based on the basic operational laws of intuitionistic fuzzy numbers (IFNs), has been put forward. However, the interval-valued IFC (IVIFC), based on the basic operational laws of interval-valued IFNs (IVIFNs), is only in the original stage. To further develop the theory of the IVIFC and make it be rigorous, the primary task is to systematically investigate the characteristics of the limits and differentials, which is a foundation of the IVIFC. Moreover, there is quite a lot of literature on IVIFNs; however, the scholars did not reveal the relationships between IFNs and the IVIFNs. To do that, we first investigate the limit of interval-valued intuitionistic fuzzy sequences, and then, we focus on investigating the limit, the continuity, and the differential of IVIFFs in detail and reveal their relationships. After that, due to the fact that the IFC and the IVIFC are based on the basic operational laws of IFNs and IVIFNs, respectively, we reveal the relationships between the IFNs and the IVIFNs via some homomorphic mappings. Finally, a case study about continuous data of IVIFNs is provided to illustrate the advantages of continuous data.

Fuzzy calculus with aplications

2013 ◽  
Vol 41 (1) ◽  
Author(s):  
Anna Chwastyk ◽  
Witold Kosiński
Keyword(s):  
Fuzzy Calculus

Elementary fuzzy calculus

1986 ◽  
Vol 18 (1) ◽  
pp. 31-43 ◽  
Author(s):  
Roy Goetschel ◽  
William Voxman
Keyword(s):  
Fuzzy Calculus

Comparing Fuzzy and Probability Calculus for Representing Imprecision in Preliminary Engineering Design

1989 ◽  
Author(s):  
K. L. Wood ◽  
E. K. Antonsson ◽  
J. L. Beck
Keyword(s):  
Fuzzy Sets ◽  
Engineering Design ◽  
Stochastic Uncertainty ◽  
Probability Calculus ◽  
Fuzzy Calculus ◽  
Design Calculations

Abstract A technique to perform design calculations on imprecise representations of parameters using the calculus of fuzzy sets has been previously developed [17]. An analogous approach to representing and manipulating uncertainty in choosing among alternatives (imprecision) using probability calculus is presented and compared with the fuzzy calculus technique. We find that the fuzzy calculus is well suited to representing and manipulating the imprecision aspect of uncertainty, and probability is best used to represent stochastic uncertainty.

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