“Fuzzy” calculus: The link between quantum mechanics and discrete fractional operators
-
Raoul R. Nigmatullin
,
Paolo Lino
Abstract
In 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.
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© 2020 Diogenes Co., Sofia
Artikel in diesem Heft
- Frontmatter
- Editorial Note
- FCAA related news, events and books (FCAA–Volume 23–3–2020)
- Survey Paper
- Why fractional derivatives with nonsingular kernels should not be used
- Fractional-order susceptible-infected model: Definition and applications to the study of COVID-19 main protease
- Generalized fractional Poisson process and related stochastic dynamics
- Research Paper
- Determination of the fractional order in semilinear subdiffusion equations
- Degenerate Kirchhoff (p, q)–Fractional systems with critical nonlinearities
- Solution of linear fractional order systems with variable coefficients
- “Fuzzy” calculus: The link between quantum mechanics and discrete fractional operators
- The green function for a class of Caputo fractional differential equations with a convection term
- Inverse problem for a multi-term fractional differential equation
- Maximum principles for a class of generalized time-fractional diffusion equations
- Multiple positive solutions for a nonlocal PDE with critical Sobolev-Hardy and singular nonlinearities via perturbation method
- Variational approximation for fractional Sturm–Liouville problem
- The 2-adic derivatives and fractal dimension of Takagi-like function on 2-series field
- Construction of fixed point operators for nonlinear difference equations of non integer order with impulses
- An averaging principle for stochastic differential equations of fractional order 0 < α < 1
- Weak solvability of the variable-order subdiffusion equation
Artikel in diesem Heft
- Frontmatter
- Editorial Note
- FCAA related news, events and books (FCAA–Volume 23–3–2020)
- Survey Paper
- Why fractional derivatives with nonsingular kernels should not be used
- Fractional-order susceptible-infected model: Definition and applications to the study of COVID-19 main protease
- Generalized fractional Poisson process and related stochastic dynamics
- Research Paper
- Determination of the fractional order in semilinear subdiffusion equations
- Degenerate Kirchhoff (p, q)–Fractional systems with critical nonlinearities
- Solution of linear fractional order systems with variable coefficients
- “Fuzzy” calculus: The link between quantum mechanics and discrete fractional operators
- The green function for a class of Caputo fractional differential equations with a convection term
- Inverse problem for a multi-term fractional differential equation
- Maximum principles for a class of generalized time-fractional diffusion equations
- Multiple positive solutions for a nonlocal PDE with critical Sobolev-Hardy and singular nonlinearities via perturbation method
- Variational approximation for fractional Sturm–Liouville problem
- The 2-adic derivatives and fractal dimension of Takagi-like function on 2-series field
- Construction of fixed point operators for nonlinear difference equations of non integer order with impulses
- An averaging principle for stochastic differential equations of fractional order 0 < α < 1
- Weak solvability of the variable-order subdiffusion equation
