On an alternative view to complex calculus

Bashirov, Agamirza E.; Norozpour, Sajedeh

doi:10.1002/mma.4827

Cited by 11 publications

(5 citation statements)

References 17 publications

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“…Particularly, Stanley [55], Córdova-Lepe [11], Slavík [10], Pap [44] and Bashirov et al [12], have called the attention of mathematicians to the topic. More recently, non-Newtonian calculi have become a hot topic in economic and finance [56], quantum calculus [54], complex analysis [57][58][59], numerical analysis [2,37,42], inequalities [40,60], biomathematics [7,61] and mathematical education [14].…”

Section: Discussionmentioning

confidence: 99%

On a Non-Newtonian Calculus of Variations

Torres

2021

Axioms

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The calculus of variations is a field of mathematical analysis born in 1687 with Newton’s problem of minimal resistance, which is concerned with the maxima or minima of integral functionals. Finding the solution of such problems leads to solving the associated Euler–Lagrange equations. The subject has found many applications over the centuries, e.g., in physics, economics, engineering and biology. Up to this moment, however, the theory of the calculus of variations has been confined to Newton’s approach to calculus. As in many applications negative values of admissible functions are not physically plausible, we propose here to develop an alternative calculus of variations based on the non-Newtonian approach first introduced by Grossman and Katz in the period between 1967 and 1970, which provides a calculus defined, from the very beginning, for positive real numbers only, and it is based on a (non-Newtonian) derivative that permits one to compare relative changes between a dependent positive variable and an independent variable that is also positive. In this way, the non-Newtonian calculus of variations we introduce here provides a natural framework for problems involving functions with positive images. Our main result is a first-order optimality condition of Euler–Lagrange type. The new calculus of variations complements the standard one in a nontrivial/multiplicative way, guaranteeing that the solution remains in the physically admissible positive range. An illustrative example is given.

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Section: Discussionmentioning

confidence: 99%

On a Non-Newtonian Calculus of Variations

Torres

2021

Axioms

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“…The orbital derivatives of the radial basis function can be used to approximate the Lyapunov function, which results in a global Lyapunov function that determines each subset of the gravitational basin [14]. Bashirov, A. E. N. S verified that transferring to the Riemann curves of complex logarithms through the domain structure of the complex system changes the meaning of derivatives and integrals and that the complex calculus can be expressed through the form of polar coordinates of the complex variables, rather than algebraic forms [15]. Fernandez A proved that the integral of the Mittag-Leffler function can be rewritten as a complex contour integral, which can be used to define the extension of analytic complex order differentiation [16].…”

Section: Introductionmentioning

confidence: 99%

A method for solving the limit of multivariate composite functions

Liang,

Wang

2024

Applied Mathematics and Nonlinear Sciences

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In this paper, according to the concept of multivariate composite function, the limit and continuity conditions of multivariate composite function are given, and the continuity relationship between the inner function and the outer function of the composite function at the same point is investigated. The relationship between the limit and continuity of a composite function composed of two functions at the same point is investigated by examples, while some limit results are explained from the perspective of composite functions, and the limit calculation law of multivariate composite functions is proposed. Using the consistent central limit theorem in the limit calculation law, the absolute value and limit of the interval of the multivariate composite function are obtained, and the iterative limit of the variational reduction approximation method is confirmed by combining the absolute value and limit of the interval of the multivariate composite function. On the basis of the application of limit operation of a multivariate composite function, the limit distribution of high-dimensional likelihood ratio statistics of multivariate composite function is analyzed by using data simulation analysis. The results show that both limit distributions have better simulation results when p are small, such as p = 5 or p = 10, but the traditional limit is better than the normal limit, indicating that the results of multivariate composite function limit calculations have good accuracy. A method for solving the limits of multivariate composite functions is proposed in this study, which is of great theoretical and practical significance in the study of higher mathematics.

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“…The Dirac system will be considered from a different perspective and the results will be compared in classical and multiplicative calculi. In the following years, the theory of complex functions in multiplicative analysis which is necessary especially while dealing with multiplicative analysis in spectral theory was defined by (Bashirov and Norozpur 2018). This is required in difficult applications of the Dirac system.…”

Section: Introductionmentioning

confidence: 99%