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In this paper, we initiate the study of the multiplicative Euclidean submanifolds, exhibiting the multiplicative counterparts of the Gauss–Weingarten formulas. Some particular types such as multiplicative conic, spherical, and totally geodesic submanifolds are analyzed. In relation to such multiplicative submanifolds, we introduce multiplicative rectifying submanifolds and characterize by means of multiplicative concurrent vector fields. The multiplicative rectifying hypersurfaces are also obtained. [ABSTRACT FROM AUTHOR]

This manuscript explores the idea of a multiplicatively P-superquadratic function and its properties. Utilizing these properties, we derive the inequalities of Hermite–Hadamard type for such functions in the framework of multiplicative calculus. In addition to this, we derive integral inequalities of Hermite–Hadamard type for the product and quotient of multiplicatively P-superquadratic and multiplicatively P-subquadratic functions. Moreover, we develop the fractional version of Hermite–Hadamard type inequalities involving midpoints and end points for multiplicatively P-superquadratic functions with respect to multiplicatively Riemann–Liouville (R.L) fractional integrals. Graphical illustrations based on specific relevant examples validate the credibility of the findings. The study is further stimulating by being pushed with potential applications in terms of moment of random variables, special means, and modified Bessel functions of the first kind. Regarding superquadraticity, the results presented in this work are new, therefore they clearly provide extensions and improvements of the work available in the literature. [ABSTRACT FROM AUTHOR]

The primary goal of this paper is to define Katugampola fractional integrals in multiplicative calculus. A novel method for generalizing the multiplicative fractional integrals is the Katugampola fractional integrals in multiplicative calculus. The multiplicative Hadamard fractional integrals are also novel findings of this research and may be derived from the special situations of Katugampola fractional integrals. These integrals generalize to multiplicative Riemann–Liouville fractional integrals and multiplicative Hadamard fractional integrals. Moreover, we use the Katugampola fractional integrals to prove certain new Hermite–Hadamard and trapezoidal-type inequalities for multiplicative convex functions. Additionally, it is demonstrated that several of the previously established inequalities are generalized from the newly derived inequalities. Finally, we give some computational analysis of the inequalities proved in this paper. [ABSTRACT FROM AUTHOR]

It is essential to solve nonlinear equations in engineering, where accuracy and precision are critical. In this paper, a novel family of iterative methods for finding the simple roots of nonlinear equations based on multiplicative calculus is introduced. Based on theoretical research, a novel family of simple root-finding schemes based on multiplicative calculus has been devised, with a convergence order of seven. The symmetry in the pie graph of the convergence–divergence areas demonstrates that the method is stable and consistent when dealing with nonlinear engineering problems. An extensive examination of the numerical results of the engineering applications is presented in order to assess the effectiveness, stability, and consistency of the recently established method in comparison to current methods. The analysis includes the total number of functions and derivative evaluations per iteration, elapsed time, residual errors, local computational order of convergence, and error graphs, which demonstrate our method's better convergence behavior when compared to other approaches. [ABSTRACT FROM AUTHOR]

Fractional-order nonlinear equation-solving methods are crucial in engineering, where complex system modeling requires great precision and accuracy. Engineers may design more reliable mechanisms, enhance performance, and develop more accurate predictions regarding outcomes across a range of applications where these problems are effectively addressed. This research introduces a novel hybrid multiplicative calculus-based parallel method for solving complex nonlinear models in engineering. To speed up the method's rate of convergence, we utilize a second-order multiplicative root-finding approach as a corrector in the parallel framework. Using rigorous theoretical analysis, we illustrate how the hybrid parallel technique based on multiplicative calculus achieves a remarkable convergence order of 12, indicating its effectiveness and efficiency in solving complex nonlinear equations. The intrinsic stability and consistency of the approach—when applied to nonlinear situations—are clearly indicated by the symmetry seen in the dynamical planes for various parameter values. The method's symmetrical behavior indicates that it produces accurate findings under a range of scenarios. Using a dynamical system procedure, the ideal parameter values are systematically analyzed in order to further improve the method's performance. Implementing the aforementioned parameter values using the parallel approach yields very reliable and consistent outcomes. The method's effectiveness, reliability, and consistency are evaluated through the analysis of numerous nonlinear engineering problems. The analysis provides a detailed comparison with current techniques, emphasizing the benefits and potential improvements of the novel approach. [ABSTRACT FROM AUTHOR]

Abstract This manuscript explores the idea of a multiplicatively P-superquadratic function and its properties. Utilizing these properties, we derive the inequalities of Hermite–Hadamard type for such functions in the framework of multiplicative calculus. In addition to this, we derive integral inequalities of Hermite–Hadamard type for the product and quotient of multiplicatively P-superquadratic and multiplicatively P-subquadratic functions. Moreover, we develop the fractional version of Hermite–Hadamard type inequalities involving midpoints and end points for multiplicatively P-superquadratic functions with respect to multiplicatively Riemann–Liouville (R.L) fractional integrals. Graphical illustrations based on specific relevant examples validate the credibility of the findings. The study is further stimulating by being pushed with potential applications in terms of moment of random variables, special means, and modified Bessel functions of the first kind. Regarding superquadraticity, the results presented in this work are new, therefore they clearly provide extensions and improvements of the work available in the literature.

Our aim was to develop a new class of bi starlike functions by utilizing the concept of subordination, driven by the idea of multiplicative calculus, specifically multiplicative derivatives. Several restrictions were imposed, which were indeed strict constraints, because we have tried to work within the current framework or the design of analytic functions. To make the study more versatile, we redefined our new class of function with Miller-Ross Poisson distribution (MRPD), in order to increase the study's adaptability. We derived the first coefficient estimates and Fekete-Szegő inequalities for functions in this new class. To demonstrate the characteristics, we have provided a few examples.

In this work, we, for the first time, establish a class of multiplicatively (P,m)-superquadratic function and look into its various features. In the light of these features, we come up with the several integer order integral inequalities in the frame of multiplicative calculus. Moreover, we develop the fractional version of Hermite–Hadamard’s type inequalities involving midpoints and end points for multiplicatively (P,m)-superquadratic function with respect to multiplicatively k-Riemann–Liouville fractional integrals. By choosing different values for the parameters of such integral operators, we acquire a simple version of integral inequalities of Hermite–Hadamard’s type as well as its fractional form via multiplicatively Riemann–Liouville fractional integrals for multiplicatively (P,m)-superquadratic function. The findings are confirmed by graphical illustration by taking appropriate examples into account. The study is further enhanced by the addition of applications of special means and first-type modified Bessel functions. The new results clearly provide extensions and improvements of the work available in the literature. [ABSTRACT FROM AUTHOR]

Our aim was to develop a new class of bi starlike functions by utilizing the concept of subordination, driven by the idea of multiplicative calculus, specifically multiplicative derivatives. Several restrictions were imposed, which were indeed strict constraints, because we have tried to work within the current framework or the design of analytic functions. To make the study more versatile, we redefined our new class of function with Miller-Ross Poisson distribution (MRPD), in order to increase the study's adaptability. We derived the first coefficient estimates and Fekete-Szegő inequalities for functions in this new class. To demonstrate the characteristics, we have provided a few examples. [ABSTRACT FROM AUTHOR]

This article describes a control approach for obtaining predefined‐time robust tracking in multiplicative systems despite positive, bounded, and unknown multiplicative disturbances. The proposed approach is distinguished by imposing predefined‐time convergence, a topic previously studied in conventional calculus in the context of multiplicative systems. Multiplicative calculus is recognized as a beneficial tool that complements standard calculus by simplifying the modeling and comprehension of numerous processes. Simulations are carried out to illustrate that the given control strategy enforces convergence before a predefined time instant and, while inducing robustness against system uncertainties. The findings of this article pave the way for further research into predefined‐time synchronization of multiplicative oscillator systems, which would bring promising implications for data encryption and secure communication. [ABSTRACT FROM AUTHOR]

In this article, we develop multiplicative fractional versions of Simpson's and Newton's formula-type inequalities for differentiable generalized convex functions with the help of established identities. The main motivation for using generalized convex functions lies in their ability to extend results beyond traditional convex functions, encompassing a broader class of functions, and providing optimal approximations for both lower and upper bounds. These inequalities are very useful in finding the error bounds for the numerical integration formulas in multiplicative calculus. Applying these results to the Quadrature formulas demonstrates their practical utility in numerical integration. Furthermore, numerical analysis provides empirical evidence of the effectiveness of the derived findings. It is also demonstrated that the newly proven inequalities extend certain existing results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we defined a new family of meromorphic functions whose analytic characterization was motivated by the definition of the multiplicative derivative. Replacing the ordinary derivative with a multiplicative derivative in the subclass of starlike meromorphic functions made the class redundant; thus, major deviation or adaptation was required in defining a class of meromorphic functions influenced by the multiplicative derivative. In addition, we redefined the subclass of meromorphic functions analogous to the class of the functions with respect to symmetric points. Initial coefficient estimates and Fekete–Szegö inequalities were obtained for the defined function classes. Some examples along with graphs have been used to establish the inclusion and closure properties. [ABSTRACT FROM AUTHOR]

In this study, conics (circle, ellipse, hyperbola) are characterized by taking into account basic multiplication operations in multiplicative space. For this purpose, firstly multiplicative axes and regions are introduced. Additionally, the multiplicative cone definition is given and visualized on the figure. General definitions and theorems of non-Newtonian conics are given. Additionally, examples were given and drawings were made to make the resulting characterizations and theorems more memorable. [ABSTRACT FROM AUTHOR]

We define a Bessel type equation in multiplicative calculus by some algebraic structures. Eigenfunctions of constructed problem are obtained by power series solution technique. While making these solutions, multiplicative Bessel functions are used strongly. We get a generator for that problem and construct integration representations for multiplicative Bessel functions. Finally, some basic spectral properties of multiplicative Bessel problem are examined in detail. [ABSTRACT FROM AUTHOR]

Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new inequalities help in finding the error bounds for different numerical integration formulas in multiplicative calculus. The use of s-convex function extends the results for convex functions and covers a large class of functions, which is the main motivation for using s-convexity. To prove the inequalities, we derive two different integral identities for multiplicative differentiable functions in the setting of multiplicative calculus. Then, with the help of these integral identities, we prove some integral inequalities of the Simpson and Ostrowski types for multiplicative generalized convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities, to show the validity of the results for multiplicative s-convex functions. We also give some applications to quadrature formula and special means of real numbers within the framework of multiplicative calculus. [ABSTRACT FROM AUTHOR]

The main purpose of this study is to examine the scattering analysis of a multiplicative Sturm-Liouville boundary value problem. Firstly, we find the Jost solution of this problem and give the Jost function to construct the scattering function of the problem. Secondly, we obtain asymptotic equations for Jost solution and investigate some properties of the Jost function. Moreover, we present other solutions of the problem by giving the relations between these solutions and Jost solution. Finally, we define the scattering function of this problem by using the Jost function and examine the basic properties of the scattering function. [ABSTRACT FROM AUTHOR]

Fractional-order nonlinear equation-solving methods are crucial in engineering, where complex system modeling requires great precision and accuracy. Engineers may design more reliable mechanisms, enhance performance, and develop more accurate predictions regarding outcomes across a range of applications where these problems are effectively addressed. This research introduces a novel hybrid multiplicative calculus-based parallel method for solving complex nonlinear models in engineering. To speed up the method’s rate of convergence, we utilize a second-order multiplicative root-finding approach as a corrector in the parallel framework. Using rigorous theoretical analysis, we illustrate how the hybrid parallel technique based on multiplicative calculus achieves a remarkable convergence order of 12, indicating its effectiveness and efficiency in solving complex nonlinear equations. The intrinsic stability and consistency of the approach—when applied to nonlinear situations—are clearly indicated by the symmetry seen in the dynamical planes for various parameter values. The method’s symmetrical behavior indicates that it produces accurate findings under a range of scenarios. Using a dynamical system procedure, the ideal parameter values are systematically analyzed in order to further improve the method’s performance. Implementing the aforementioned parameter values using the parallel approach yields very reliable and consistent outcomes. The method’s effectiveness, reliability, and consistency are evaluated through the analysis of numerous nonlinear engineering problems. The analysis provides a detailed comparison with current techniques, emphasizing the benefits and potential improvements of the novel approach.

It is essential to solve nonlinear equations in engineering, where accuracy and precision are critical. In this paper, a novel family of iterative methods for finding the simple roots of nonlinear equations based on multiplicative calculus is introduced. Based on theoretical research, a novel family of simple root-finding schemes based on multiplicative calculus has been devised, with a convergence order of seven. The symmetry in the pie graph of the convergence–divergence areas demonstrates that the method is stable and consistent when dealing with nonlinear engineering problems. An extensive examination of the numerical results of the engineering applications is presented in order to assess the effectiveness, stability, and consistency of the recently established method in comparison to current methods. The analysis includes the total number of functions and derivative evaluations per iteration, elapsed time, residual errors, local computational order of convergence, and error graphs, which demonstrate our method’s better convergence behavior when compared to other approaches.

Motivated by the notion of multiplicative calculus, more precisely multiplicative derivatives, we used the concept of subordination to create a new class of starlike functions. Because we attempted to operate within the existing framework of the design of analytic functions, a number of restrictions, which are in fact strong constraints, have been placed. We redefined our new class of functions using the three-parameter Mittag–Leffler function (Srivastava–Tomovski generalization of the Mittag–Leffler function), in order to increase the study's adaptability. Coefficient estimates and their Fekete-Szegő inequalities are our main results. We have included a couple of examples to show the closure and inclusion properties of our defined class. Further, interesting bounds of logarithmic coefficients and their corresponding Fekete–Szegő functionals have also been obtained. [ABSTRACT FROM AUTHOR]

In this article, we develop multiplicative fractional versions of Simpson’s and Newton’s formula-type inequalities for differentiable generalized convex functions with the help of established identities. The main motivation for using generalized convex functions lies in their ability to extend results beyond traditional convex functions, encompassing a broader class of functions, and providing optimal approximations for both lower and upper bounds. These inequalities are very useful in finding the error bounds for the numerical integration formulas in multiplicative calculus. Applying these results to the Quadrature formulas demonstrates their practical utility in numerical integration. Furthermore, numerical analysis provides empirical evidence of the effectiveness of the derived findings. It is also demonstrated that the newly proven inequalities extend certain existing results in the literature.

In this paper, we defined a new family of meromorphic functions whose analytic characterization was motivated by the definition of the multiplicative derivative. Replacing the ordinary derivative with a multiplicative derivative in the subclass of starlike meromorphic functions made the class redundant; thus, major deviation or adaptation was required in defining a class of meromorphic functions influenced by the multiplicative derivative. In addition, we redefined the subclass of meromorphic functions analogous to the class of the functions with respect to symmetric points. Initial coefficient estimates and Fekete–Szegö inequalities were obtained for the defined function classes. Some examples along with graphs have been used to establish the inclusion and closure properties.

Abstract In this paper, we introduced the concept of multiplicatively p-convex functions and established Hermite-Hadamard type integral inequalities in the setting of multiplicative calculus for this newly created class of functions. We also gave integral inequalities of Hermite-Hadamard type for product and quotient of multiplicatively p-convex functions. Furthermore, we obtained novel multiplicative integral-based inequalities for the product and quotient of convex and multiplicatively p-convex functions. Additionally, we derived certain upper limits for this new class of functions. The findings we proved are generalizations of the results in the literature. The results obtained in this study may inspire further research in various scientific areas.

Abstract In this work, the notion of a multiplicative harmonic convex function is examined, and Hermite–Hadamard inequalities for this class of functions are established. Many inequalities of Hermite–Hadamard type are also taken into account for the product and quotient of multiplicative harmonic convex functions. In addition, new multiplicative integral-based inequalities are found for the quotient and product of multiplicative harmonic convex and harmonic convex functions. In addition, we provide certain upper limits for such classes of functions. The obtained results have been verified by providing examples with included graphs. The findings of this study may encourage more research in several scientific areas.

After its introduction by Grossman and Katz, the geometry part of geometric calculus has been mostly left unattended. In this paper, we give the elementary theorems to build the geometry on geometric space. We apply the well‐known theorems for inner product in geometric space. The terms geometric determinant and geometric vector product are introduced, and their properties are studied. We give the definitions of the circle, the plane, and the sphere and visualize them along with a line in geometric space. Finally, we discuss the Gram–Schmidt process in geometric space. [ABSTRACT FROM AUTHOR]

Digital imaging is used in variety of applications. Together with the improvements in artificial intelligence and its sub-fields, improving computer vision methods to address inter- and multi-disciplinary problems is possible. Especially in medical applications, there are significant improvements related to imaging in the last decades. Digital image interpolation is a key operation in digital image processing where there are no sufficient samples during the acquisition process. Using the available samples in hand, digital interpolation techniques are predicting the missing samples. The paper addresses the problem of digital image interpolation and proposes a novel algorithm using multiplicative calculus. The main contribution of the paper is the application of multiplicative Lagrange interpolation to accomplish image interpolation task. The proposed method is tested on several datasets, and the results are comparable to the state-of-the-art methods. The paper presents encouraging results to the literature, and the proposed method is open for further improvements. [ABSTRACT FROM AUTHOR]

Backpropagation is one of the most widely used algorithms for training feedforward deep neural networks. The algorithm requires a differentiable activation function and it performs computations of the gradient proceeding backwards through the feedforward deep neural network from the last layer through to the first layer. In order to calculate the gradient at a specific layer, the gradients of all layers are combined via the chain rule of calculus. One of the biggest disadvantages of the backpropagation is that it requires a large amount of training time. To overcome this issue, this paper proposes a modified backpropagation algorithm with multiplicative calculus. Multiplicative calculus provides an alternative to the classical calculus and it defines new kinds of derivative and integral forms in multiplicative form rather than addition and subtraction forms. The performance analyzes are discussed in various case studies and the results are given comparatively with classical backpropagation algorithm. It is found that the proposed modified backpropagation algorithm converges in less time to the solution and thus provides fast training in the given case studies. It is also shown that the proposed algorithm avoids the local minima problem.

Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new inequalities help in finding the error bounds for different numerical integration formulas in multiplicative calculus. The use of s-convex function extends the results for convex functions and covers a large class of functions, which is the main motivation for using s-convexity. To prove the inequalities, we derive two different integral identities for multiplicative differentiable functions in the setting of multiplicative calculus. Then, with the help of these integral identities, we prove some integral inequalities of the Simpson and Ostrowski types for multiplicative generalized convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities, to show the validity of the results for multiplicative s-convex functions. We also give some applications to quadrature formula and special means of real numbers within the framework of multiplicative calculus.

The purpose of this paper is to investigate some properties of multiplicative regular and periodic Sturm-Liouville problems given in general form. We first introduce regular and periodic Sturm-Liouville (S-L) problems in multiplicative analysis by using some algebraic structures. Then, we discuss the main properties such as orthogonality of different eigenfunctions of the given problems. We show that the eigenfunctions corresponding to same eigenvalues are unique modulo a constant multiplicative factor and reality of the eigenvalues of multiplicative regular S-L problems. Finally, we present some examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we introduced the concept of multiplicatively p-convex functions and established Hermite-Hadamard type integral inequalities in the setting of multiplicative calculus for this newly created class of functions. We also gave integral inequalities of Hermite-Hadamard type for product and quotient of multiplicatively p-convex functions. Furthermore, we obtained novel multiplicative integral-based inequalities for the product and quotient of convex and multiplicatively p-convex functions. Additionally, we derived certain upper limits for this new class of functions. The findings we proved are generalizations of the results in the literature. The results obtained in this study may inspire further research in various scientific areas. [ABSTRACT FROM AUTHOR]

In this work, the notion of a multiplicative harmonic convex function is examined, and Hermite–Hadamard inequalities for this class of functions are established. Many inequalities of Hermite–Hadamard type are also taken into account for the product and quotient of multiplicative harmonic convex functions. In addition, new multiplicative integral-based inequalities are found for the quotient and product of multiplicative harmonic convex and harmonic convex functions. In addition, we provide certain upper limits for such classes of functions. The obtained results have been verified by providing examples with included graphs. The findings of this study may encourage more research in several scientific areas. [ABSTRACT FROM AUTHOR]

In this paper, we establish an integral equality involving a multiplicative differentiable function for the multiplicative integral. Then, we use the newly established equality to prove some new Simpson's and Newton's inequalities for multiplicative differentiable functions. Finally, we give some mathematical examples to show the validation of newly established inequalities.

When exponentials are employed to model procedures and efficacies appearing in real life, an additive derivative of this type of function does not exist. From this perspective, we define the Legendre equation in multiplicative analysis by several algebraic structures. Multiplicative Legendre polynomials of constituted problem are obtained by the power series solution method. Moreover, for the multiplicative Legendre equation, the generating function is obtained, and an integral presentation is constructed. Eventually, some fundamental spectral features of the multiplicative Legendre problem are analysed.

We define weighted geometric mean method of convergence for sequences in [Formula: see text] by using multiplicative calculus and obtain necessary and sufficient conditions under which convergence of sequences in [Formula: see text] follows from convergence of their weighted geometric means. We also obtain multiplicative analogues of Schmidt type slow oscillation condition and Landau type two-sided condition for the convergence in particular. Besides, we introduce the concepts of [Formula: see text]convergence, [Formula: see text]convergence, [Formula: see text]convergence, [Formula: see text]convergence for sequences of intuitionistic fuzzy numbers (IFNs) and apply the aforementioned conditions to achieve convergence in intuitionistic fuzzy number space. Examples of sequences are also given to illustrate the proposed methods of convergence.

This paper offers several approaches for solving multiplicative second-order linear differential equations with variable exponentials, such as normalization and reduction to Riccati equations. In addition, in this paper, the multiplicative version of the Airy equation, which emerges in fluid mechanics, geophysics, and atomic physics, is solved using the multiplicative power series solution method. [ABSTRACT FROM AUTHOR]

Backpropagation (BP) is one of the most widely used algorithms for training feedforward deep neural networks (NNs). The algorithm requires a differentiable activation function and it performs computations of the gradient proceeding backward through the feedforward deep NN from the last layer through to the first layer. To calculate the gradient at a specific layer, the gradients of all layers are combined using the chain rule of calculus. One of the biggest disadvantages of BP is that it requires a large amount of training time. To overcome this issue, this paper proposes a modified BP algorithm with multiplicative calculus. Multiplicative calculus provides an alternative to the classical calculus, and it defines new kinds of derivative and integral forms in multiplicative form rather than addition and subtraction forms. The performance analyses are discussed in various case studies, and the results are given comparatively with the classical BP algorithm. It is found that the proposed modified BP algorithm converges in less time to the solution and thus provides fast training in the given case studies. It is also shown that the proposed algorithm avoids the problem of local minima. [ABSTRACT FROM AUTHOR]

Motivated by the notion of multiplicative calculus, more precisely multiplicative derivatives, we used the concept of subordination to create a new class of starlike functions. Because we attempted to operate within the existing framework of the design of analytic functions, a number of restrictions, which are in fact strong constraints, have been placed. We redefined our new class of functions using the three-parameter Mittag–Leffler function (Srivastava–Tomovski generalization of the Mittag–Leffler function), in order to increase the study’s adaptability. Coefficient estimates and their Fekete-Szegő inequalities are our main results. We have included a couple of examples to show the closure and inclusion properties of our defined class. Further, interesting bounds of logarithmic coefficients and their corresponding Fekete–Szegő functionals have also been obtained.

For solving non-linear equations, iterative root-finding methods are important because of the broad range of applications in science and engineering. We have constructed an iterative method based on multiplicative calculus in this paper. Some numerical results are performed to exposed the efficiency of proposed and earlier method. [ABSTRACT FROM AUTHOR]

In this paper, we suggest and analyze a new family of two-step predictor corrector type iterative schemes for solving nonlinear equations in the framework of multiplicative calculus. We also discuss the convergence criteria of these newly developed iterative methods. Some numerical examples will be given to illustrate the efficiency and performance of derived methods. [ABSTRACT FROM AUTHOR]

We introduce calculus and calculus for intuitionistic fuzzy values and prove some basic theorems by using multiplicative calculus which has useful tools to represent the concepts of introduced calculi. Besides, we construct some isomorphic mappings to interpret the relationships between calculus and calculus. This paper reveals also new calculi for fuzzy sets in particular. [ABSTRACT FROM AUTHOR]

In this paper, we establish an integral equality involving a multiplicative differentiable function for the multiplicative integral. Then, we use the newly established equality to prove some new Simpson's and Newton's inequalities for multiplicative differentiable functions. Finally, we give some mathematical examples to show the validation of newly established inequalities. [ABSTRACT FROM AUTHOR]

We define a Dirac system in multiplicative calculus by some algebraic structures. Asymptotic estimates for eigenfunctions of the multiplicative Dirac system are obtained. Eventually, some fundamental properties of the multiplicative Dirac system are examined in detail. [ABSTRACT FROM AUTHOR]

We define the multiplicative derivative and its properties on time scales. Then, we restate many concepts for multiplicative analysis such as derivative, Rolle's theorem, mean value theorem, and increasing‐decreasing property on time scales. We aim to create important fields of study by carrying this most important issue of multiplicative analysis, which has applications in economics, finance and many other fields, to time scale calculus. [ABSTRACT FROM AUTHOR]

When exponentials are employed to model procedures and efficacies appearing in real life, an additive derivative of this type of function does not exist. From this perspective, we define the Legendre equation in multiplicative analysis by several algebraic structures. Multiplicative Legendre polynomials of constituted problem are obtained by the power series solution method. Moreover, for the multiplicative Legendre equation, the generating function is obtained, and an integral presentation is constructed. Eventually, some fundamental spectral features of the multiplicative Legendre problem are analysed. [ABSTRACT FROM AUTHOR]

We define weighted geometric mean method of convergence for sequences in R + by using multiplicative calculus and obtain necessary and sufficient conditions under which convergence of sequences in R + follows from convergence of their weighted geometric means. We also obtain multiplicative analogues of Schmidt type slow oscillation condition and Landau type two-sided condition for the convergence in particular. Besides, we introduce the concepts of ⊕ convergence, ⊗ convergence, (N ¯ , p) − ⊕ convergence, (G ¯ , p) − ⊗ convergence for sequences of intuitionistic fuzzy numbers (IFNs) and apply the aforementioned conditions to achieve convergence in intuitionistic fuzzy number space. Examples of sequences are also given to illustrate the proposed methods of convergence. [ABSTRACT FROM AUTHOR]

Multiplicative calculus (MUC) measures the rate of change of function in terms of ratios, which makes the exponential functions significantly linear in the framework of MUC. Therefore, a generally non-linear optimization problem containing exponential functions becomes a linear problem in MUC. Taking this as motivation, this paper lays mathematical foundation of well-known classical Gauss-Newton minimization (CGNM) algorithm in the framework of MUC. This paper formulates the mathematical derivation of proposed method named as multiplicative Gauss-Newton minimization (MGNM) method along with its convergence properties. The proposed method is generalized for n number of variables, and all its theoretical concepts are authenticated by simulation results. Two case studies have been conducted incorporating multiplicatively-linear and non-linear exponential functions. From simulation results, it has been observed that proposed MGNM method converges for 12972 points, out of 19600 points considered while optimizing multiplicatively-linear exponential function, whereas CGNM and multiplicative Newton minimization methods converge for only 2111 and 9922 points, respectively. Furthermore, for a given set of initial value, the proposed MGNM converges only after 2 iterations as compared to 5 iterations taken by other methods. A similar pattern is observed for multiplicatively-non-linear exponential function. Therefore, it can be said that proposed method converges faster and for large range of initial values as compared to conventional methods. [ABSTRACT FROM AUTHOR]

In this paper, some integral inequalities of Hermite-Hadamard type for multiplicatively preinvex functions are established. Also, new inequalities involving multiplicative integrals are obtained for product and quotient of preinvex and multiplicatively preinvex functions.

In this study, *-directional derivative and *-subgradient are defined using the multiplicative derivative, making a new contribution to non-Newtonian calculus for use in non-smooth analysis. As for directional derivative and subgradient, which are used in the non-smooth optimization theory, basic definitions and preliminary facts related to optimization theory are stated and proved, and the *-subgradient concept is illustrated by providing some examples, such as absolute value and exponential functions. In addition, necessary and sufficient optimality conditions are obtained for convex problems.

In this paper, we firstly obtain two identities for multiplicative differentiable functions. Then by using these identities, we establish Ostrowski and Simpson type inequalities for multiplicative integrals. At the end we give detail applications of our main results. [ABSTRACT FROM AUTHOR]