# Non-Newtonian Calculus

... In the light of these results, now there is a need to define the concepts of ⊕ limit and ⊗ limit for intuitionistic fuzzy valued functions(IFVF) in order to extend the aforementioned advantages to intuitionistic fuzzy calculus, and a need to construct corresponding calculi. The aim of this paper is to define ⊕ limit and ⊗ limit for IFVFs and construct corresponding intuitionistic fuzzy calculi by utilizing the tools of multiplicative calculus [8,17] which has close relation with the new calculi. The constructed calculi reveals also a new calculi for fuzzy sets in the absence of hesitancy. ...

... Besides, many other operations on IFVs such as integrals [1,13], intuitionistic fuzzy aggregation operators [18,19], convergence methods [20], infinite series and products [22] include again multiplication and division of membership-nonmemberships. On the other hand, multiplication and division operations are also crucial in multiplicative calculus and the tools of multiplicative calculus are useful to represent and to handle some intuitionistic fuzzy concepts. For this reason, we here give some basic concepts of multiplicative calculus [5,8,17] which will be used in Sections 2-4. Definition 1.6. ...

... Here, the tools of multiplicative calculus [8,17] may be useful to represent (2.2). Besides, we have ...

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.

... In the light of these results, now there is a need to define the concepts of ⊕ limit and ⊗ limit for intuitionistic fuzzy valued functions (IFVF) in order to extend the aforementioned advantages to intuitionistic fuzzy calculus, and a need to construct corresponding calculi. The aim of this paper is to define ⊕ limit and ⊗ limit for IFVFs and construct corresponding intuitionistic fuzzy calculi by utilizing the tools of multiplicative calculus [8,17] which has close relation with the new calculi. The constructed calculi reveals also a new calculi for fuzzy sets in the absence of hesitancy. ...

... Besides, many other operations on IFVs such as integrals [1,13], intuitionistic fuzzy aggregation operators [18,19], convergence methods [20], infinite series and products [22] include again multiplication and division of membership-nonmemberships. On the other hand, multiplication and division operations are also crucial in multiplicative calculus and the tools of multiplicative calculus are useful to represent and to handle some intuitionistic fuzzy concepts. For this reason, we here give some basic concepts of multiplicative calculus [5,8,17] which will be used in Sections 2-4. ...

... Here, the tools of multiplicative calculus [8,17] may be useful to represent (2.2). Besides, we have ...

... (1. 6) Its solution is the function ...

... 5. Extension of the solutions. 6. Upper and lower bounds of the unknown solutions and existence of maximal and minimal solutions. ...

... Problem 1. 6.1 Determine the order of the following MDEs 1. y * * * − * sin * x · * y * * + * e x · * y = (cos * x) 2 * , x ∈ R * . ...

In this Book, we give a definition for multiplicative differential equation(shortly MDE), order of MDE and solution of MDE. We give a classification of the MDEs and they are described the basic problems for the MDEs.

... Definition 1. 6. For x ∈ R , x = 0 , the number ...

... We have 1 + cos e 2 · x / e 2 = e + e cos(log(e 2 · x)) / e 2 This completes the proof. 6. cos x + cos y = e 2 · cos (x + y) / e 2 · cos (x − y) / e 2 . ...

... Problem 2.12. Let P (6,9), N = e 2. Let X(2, 11). Find Ω l X. ...

... In the light of these results, now there is a need to define the concepts of ⊕ limit and ⊗ limit for intuitionistic fuzzy valued functions(IFVF) in order to extend the aforementioned advantages to intuitionistic fuzzy calculus, and a need to construct corresponding calculi. The aim of this paper is to define ⊕ limit and ⊗ limit for IFVFs and construct corresponding intuitionistic fuzzy calculi by utilizing the tools of multiplicative calculus [12,13] which has close relation with the new calculi. The constructed calculi reveals also a new calculi for fuzzy sets in the absence of hesitancy. ...

... The concepts of multiplicative calculus are given below(see [12,13,18]). Definition 1.6. ...

... Here, the tools of multiplicative calculus [12,13] may be useful to represent (2). Besides, we have ...

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. This paper reveals also new calculi for fuzzy sets in particular.

... Grossman and Katz [8] introduced non-Newtonian calculus as an alternative to classical calculus in the period from 1967 till 1972. They defined an infinite family of calculus which includes some special calculi such as geometric calculus, harmonic calculus, bigeometric calculus, anageometric calculus. ...

... 1. ı is one-to-one. 2. ı is on A and onto B. p [1,8]. ...

... If a lim m!1 S m ¼ S, then it is written a P 1 n¼1 a n ¼ S: If the limit a lim m!1 S m is not available or equal to À1 or þ1, then it is said that the series a P 1 n¼1 a n is aÀdivergent [5]. The concepts of aÀconvergent sequence, non-Newtonian metric space, non-Newtonian completeness, non-Newtonian upper bound, non-Newtonian supremum, non-Newtonian open set, non-Newtonian closed set are discussed in detail in [1,5,8,11]. ...

In this article, we study the non-Newtonian version of C*-algebras. Further, we generalize some results which hold for the classical C*-algebras. We also discuss some illustrative examples to show accuracy and effectiveness of the new findings. If we take the identity function I instead of the generators a and b in the construction of the set C(N), then non-Newtonian C*-algebras turn into the classical C*-algebras, so our results are stronger than some knowledge and facts in the most existing literature.

... Multiplicative calculus [1,2] is alternative to classical calculus and uses ratios instead of differences in order to measure deviations and compare numbers. The operations multiplication and division are crucial in multiplicative calculus and many concepts such as differentiation and integration are based on these operations. ...

... A sequence (u n ) in (R + , | · | * ) is said to be * convergent to a ∈ R + if for all ε > 1 there exists n 0 ∈ N such that d * (u n , a) = | u n a | * < ε whenever n > n 0 , and denoted by u n * → a. Sequence (u n ) is said to be * bounded if there exists B > 1 such that |u n | * < B for all n ∈ N. For further concepts such as multiplicative derivative, multiplicative differential equations and the Newtonian counterparts, see [1,2,[6][7][8][9][10][11]. ...

... Let X be a non-empty set. Then, an Atanassov's intuitionistic fuzzy set(A-IFS) [13] has the following form: 1] is called membership function and ν : X → [0, 1] is called non-membership function. For any x ∈ X, 0 ≤ μ A (x) + ν A (x) ≤ 1. ...

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.

... The non-Newtonian calculus comprising of the branches of geometric, harmonic, quadratic, bigeometric, biharmonic and biquadratic calculus introduced and studied by Grossman and Katz [13]. Bigeometric calculus which is one of the most popular non-Newtonian calculus is worked by many researchers. ...

... In other words, one uses geometric arithmetic on function arguments and values in the bigeometric calculus. [2,13,15]. ...

... 13,15]. ...

... For details and a proof we refer to [35]. If one is familiar with non-Newtonian calculus [22], Equation (1) can be recognized as a φ-arithmetic operation, a φ-sum speci cally [31]. This suggests that a bivariate Archimedean copula endows the interval [ , ] with a semi-group structure. ...

... where Φ and Φ − are respectively the cumulative distribution and the quantile function of a standard normal, and σ > is a scale parameter (equal to the standard deviation of log(X)). Equation (22) can be obtained by substituting y = F − (x) in Equation (4), and then by using the properties of the Gaussian integral in Φ, when multiplied by an exponential function [40]; nally, reverting the substitution y = F − (x) leads to the desired result. Figure 1 shows some examples of the lognormal generator for di erent values of σ. ...

... Notice that the quantity Φ(Φ − (y) − σ), with σ > , is a well-known distortion function in actuarial mathematics, usually called Wang transform, and it has powerful applications in the elds of asset pricing, risk theory and utility theory [29,50]. Furthermore, in terms of non-Newtonian calculus [22], it represents the pseudo-di erence of a variable y and a constant σ: just notice that, given the continuity of Φ − , one has [31,39]. ...

A novel generating mechanism for non-strict bivariate Archimedean copulas via the Lorenz curve of a non-negative random variable is proposed. Lorenz curves have been extensively studied in economics and statistics to characterize wealth inequality and tail risk. In this paper, these curves are seen as integral transforms generating increasing convex functions in the unit square. Many of the properties of these “Lorenz copulas”, from tail dependence and stochastic ordering, to their Kendall distribution function and the size of the singular part, depend on simple features of the random variable associated to the generating Lorenz curve. For instance, by selecting random variables with a lower bound at zero it is possible to create copulas with asymptotic upper tail dependence. An “alchemy” of Lorenz curves that can be used as general framework to build multiparametric families of copulas is also discussed.

... If is chosen to be the identity function, then one gets the standard arithmetic, from which the traditional (Newton-Leibniz) calculus is developed. For other choices of , we can get an infinitude of other arithmetics from which Grossman and Katz produced a series of non-Newton calculi, compiled in the seminal book of 1972 [1]. Choosing exp, the geometric arithmetic are obtained, respectively, and also, we have [15] helps to prove i). ...

... The set of numbers such that x and is the square root of x , denoted by . x[1,10]. The n Fibonacci number n F is defined recursively by In[7] we introduced non-Newtonian versions of Fibonacci and Lucas numbers as follows respectively:Proof.ii) ...

... Besides, many other operations on IFVs such as integrals [18,19], intuitionistic fuzzy aggregation operators [16,17], convergence methods [9], infinite series and products [8] include again multiplication and division of membership-nonmemberships. On the other hand, multiplication and division operations are also crucial in multiplicative calculus and the tools of multiplicative calculus are useful to represent and to handle some intuitionistic fuzzy concepts. For this reason, we here give some basic concepts of multiplicative calculus [12,13,20] which will be used in Sections 2-4. Definition 1.6. ...

... Here, the tools of multiplicative calculus [12,13] may be useful to represent (2.2). Besides, we have ...

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.

... In 1972, Grossman and Katz [4] laid the foundations of non-newtonian calculus which modify the calculi initiated by Gottfried Wilhelm Leibnitz and Isaac Newton in the 17th century. A generator is a one-to-one function : A →. ...

... Now, we briefly mention several known concepts on non-Newtonian calculus. The details can be found in [4,5,25]. ...

In this paper, we deal with complex and bicomplex numbers with respect to the geometric calculus, and we obtain the set of complex numbers with respect to the geometric calculus C(GC) is a field and the set of bicomplex numbers with respect to the geometric calculus BC(GC) is a vector space on the field C(GC) by defining addition and multiplication operations on the sets of such numbers. Also, we give the concepts of norm, metric, sequence, convergence of a sequence, Cauchy sequence and completeness in the settings C(GC) and BC(GC). Moreover, we discuss bicomplex versions with respect to geometric calculus of some well-known inequalities. This paper is a new and important addition to the current literature thanks to its applications in different areas and the obtained results unify, private and complement the corresponding results.

... If is chosen to be the identity function, then one gets the standard arithmetic, from which the traditional (Newton-Leibniz) calculus is developed. For other choices of , we can get an infinitude of other arithmetics from which Grossman and Katz produced a series of non-Newton calculi, compiled in the seminal book of 1972 [1]. Choosing exp, the geometric arithmetic are obtained, respectively, and also, we have ...

... It turns out that ι (x) = β α −1 (x) for every number x in A and that ι (ṅ) =n for every integer n [1]. ...

... Therefore, using multiplicative analysis instead of classical analysis provides a better physical evaluation of such events. The multiplicative analysis also gives better-thannormal results in many fields, including image processing and artificial intelligence, including computer science, finance, economics, biology, and demography [1][2][3][4][5][6][7][8][9][10]. In recent years, many researchers have conducted various studies on multiplicative analysis and have obtained effective results [11][12][13][14][15][16][17][18][19][20][21][22]. ...

In this work, the definition of the first and second types of multiplicative
Chebyshev differential equations is given and the solutions of these
equations are investigated with the help of multiplicative power series.
Also, the properties of first and second type multiplicative Chebyshev
polynomials are given and proved. Finally, these studies are supported with
numerical examples.

... However, here we do not describe this alternative construction because we do not use it for building formulas from QFT explored in this paper. (Burgin & Czachor, 2020) or in (Grossman & Katz, 1972). ...

The problem of infinities in quantum field theory (QFT) is a longstanding problem in particle physics. To solve this problem, different renormalization techniques have been suggested but the problem persists. Here we suggest another approach to the elimination of infinities in QFT, which is based on non-Diophantine arithmetics – a novel mathematical area that already found useful applications in physics, psychology, and other areas. To achieve this goal, new non-Diophantine arithmetics are constructed and their properties are studied. In addition, non-Diophantine integration is developed in these arithmetics. These constructions allow using constructed non-Diophantine arithmetics for computing integrals associated with Feynman diagrams. Although in the conventional QFT such integrals diverge, their non-Diophantine counterparts are convergent and rigorously defined. As the result, QFT becomes consistent with quantum experiments.

... In seventeenth century, Newton and Leibnitz created the differential and integral calculus concept based on subtraction and addition operation. Later on 1970's, Grossman and Katz [1] developed a different definition of differential and integral calculus that utilize the multiplication and division operation instead of addition and subtraction. This definition of differential and integral calculus named as Multiplicative Calculus. ...

Problems like population growth, continuous stirred tank reactor (CSTR) and ideal gas are studied from the last four decades in the field medicine science, Engineering and applied science, respectively. One of the main motivation was to understand the pattern of such issues and how to fix them. With the help of applied Mathematics, such problems can be converted or modeled by nonlinear expressions with similar properties and the required solution can be obtained by iterative techniques. In this manuscript, we proposed a new iterative scheme for multiple roots (without prior knowledge of multiplicity m ) by adopting multiplicative calculus rather than the standard calculus. The base of our scheme is on the well-known Schröder method and we retain the same second-order of convergence. In addition, we extend the order of convergence from second to fourth by constructing a two-step joint Schröder scheme with hybrid approach of ordinary and multiplicative calculus. Some numerical examples are tested to find the roots of nonlinear equations and results are found to be competent as compared to ordinary derivative methods. Finally, the convergence of schemes is also analyzed by basin of attractions that also support the theoretical aspects.

... N. Yalcin and M. Dedeturk give solutions of multiplicative differential equations via the multiplicative differential transform method [12] and via the multiplicative power series method [13] . More details on the multiplicative analysis topic can be found in [14][15][16][17][18][19][20][21][22][23][24][25][26]. In this work, we give solutions of MIE via the multiplicative power series method (MPSM) ...

In this study, definitions of types of multiplicative integral equations are given. And solutions of different types of multiplicative integral equations are investigated using the multiplicative power series method. These are supported by numerical examples.

... Proof 2.5. 6 We have ...

... There exist too many studies on fixed-point theory in different spaces [1][2][3][4][5][6][7][8][9][10][11][12]. Also, there are many applications of the theory and mappings that meet certain conditions of contraction and have been a crucial area of different research works. ...

TThe work of non-Newtonian calculus was begun in 1972. This calculus provides a different area to the classical one. Non-Newtonian metric concept was defined in 2002 by Basar and Cakmak. Then Binbaşıoğlu et al. had given the metric spaces of non-Newtonian in 2016. Also, they started to the fixed-point theory by defining some topological properties in non-Newtonian metric spaces. In this work, we give some fixed-point theorems and results for self-mappings satisfying certain conditions in the non-Newtonian metric spaces.

... Geometric calculus is an alternative to the usual calculus of Newton and Leibniz introduced by Grossman and Katz [9]. It provides differentiation and integration tools based on multiplication instead of addition. ...

The aim of this paper is to investigate relation between PC matrices and G-orthogonal matrices. For this purpose, first of all, we define G- orthogonal and G-skew-symmetric matrices according to geometric calculus. Then, we give a relation between G-orthogonal matrices and PC matrices using the useful diagram constructed by Cayley formula in real mean. Finally we define G-Cayley formula according to G-calculus and we verify our theory with some examples.

... İlk önce klasik analiz, geometrik analiz, harmonik analiz ve kuadratik analizi daha sonra ise bigeometrik, biharmonik ve bikuadratik analizleri tanımlamışlardır. Bu analizlerin temel tanım ve teoremlerini 1972 yılında bir kitap çerçevesinde tamamlamışlardır (Grossman and Katz, 1972). Klasik analizde geçen tüm kavramların Non-Newtonian analizde bir karşılığı vardır. ...

... Daha sonra ünlü bilim insanı olan Michael Grossman ve Robert Katz 1967 ile 1970 yılları arasında yaptıkları çalışmalar sonucunda geometrik analiz, harmonik analiz, quadratik analiz, bigeometrik analiz, biharmonik analiz, biquadratik analiz ve anageometrik analiz olarak adlandırılan yeni analizler tanımlamışlardır. Bu yeni analizlerin hepsi birden Newtonyen olmayan analiz olarak da adlandırılmış ve bunlarla ilgili bazı temel tanım ve teoremler verilmiştir (Grossman and Katz, 1972). Newtonyen olmayan analizler içinde önemli bir tanesi ise geometrik analizdir. ...

... In 1960s, multiplicative analysis was firstly presented as an alternative to usual analysis [1,2]. At the same time, this analysis is entitled Geometric analysis, which is one of the sub-branches of Non-Newtonian analysis. ...

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.

... The belief is at the very heart of various proofs of the fundamental security of quantum cryptography, but it has been recently challenged in a series of papers 2020a, b;Czachor & Nalikowski, 2021). The claim is that it is possible to reconstruct singlet-state probabilities in a local hidden-variable way if one assumes that the hidden-variable model is constructed by means of a non-Newtonian calculus (Grossman & Katz, 1972;Pap, 1993;Burgin & Czachor, 2020), a possibility not taken into account so far. Non-Newtonian calculus involves derivatives and integrals that are linear with respect to projective arithmetics of real numbers (Burgin & Czachor, 2020;Burgin, 2010), but typically are not linear with respect to the usual (Diophantine) arithmetic of reals (Mesiar, 1995;Pap, 2002). ...

A class of quantum probabilities is reformulated in terms of non-Newtonian calculus and projective arithmetic. The model generalizes spin-1/2 singlet state probabilities discussed in Czachor (Acta Physica Polonica:139 70–83, 2021) to arbitrary spins s. For s→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\rightarrow \infty$$\end{document} the formalism reduces to ordinary arithmetic and calculus. Accordingly, the limit “non-Newtonian to Newtonian” becomes analogous to the classical limit of a quantum theory.

... In 1967, Grossman and Katz, created the first non-Newtonian calculation system, called geometric calculation. Over the next few years they had created an infinite family of non-Newtonian calculi, thus modifying the classical calculus introduced by Newton and Leibniz in the 17th century each of which differed markedly from the classical calculus of Newton and Leibniz known today as the non-Newtonian calculus or the multiplicative calculus, where the ordinary product and ratio are used respectively as sum and exponential difference over the domain of positive real numbers see [8]. This calculation is useful for dealing with exponentially varying functions. ...

In this paper, we first prove two new identities for multiplicative differentiable functions. Based on this identity, we establish a midpoint and trapezoid type inequalities for multiplicatively convex functions. Applications to special means are also given.

... A hierarchy of arithmetics leads to a hierarchy of calculi. In the terminology of Grossman and Katz (Grossman and Katz 1972) the calculi are non-Newtonian. In particular, a derivative of a function A ∶ ℝ → ℝ reads where R k is the arithmetic of the domain, and R l is the arithmetic of the codomain. ...

Local hidden-variable model of singlet-state correlations discussed in Czachor (Acta Phys Polon A 139:70, 2021a) is shown to be a particular case of an infinite hierarchy of local hidden-variable models based on an infinite hierarchy of calculi. Violation of Bell-type inequalities can be interpreted as a ‘confusion of languages’ problem, a result of mixing different but neighboring levels of the hierarchy. Mixing of non-neighboring levels results in violations beyond the Tsirelson bounds.

... Two operations of multiplicative calculus are multiplicative derivative and multiplicative integral. We refer to Grossman and Katz [1], Stanley [2], Campbell [3], Grossman [4,5], Jane Grossman [6,7], for different types of Non-Newtonian calculus and its applications. Bashirov et al. [8], gave the complete mathematical description of multiplicative calculus. ...

Multiplicative analysis, whic is an alternative to the classical analysis defined by the additive arithmetic, and built on the multiplicative arithmetic, offers a new perspective to the mathematical problems encountered in science and engineering. In this paper, using matrix methods, we obtained rotation pole in multiplicative one-parameter motion on the plane kinematics in multiplicative motions and multiplacative pole orbits, multiplacative accelerations and multiplacative combinations of accelerations, multiplicative acceleratıon poles of the motıons. Moreover, some new theorems are given.

... An explanation of the non-linearity of interactions within complex systems is their non-homogenous and non-additive attributes [113,114]. For a simple example, consider function f that is regarded homogeneous if it exhibits the following property: ...

The quest for simple solutions is not new in machine learning (ML) and related methods such as genetic programming (GP). GP is a nature-inspired approach to the automatic programming of computers used to create solutions to a broad range of computational problems. However, the evolving solutions can grow unnecessarily complex, which presents considerable challenges. Typically, the control of complexity in GP means reducing the sizes of the evolved expressions – known as bloat-control. However, size is a function of solution representation, and hence it does not consistently capture complexity across diverse GP applications. Instead, this thesis proposes to estimate the complexity of the evolving solutions by their evaluation time – the computational time required to evaluate a GP evolved solution on the given task. After all, the evaluation time depends not only on the size of the evolved expressions but also on other aspects such as their composition, thus acting as a more nuanced measure of model complexity than the expression size alone. Also, GP evaluates all the solutions in a population identically to determine their relative performance, for example, with the same dataset. Therefore, evaluation time can consistently compare the relative complexity.
To discourage complexity using the proposed evaluation time, two approaches are used. The first approach explicitly penalises models with long evaluation times by customising well-tested techniques that traditionally control the size. The second uses a novel technique that implicitly discourages long evaluation times by incorporating a race condition in the GP process. The proposed methods yield accurate yet simple solutions; furthermore, the implicit method improves the runtime and training speed of GP.
Across a diverse suite of GP applications, the evaluation time methods proffer several qualitative advantages over the bloat-control methods. They effectively manage the functional complexity of regression models to enable them to predict unseen data (generalise) better than those produced by bloat-control. In two feature engineering applications, they decrease the number of features – principally responsible for model complexity – while bloat-control does not. In a robot control application, they evolve accurate and efficient routines – efficient routines use fewer time steps to complete their tasks; bloat-control could not detect the efficiency of the programs. In Boolean logic problems where size emerges as the major cause of complexity, these methods are not hindered and perform at least as well as bloat-control. Overall, the proposed system characterises and manages various forms of complexity; also, it is broadly applicable and, hence, suitable for an automatic programming system.

... The first traces of the pseudo-analysis goes to Grossman and Katz [1] and Burgin [2] (what today is called g-calculus, see [3]), then Maslov [4] (what today is called idempotent analysis). These previous results were a starting point to develope a complete unified theory under the name pseudo-analysis [5][6][7][8][9][10][11], and as a special case the g-calculus [3]. ...

The theory of the pseudo-analysis is based on a special real semiring (called also tropical semiring). This theory enables a unified approach to three important problems as nonlinearity, uncertainty and optimization, with many applications. There are presented applications in fuzzy logics and fuzzy sets, utility theory, Cumulative Prospect theory of partial differential equations.

... Secondly, let's consider the multiplicative calculus theory. Multiplicative calculus was firstly presented as an alternative to usual calculus by Grossman and Katz [25,26]. This is also entitled simultaneously Geometric calculus which is one of sub-branches of non-Newtonian calculus. ...

In multiplicative fractional calculus, the well-known Dirac system in fractional calculus is redefined. The aim 5 of this study is to analyze some spectral properties such as self-adjointness of the operator, structure of all eigenvalues, 6 orthogonality of distinct eigenfunctions, etc. for this system. Moreover, Green’s function in multiplicative case is
7 reconstructed for this system.

... Recently, it has been shown that the non-Newtonian/multiplicative calculus introduced by Grossman and Katz [1] is very useful in some problems of actuarial science, finance, economics, biology, demography, pattern recognition, signal processing, thermostatistics, and quantum information theory [2][3][4][5][6]. Additionally, a mathematical problem, which is difficult or impossible to solve in one calculus, can be easily revealed through another calculus [7,8]. ...

The universal principle obtained by Emmy Noether in 1918 asserts that the invariance of a variational problem with respect to a one-parameter family of symmetry transformations implies the existence of a conserved quantity along with the Euler–Lagrange extremals. Here, we prove Noether's theorem for the recent non-Newtonian calculus of variations. The proof is based on a new necessary optimality condition of DuBois–Reymond type.

... Recently, it has been shown that the non-Newtonian/multiplicative calculus introduced by Grossman and Katz [1] is very useful in some problems of actuarial science, finance, economics, biology, demography, pattern recognition, signal processing, thermostatistics, and quantum information theory [2][3][4][5][6]. Additionally, a mathematical problem, which is difficult or impossible to solve in one calculus, can be easily revealed through another calculus [7,8]. ...

The universal principle obtained by Emmy Noether in 1918, asserts that the invariance of a variational problem with respect to a one-parameter family of symmetry transformations implies the existence of a conserved quantity along the Euler-Lagrange extremals. Here we prove Noether's theorem for the recent non-Newtonian calculus of variations. The proof is based on a new necessary optimality condition of DuBois-Reymond type.

... for every number x in A and that () ı n n for every integer n (Grossman and Katz, 1972 (Sağır and Erdoğan, 2019a). ...

On_-Continuity_and_-Uniform_Continuity_of_Some_Non-Newtonian_Superposition_Operators

... Multiplicative calculus was introduced by Grossman & Katz (Grossman, 1979;Grossman & Katz, 1972) in 1967 as an alternative to classical calculus. This type of calculus is also known as non-Newtonian because of its difference from classical calculus of Newton and Leibniz. ...

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

... The concept of multiplicative calculus, in which the role of addition and subtraction are replaced by multiplication and division, was not the interest of researchers for a long time, even though it was defined by Grossman and Katz [8] in the period from 1967 till 1970 (published a book called Non-Newtonian Calculus in 1972), and Stanley [20] published a paper 'A multiplicative calculus' in 1999. But in 2008, Bashirov et al. [3] draw the attention of researchers especially in the field of analysis by highlighting various properties like multiplicative derivatives, multiplicative integrals, etc. ...

In this paper, we discuss some common fixed point theorems for compatible mappings of type (E) and R−weakly commuting mappings of type (P) of a complete b−multiplicative metric space along with some examples. As an application, we establish an existence and uniqueness theorem for a solution of a system of multiplicative integral equations. In the last section, we introduce the concept of R−multiplicative metric space by giving some examples and at the end of the section, we give an open question. Keywords: compatible mappings of type (E), R−weakly commuting mappings of type (P), multiplicative metric space, b−multiplicative metric space, common fixed point, compatible mapping, R−metric space, orthogonal set. 2010 MSC: Primary 47H10; Secondary 54H25.

... A complete account on non-Diophantine arithmetics can be found in the recent book by Burgin and Czachor [11]. There, for example, the authors show how non-Diophantine arithmetics are crucial for the development and application of different kinds of non-Newtonian calculi [3,6,12]. ...

We present three classes of abstract prearithmetics, $\{\mathbf{A}_M\}_{M \geq 1}$, $\{\mathbf{A}_{-M,M}\}_{M \geq 1}$, and $\{\mathbf{B}_M\}_{M > 0}$. The first one is weakly projective with respect to the nonnegative real Diophantine arithmetic $\mathbf{R_+}=(\mathbb{R}_+,+,\times,\leq_{\mathbb{R}_+})$, the second one is weakly projective with respect to the real Diophantine arithmetic $\mathbf{R}=(\mathbb{R},+,\times,\leq_{\mathbb{R}})$, while the third one is projective with respect to the extended real Diophantine arithmetic $\overline{\mathbf{R}}=(\overline{\mathbb{R}},+,\times,\leq_{\overline{\mathbb{R}}})$. In addition, we have that every $\mathbf{A}_M$ and every $\mathbf{B}_M$ are a complete totally ordered semiring, while every $\mathbf{A}_{-M,M}$ is not. We show that the projection of any series of elements of $\mathbb{R}_+$ converges in $\mathbf{A}_M$, for any $M \geq 1$, and that the projection of any non-oscillating series series of elements of $\mathbb{R}$ converges in $\mathbf{A}_{-M,M}$, for any $M \geq 1$, and in $\mathbf{B}_M$, for all $M > 0$. We also prove that working in $\mathbf{A}_M$ and in $\mathbf{A}_{-M,M}$, for any $M \geq 1$, and in $\mathbf{B}_M$, for all $M>0$, allows to overcome a version of the paradox of the heap.

... Under these considerations the resulting arithmetic is called proportional arithmetic. Grossman and Katz used this way to operate and they presented the Bigeometric Calculus (see [21]). Similarly, in [2] and [3], Bashirov in-troduces the Multiplicative Calculus and its applications. ...

On the set of positive real numbers, multiplication, represented by $\oplus$, is considered as an operation associated with the notion of sum, and the operation $a\odot b=a^{ln(b)}$ represents the meaning of the traditional multiplication. The triple $(\mathbb{R}^{+},\oplus,\odot)$ forms an ordered and complete field in which derivative and integration operators are defined analogously to the Differential and Integral Calculus. In this article, we present the proportional arithmetic and we construct the theory of ordinary proportional differential equations. A proportional version of Gronwall inequality, Gompertz’s function, the q-Periodic functions, proportional heat, and wave equations as well as a proportional version of Fourier’s series are presented. Furthermore, a non-Newtonian logistic growth model is proposed.

Our aim is to give concepts of density, statistical convergence, statistically Cauchy sequence, Cesàro summability and \(\lambda -\)statistical convergence on non-Newtonian calculus. Especially, if a generator on this calculus is determined as identity function, all results in the classical case are reached.

Problems such as population growth, continuous stirred tank reactor (CSTR), and ideal gas have been studied over the last four decades in the fields of medical science, engineering, and applied science, respectively. Some of the main motivations were to understand the pattern of such issues and how to obtain the solution to them. With the help of applied mathematics, these problems can be converted or modeled by nonlinear expressions with similar properties. Then, the required solution can be obtained by means of iterative techniques. In this manuscript, we propose a new iterative scheme for computing multiple roots (without prior knowledge of multiplicity m) based on multiplicative calculus rather than standard calculus. The structure of our scheme stands on the well-known Schröder method and also retains the same convergence order. Some numerical examples are tested to find the roots of nonlinear equations, and results are found to be competent compared with ordinary derivative methods. Finally, the new scheme is also analyzed by the basin of attractions that also supports the theoretical aspects.

Grossman and Katz (five decades ago) suggested a new definition of differential and integral calculus which utilizes the multiplicative and division operator as compared to addition and subtraction. Multiplicative calculus is a vital part of applied mathematics because of its application in the areas of biology, science and finance, biomedical, economic, etc. Therefore, we used a multiplicative calculus approach to develop a new fourth-order iterative scheme for multiple roots based on the well-known King’s method. In addition, we also propose a detailed convergence analysis of our scheme with the help of a multiplicative calculus approach rather than the normal one. Different kinds of numerical comparisons have been suggested and analyzed. The obtained results (from line graphs, bar graphs and tables) are very impressive compared to the earlier iterative methods of the same order with the ordinary derivative. Finally, the convergence of our technique is also analyzed by the basin of attractions, which also supports the theoretical aspects.

Citation: Moumen, A.; Boulares, H.; Meftah, B.; Shafqat, R.; Alraqad, T.; Ali, E.E.; Khaled, Z. Multiplicatively Simpson Type Inequalities via Fractional Integral. Symmetry 2023, 15, 460. https://doi.org/10.3390/ sym15020460 Academic Editors: Hassen Fourati, Abstract: Multiplicative calculus, also called non-Newtonian calculus, represents an alternative approach to the usual calculus of Newton (1643-1727) and Leibniz (1646-1716). This type of calculus was first introduced by Grossman and Katz and it provides a defined calculation, from the start, for positive real numbers only. In this investigation, we propose to study symmetrical fractional multiplicative inequalities of the Simpson type. For this, we first establish a new fractional identity for multiplicatively differentiable functions. Based on that identity, we derive new Simpson-type inequalities for multiplicatively convex functions via fractional integral operators. We finish the study by providing some applications to analytic inequalities.

Various scholars have lately employed a wide range of strategies to resolve specific types
of symmetrical fractional differential equations. In this paper, we propose a new fractional identity
for multiplicatively differentiable functions; based on this identity, we establish some new fractional
multiplicative Bullen-type inequalities for multiplicative differentiable convex functions. Some
applications of the obtained results are given.

This book is devoted to the multiplicative Euclidean and non-Euclidean geometry. It summarizes the most recent contributions in this area. The book is intended
for senior undergraduate students and beginning graduate students of engineering
and science courses. The book contains eight chapters. The chapters in the book
are pedagogically organized. Each chapter concludes with a section with practical
problems.

In this study, the Volterra integral equations are defined in the non-Newtonian calculus with the aid of ∗-integral. The focus of this study is obtaining the solution of linear non-Newtonian integral equations. The ∗-resolvent kernel method, successive approximations method and Adomian decomposition method are applied for solving non-Newtonian integral equations in the non-Newtonian sense. Some numerical examples are presented to explain the procedure of these methods. Finally, a model that describes a type of the exponential growth model is proposed in the sense of non-Newtonian calculus as an example.

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.

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.

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