Applications and Reception

This website concerns the systems of non-Newtonian calculus, multiplicative calculus, and nonclassical arithmetic created by Michael Grossman and Robert Katz between 1967 and 1970. The website was last edited on 24 June 2021. 

In their Non-Newtonian Calculus, Michael Grossman and Robert Katz were apparently the first to use some significant mathematical systems: the nonclassical arithmetics ( i.e., the complete ordered fields distinct from the real number system), which they used to construct the non-Newtonian calculi. In fact, as indicated in Appendix 3, the distinguished mathematician Mark Burgin (University of California at Los Angeles) asserted that the construction of the non-Newtonian calculi was indeed the first application of the nonclassical arithmetics. The nonclassical arithmetics are alternatives to the real number system. Like the non-Newtonian calculi, the nonclassical arithmetics provide a wide variety of mathematical tools for use in science, engineering, and mathematics. Various applications of nonclassical arithmetics are discussed below.

Non-Newtonian calculus was recommended as a featured topic for the 21st-century college-mathematics-curriculum, in the keynote speech at the 27th International Conference on Technology in Collegiate Mathematics (ICTCM) on 13 March 2015. The keynote speaker was the mathematics-educator Eric Gaze. His speech is entitled "Complexity, Computation, and Quantitative Reasoning: A Mathematics Curriculum for the 21st Century". (The conference was sponsored by Pearson PLC, the largest education company and the largest book publisher in the world; and the Electronic Proceedings of the conference were hosted by Math Archives (archives.math.utk.edu) with partial support provided by the National Science Foundation.) [224]

A special-session (mini-symposium) called "Non-Newtonian Calculus" was held at the 17th International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE), 4-8 July 2017, at Rota, Cadiz - Spain. The 2017 CMMSE Scientific Committee included over 50 distinguished scholars in a variety of fields, and CMMSE conferences are attended by numerous scholars worldwide. The special-session on non-Newtonian calculus was organized by Fernando Córdova-Lepe and Marco Mora, both from Universidad Católica del Maule in Chile. Included among the papers delivered at the special session on non-Newtonian calculus: "Non-Newtonian Calculus. Theory and Applications" by Fernando Cordova-Lepe and Marco Mora (both from Universidad Católica del Maule in Chile); "Introductory elements for the development of a multiplicative statistic" by Carol Pavez Rojas, Fernando Córdova-Lepe, and Karina Vilches Ponce (all from Universidad Católica del Maule in Chile); "Linearity and its algebra in the bi-geometrical context" (i.e., multiplicative linear-algebra) by Fernando Córdova-Lepe, Rodrigo del Valle, and Karina Vilches Ponce (all from Universidad Católica del Maule in Chile); and "Bigeometric Complex Calculus" by Agamirza E. Bashirov and Sajedeh Norozpour (both from Eastern Mediterranean University in North Cyprus). "Bigeometric Complex Calculus" was also presented at the special-session "Mathematical Modeling and Computational PDE". [324, 383, 384, 385]  From a translation (from Spanish) of the conference-announcement: "This mini symposium [on non-Newtonian calculus] will be one of the first international meetings of a dispersed scientific community that has worked or is working on this topic, and [is a landmark] in the history of non-Newtonian calculus. Taking into account the novelty of the subject, all topics related to non-Newtonian calculus (theory and applications) are welcome."

The first application of non-Newtonian calculus was made by James R. Meginniss (Claremont Graduate School and Harvey Mudd College). In his 1980 article "Non-Newtonian calculus applied to probability, utility, and Bayesian analysis" [16], he created a theory of probability that is adapted to human behavior and decision making. In his article [16] he asserts: "A substantial body of evidence indicates that people really don’t process probability information in the way that the classical or Bayesian models suggest. ... The purpose of this paper is to present a new theory of probability that is adapted to human behavior and decision making. Two basic ideas will be used: that probabilities do not obey the laws of ordinary arithmetic and calculus, but instead are governed by the laws of one of the non-Newtonian calculi and its corresponding arithmetic; and that ... . With these two ideas it is possible to reform the calculus of probabilities without abandoning the basic formalism of probability.”

Non-Newtonian Calculus [15] is mentioned in the book The Rainbow of Mathematics: A History of the Mathematical Sciences by the eminent mathematics-historian Igor Grattan-Guinness. [6].  

Wave physics, fractals, arithmetics, and non-Newtonian calculus are the central topics in the article “Waves along fractal coastlines: from fractal arithmetic to wave equations” by Marek Czachor (Gdańsk University of Technology in Poland). [487]  In the article he devised a nonclassical arithmetic intrinsic to a Koch-type fractal curve. He then used that arithmetic to construct a non-Newtonian calculus, yielding what “seems to be the first example of a truly intrinsic description of wave propagation along a fractal curve”. From the article:  “Beginning with addition and multiplication intrinsic to a Koch-type curve, we formulate and solve wave equation describing wave propagation along a fractal coastline. ... This seems to be the first example of a truly intrinsic description of wave propagation along a fractal curve. ... This counterintuitive possibility opened by the non-Newtonian calculus is especially useful in fractal applications. ... The arithmetic perspective is simultaneously applicable to all the other aspects of mathematical modeling, including algebraic or probabilistic methods. The freedom of choice of arithmetic plays a role of a universal symmetry of any mathematical model. ... It would not be very surprising if our fractal calculus found applications also in other branches of physics, where finite physical results are buried in apparently infinite theoretical predictions." ... It turns out that in 2014, Marek Czachor independently discovered non-Newtonian calculus, and generalized the earlier work of Grossman and Katz. [510]

Marek Czachor (Gdańsk University of Technology in Poland) has used nonclassical arithmetics and non-Newtonian calculus to study dark matter and dark energy in physics. From his 2020 article “Non-Newtonian mathematics instead of non-Newtonian physics: Dark matter and dark energy from a mismatch of arithmetics” [495]: “Newtonian physics is based on Newtonian calculus applied to Newtonian dynamics. ... However, calculus is dependent on arithmetic, that is the way we add and multiply numbers. For example, in special relativity we add velocities by means of addition x ⊕ y = tanh(arctanh(x) + arctanh(y)). ... The new arithmetic allows us to define the corresponding derivative and integral, and thus a new calculus, which is non-Newtonian in the sense of Grossman and Katz (1972). ... M. Grossman and R. Katz worked out a form of calculus which culminated in their book Non-Newtonian Calculus. It went basically unnoticed by the main-stream mathematical community, and was completely ignored by physicists. ... The problems with dark energy and dark matter look like an experimental indication that the arithmetic we all work with [i.e., the real number system] is not necessarily the one preferred by Nature. ...  The new arithmetic allows us to define the corresponding derivative and integral, and thus a new calculus which is non-Newtonian in the sense of Grossman and Katz. ... We do not yet know if the proposed generalization ultimately removes any need of dark energy and dark matter, but it will certainly change estimates of their parameters. ... Putting it more modestly, even if the arithmetic perspective will not entirely eliminate the need for dark matter or energy, it should at least change theoretical estimates for their parameters.”

In 2020 and 2021, Marek Czachor (Politechnika Gdańska in Poland) wrote articles with application of NNC to quantum mechanics [502, 503, 518]. The article [518] was supported by The National Oceanic and Atmospheric Administration’s [NOAA] CI TASK grant ‘Non-Newtonian calculus with interdisciplinary applications’. From [502]: “Diophanltine arithmetics and non-Newtonian calculus seem counterintuitive only at a first encounter. The are as natural as non-Euclidean geometry, non-Boolean logic, or non-Kolmogorovian probability.” From [503]: “An explicit model shows that classical local hidden-variable theories based on a generalized arithmetic can reconstruct quantum probabilities typical of two-particle singlet states. The model is classical, realistic, local, rotationally invariant, detectors are perfect, and observers have freewill. ... Non-Diophantine arithmetics imply non-Newtonian calculi, in particular a harmonic one. A harmonic derivative of a function A : R → R is defined in the usual way by means of the harmonic arithmetic.” From [518]: “The power and efficiency of the non-Newtonian approach lies in its low-level starting point — the arithmetic.”

Non-Newtonian calculus is one of the applications presented in the 2020 book Non-Diophantine Arithmetics in Mathematics, Physics and Psychology by Mark Burgin (University of California, Los Angeles, USA) and Marek Czachor (Politechnika Gdańska, Poland). Intended for researchers and graduate students in mathematics, physics, psychology, and philosophy, this masterful book provides a detailed exposition of the theory of non-Diophantine arithmetics and its various applications. One of the chapters is called “From Non-Diophantine Arithmetic to Non-Newtonian Calculus”. The non-Diophantine arithmetics were created by Mark Burgin in 1977 and have been used extensively by him and Marek Czachor. From the book: “The credit for inventing an arithmetic-inspired non-Newtonian calculus is due to Michael Grossman and Robert Katz. ... Their little book Non-Newtonian Calculus(Grossman and Katz, 1972) went, unfortunately, basically unnoticed by the mainstream mathematical community. ... The non-Newtonian calculus of Grossman and Katz has many applications in different areas including decision making, dynamical systems, differential equations, chaos theory, economics, marketing, finance, fractal geometry, image analysis, and electrical engineering.  … In 2014, [Marek Czachor] independently discovered non-Newtonian calculus, generalizing the earlier formalisms of Grossman and Katz, and Pap, and applying it to problems of fractal analysis, dark energy, and Bell’s theorem.” [510]

It's worth noting that in Grossman and Katz’s publications on non-Newtonian calculus, “arithmetic” means any complete ordered field whose elements are real numbers. (They defined an “arithmetic” to be a complete ordered field whose realm is a subset of R.) However, there was no need to restrict attention to real numbers. If they had simply defined an “arithmetic” to be any complete ordered field, all their results would still be valid.

Non-Newtonian Calculus is the subject of Marek Czachor’s (Politechnika Gdańska, Poland) article “Unifying aspects of generalized calculus” [524]. The article was published in the journal Entropy in 2020. Here are the first four paragraphs of the article, followed by the last two:

      "Non-Newtonian calculus naturally unifies various ideas that have occurred over the years in the field of generalized thermostatistics, or in the borderland between classical and quantum information theory. The formalism, being very general, is as simple as the calculus we know from undergraduate courses of mathematics. Its theoretical potential is huge, and yet it remains unknown or unappreciated."

      “Studies of a calculus based on generalized forms of arithmetic were initiated in the late 1960s by Grossman and Katz, resulting in their little book Non-Newtonian Calculus [1–3]. Some twenty years later the main construction was independently discovered in a differentcontext, and pushed in a different direction, by Pap [4–6]. After another two decades the same idea, but in its currently most general form, was rediscovered by myself [7–15]. In a wider perspective, non-Newtonian calculus is conceptually related to the works ofRashevsky[16] and Burgin [17–20] on non-Diophantine arithmetics of natural numbers, andto Benioff’s attempts [21–25] of basing physics and mathematics on a common fundamental ground. Traces of non-Newtonian and non-Diophantine thinking can be found in the works of Kaniadakis on generalized statistics [26–34]. A relatively complete account of the formalism can be found in the forthcoming monograph [35]."

      “In the paper, we will discuss links between generalized arithmetics, non-Newtoniancalculus, generalized entropies, and classical, quantum, and escort probabilities. As we willsee, certain constructions such as Rényi entropies or exponential families of probabilitieshave direct relations to generalized arthmetics and calculi. Some of the constructions onefinds in the literature are literally non-Newtonian. Some others only look non-Newtonian, but closer scrutiny reveals formal inconsistencies, at least from a strict non-Newtonian perspective. ''

      “Our goal is to introduce non-Newtonian calculus as a sort of unifying principle, simultaneously sketching new theoretical directions and open questions.”

      "Non-Newtonian calculus, and non-Diophantine arithmetics behind it, are as simple as the undergraduate arithmetic and calculus we were taught at schools. Their conceptual potential is immense but basically unexplored and unappreciated. Apparently, physicists in general do not feel any need of going beyond standard Diophantine arithmetic operations, in spite of the fact that the two greatest revolutions of the 20th century physics were, in their essence, arithmetic (relativistic addition of velocities, quantum mechanical addition of probabilities). It is thus intriguing that two of the most controversial issues of modern science, dark energy and Bell’s theorem, reveal new aspects when reformulated in generalized arithmetic terms."

     "One should not be surprised that those who study generalizations of Boltzmann-Gibbs statistics are naturally more inclined to accept non-aprioric rules of physical arithmetic. Anyway, the very concept of nonextensivity, the core of many studies on generalized entropies, is implicitly linked with generalized forms of addition, multiplication, and differentiation [53, 58–60]."

In their article "Optimization on fractal sets", Nizar Riane (Sorbonne University in France) and Claire David (Sorbonne University in France) asserted: " One can find other ways of dealing with fractals, for instance, a very original and interesting one uses non-Newtonian calculus in the case of the Sierpinski-type fractal." [530]

On the World Wide Web, videos on the Marek Czachor YouTube channel feature applications of NNC: https://www.youtube.com/channel/UCXSA1laWkySSbavtMbXDspg/videos . [556]

Non-Newtonian calculus is used in the article “The Renyi entropies operate in positive semifields” by Francisco J. Valverde-Albacete and Carmen Peláez-Moreno (both from Universidad Carlos III de Madrid, in Spain). [493] The article was published in the journal Entropy in 2019. Keywords: shifted Renyi entropy, Pap’s g-calculus, non-Newtonian calculus, positive commutative semifields, idempotent semifields, artificial intelligence, machine learning, computational intelligence. From the article: “In any case it would be interesting to follow up on Grossman and Katz’s approach to particular “positive fields”. ... We leave this for future work.” From the Acknowledgement: "We would also like to acknowledge the reviewers of previous versions of the paper for their timely criticism and suggestions, and in particular for the references, previously unknown to us, to Burgin’s and Grossman’s and Katz’s work."

The applicability of non-Newtonian calculus in quantitative finance and financial engineering is discussed in “Peter Carr's Hall of Mirrors”, an article (12 May 2017) about the distinguished financial engineer Peter Carr, written by Dan Tudball. [481] The article explains, among other things, why the bigeometric derivative is a useful tool for working with the widely used Black-Scholes model in financial engineering. From the article:

      “Carr’s zeal was sparked when he stumbled across an idea, developed by two researchers [Grossman and Katz] that might be easily dismissed as nuts. As a preamble to introducing this idea, Carr starts with the familiar: ‘So, a standard derivative in [classical] calculus is a limiting quotient of [differences]. ... .’

      “‘So, the idea is: what if you [replace] the differences [with] ratios?  At first, when you hear this you think it’s nuts,’ laughs Carr, ‘but I’ve totally bought into this philosophy.’

      “Authors Robert Katz and Michael Grossman, in their book Non-Newtonian Calculus, have invented [among other things] three new types of derivatives; in the first, there’s still a difference in the numerator, but now you have a ratio in the denominator. The next type does it the other way around and we put the ratio in the numerator position instead. Finally, you could have a third derivative that has a ratio for both numerator and denominator.

      ".... but this led him to think about lognormal distributions. ‘Well, you are not going to believe this but I did a calculation and it turns out that if you take that last kind of new derivative [bigeometric derivative], which is a ratio over a ratio, and you apply it twice to this log normal density, you actually get a constant, which is like kind of shocking!

      “‘I’m basically doing a novel kind of second derivative,’ Carr continues. ‘If you take the usual kind [classical] of second derivative, you would not get a constant. If you take this unusual derivative [bigeometric] and apply it twice to a lognormal density, you actually get a constant. In this sense, lognormal is like a quadratic function because if you take a standard second derivative and apply it to a quadratic function, you get a constant so I thought that was cool.

      "‘The lognormal distribution is something that everybody in mathematical finance, like option pricing theories, is intimately familiar with. It’s just like, you talk about it every day and I don’t think that anybody realizes that it has that strange property because nobody ever thought of that strange derivative [bigeometric derivative]. I was kind of happy to discover that recently.’

      “This takes Carr back to symmetry. A standard [classical] derivative has translation invariance in both the numerator and the denominator, so, for example, x2 with respect to x is 2x and the first derivative of x2 plus a constant with respect to x is also 2x.

      “These alternative [bigeometric] derivatives have scale invariance, meaning that if you were to take some function and then double it, and then compare the derivatives before and after doubling, they’d be the same.

      “‘In some applications, that’s really important,’ Carr stresses. “Getting back to lognormal, that’s actually what’s key; this Black–Scholes model is all about scale invariance [ratios] and not about translation invariance [differences].

      “‘It turns out that things are a lot simpler if you use the right tools for the job, and these guys [Grossman and Katz] proposed useful tools when you have scale invariance in a problem like you do in Black–Scholes; and yet, no one is really aware of their work, ... .’” [The famous Black-Scholes model, also known as the Black-Scholes-Merton model, is a model of price variation over time of financial instruments such as stocks.]

From the article "Fixed points of non-Newtonian contraction mappings on non-Newtonian metric spaces" [254] by Demet Binbaşıoǧlu (Gaziosmanpaşa University in Turkey), Serkan Demiriz (Gaziosmanpaşa University in Turkey), and Duran Türkoǧlu (Gazi University in Turkey): "Non-Newtonian calculus has many applications in different areas including fractal geometry, image analysis (e.g.,in biomedicine), growth/decay analysis (e.g.,in economic growth, bacterial growth and radioactive decay), finance (e.g.,rates of return), the theory of elasticity in economics, marketing, the economics of climate change, atmospheric temperature, signal processing (electrical engineering), wave theory in physics, quantum physics and gauge theory, information technology, pathogen counts in treated water, actuarial science, tumor therapy and cancer-chemotherapy in medicine, materials science/engineering, demographics, differential equations (including a multiplicative Lorenz system and Runge–Kutta methods), calculus of variations, finite-difference methods, averages of functions, means of two positive numbers, weighted calculus, meta-calculus, approximation theory, least-squares methods, multivariable calculus, complex analysis, functional analysis, probability theory, utility theory, Bayesian analysis, stochastics, decision making, dynamical systems, chaos theory, and dimensional spaces."

The geometric calculus was used in an article concerning wave physics, partial differential equations, complex multiplicative calculus, and multiplicative vector spaces, by Max Cubillos (California Institute of Technology). The article is called "Modelling wave propagation without sampling restrictions using the multiplicative calculus I: Theoretical considerations". [391] From the article: "We exploit this fact to show that some partial differential equations (PDE) can be solved far more efficiently using techniques based on the multiplicative [geometric] calculus. ... The calculus developed by Newton and Leibniz is one of most significant breakthroughs in mathematics but an infinite number of other versions of calculus are possible. The treatise [Non-Newtonian Calculus] by Grossman and Katz is perhaps the earliest comprehensive work on other so-called non-Newtonian calculi ... Recent contributions have expanded on the ideas of non-Newtonian calculi and have shown some applications, particularly using the multiplicative calculus. These include significant extensions of the multiplicative calculus to complex numbers, contributions on numerical algorithms in the multiplicative calculus and applications to specific problems of scientific interest. However, to the authors’ knowledge there have not been any numerical applications to the partial differential equation (PDE) of mathematical physics. This paper is the first in a series of articles that aims to bridge that gap, by applying techniques of the multiplicative calculus to solve problems in mathematical physics far more efficiently than current methods. ... Because the main challenge in high frequency wave physics is the sampling constraint imposed by a fundamental carrier wave, we expect that the multiplicative calculus will be well suited to these problems." (The expression "multiplicative calculus" refers here to the geometric calculus.)

Non-Newtonian calculus was used in the 2016 doctoral dissertation of Michael Valenzuela at the University of Arizona in the United States. The dissertation is entitled "Machine learning, optimization, and anti-training with sacrificial data". (In computer science, machine learning is a branch of artificial intelligence.) [279] From the dissertation: "Grossman and Katz [Non-Newtonian Calculus] mention several alternative calculi including: geometric, anageometric, bigeometric, quadratic, anaquadratic, biquadratic, harmonic, anaharmonic, and biharmonic. ... Non-Newtonian calculus has been used to derive optimization algorithms that perform better than traditional Newton based methods for Expectation-Maximization algorithms. However, non-Newtonian calculus goes beyond simply being useful for optimization, it is useful for the other half of learning: modeling. ... Here are a few rules of thumb for non-Newtonian models. If a meta-model is primarily concerned with learning probabilities, non-parametric distributions, or anything else where the multiplication is the primary operation, then the geometric calculi may be of interest. If working in a domain where the squares are additive, as is common the case when estimating the variance of a sum of independent random variables, then the quadratic calculi may produce meaningful models."

The geometric calculus was used by Ali Uzer (Fatih University in Turkey) in an article on wave physics: "Improvement of the diffraction coefficient of GTD  by using multiplicative calculus". [185] In the abstract to the article he states: "The given approximation techniques [based on the geometric calculus] ... can be extended to other high frequency techniques in electromagnetics." (The expression "multiplicative calculus" refers here to the geometric calculus.)

Non-Newtonian calculus is used in the 2020 article “From AMOEBA to BRAIN: How Nature process Information” by Ignacio Ozcariz, a researcher in artificial intelligence and quantum computing from Universidad Politécnica de Madrid in Spain. Professor Ozcariz is affiliated with RQuantech (Geneva, Switzterland) and Criptosasun (Madrid, Spain). The author's theory: living organisms all share a common way of processing information. From the Abstract: "The paper presents a new Information Processing paradigm based on the dynamics encountered in life from organisms as different as Amoebas and the Mammalian brain. Our thesis contemplates that life supports information processing via metabolic dynamics in self-organized enzymatic networks which have the capacity to represent functional catalytic patterns that can be instantiated by specific input stimuli. Furthermore, the information patterns can be transferred from the functional dynamics of the metabolic networks to the biochemical enzymatic activity information encoded by DNA. The metabolic dynamics are governed by fractional dynamics that evolve in topological fractal spaces with multiscale time parameters generating complex attractors. The complete dynamic information process is driven by the shorterm process of metabolic dynamics and the longterm process of DNA expression via epigenetic mechanisms.” From the article: “The key point is that dynamics get the non-locality and historical properties from fractional dynamics, and power law “additivity” via the multiplicative geometric and bigeometric calculus. ... Seminal book for non-Newtonian calculus is Non-Newtonian Calculus."

Non-Newtonian calculus was used by Ali Uzer (Fatih University in Turkey) to develop a multiplicative type of calculus for complex-valued functions of a complex variable, which he applied to wave physics. [78, 158] (The expression "multiplicative type of calculus" refers here to the geometric calculus.)

The geometric calculus was used by Ali Uzer (Regent University College of Science and Technology in Ghana) in his article on wave physics entitled "Evaluations of a class of integrals that arise in wave scattering problems by using the Taylor product theorem". [394] From the article "The Taylor product theorem is a theorem of a new branch of calculus called multiplicative calculus. It is analogous to the conventional Taylor series theorem of the classical calculus ... ." (The expression "multiplicative calculus" refers here to the geometric calculus.)

In the abstracts to his two seminars both called "Non-Newtonian calculus for the dynamics of random fractal structures", Wojbor Woyczynski (Case Western Reserve University) asserted: "Many natural phenomena, from microscopic bacteria growth, through macroscopic turbulence, to the large scale structure of the Universe, display a fractal character. For studying the time evolution of such "rough" objects, the classical, "smooth" Newtonian calculus is not enough." In an abstract to another seminar, Professor Woyczynski asserted: "Random fractals, a quintessentially 20th century idea, arise as natural models of various physical, biological (think your mother's favorite cauliflower dish), and economic (think Wall Street, or the Horseshoe Casino) phenomena, and they can be characterized in terms of the mathematical concept of fractional dimension. Surprisingly, their  time evolution can be analyzed by employing a non-Newtonian calculus utilizing integration and differentiation of fractional order." [90, 104, 146]

Non-Newtonian calculus is used by Martin Ostoja-Starzewski and his research team in their work on fractal materials at the University of Illinois at Urbana-Champaign. [163] From Professor Ostoja-Starzewski's 2013 media-upload "The inner workings of fractal materials", University of Illinois at Urbana-Champaign: "Together with a small, highly focused research team, Ostoja-Starzewski is working across disciplines to unite methods from solid mechanics, advanced continuum mechanics, statistical physics and mathematics. Some of the specific mathematical theories they use include probability theory and non-Newtonian calculus. These approaches allow them to focus on different fractal structures, including morphogenesis of fractals at elastic-inelastic transitions in solids, composites and soils, as well as materials that have anomalous heat conduction properties and fractal patterns that are seen in biological materials."

The bigeometric calculus was used in an article on fractals and multiplicative dynamical systems by Dorota Aniszewska and Marek Rybaczuk (both from Wroclaw University of Technology in Poland). [131] In that article they state: "Describing the evolution of defects [in materials] treated as fractals implies usage of the multiplicative derivative, because the ordinary [classical] additive derivative of a function depending on fractal dimension or measure does not exist. ... The goal of this paper is chaos examination in multiplicative dynamical systems described with the multiplicative derivative." (The expression "multiplicative derivative" refers here to the bigeometric derivative.)

The bigeometric calculus was used in an article on fractals and material science by M. Rybaczuk and P. Stoppel (both from Wroclaw University of Technology in Poland). [18]

The bigeometric calculus was used in the article "Critical growth of fractal patterns in biological systems"  by Marek Rybaczuk of Wroclaw University of Technology in Poland. [238]

The bigeometric calculus was used in an article on fractal dimension and dimensional spaces by Marek Rybaczuk (Wroclaw University of Technology in Poland), Alicja Kedziab (Medical Academy of Wroclaw in Poland), and Witold Zielinskia (Wroclaw University of Technology in Poland). [132]

The bigeometric calculus was used in the book Measurements, Dimensions, Invariant Models and Fractals by Wacław Kasprzak, Bertold Lysik, Marek Rybaczuk (all from Wroclaw University of Technology in Poland). [203]

The bigeometric calculus was used in the article "Physical stability and critical effects in models of fractal defects evolution based on single fractal approximation" by Dorota Aniszewska and Marek Rybaczuk (both from Wroclaw University of Technology in Poland). [228]

It turns out that the so-called fractal derivative is a non-Newtonian derivative. The fractal derivative is used in applied mathematics and mathematical analysis. [460]

The bigeometric calculus was used in an article [126] on chaos in multiplicative dynamical systems by Dorota Aniszewska and Marek Rybaczuk, both from the Wroclaw University of Technology in Poland. They showed that "all classical conditions concerning chaotic behavior can be extended to multiplicative [dynamical] systems". Their work involves one-dimensional multiplicative versions of logistic equations, and multi-dimensional nonlinear dynamical systems described by means of the bigeometric derivative. A multiplicative version of the classical Lorenz system (as well as the Lyapunov exponent and the Runge-Kutta method) was used "for analysis of stability and chaotic behavior".

The bigeometric calculus was used in an article on multiplicative differential equations by Dorota Aniszewska (Wroclaw University of Technology in Poland). [1, 129]

The bigeometric calculus was used in an article on a multiplicative Lorenz system by Dorota Aniszewska and Marek Rybaczuk (both from Wroclaw University of Technology in Poland). [130]

The bigeometric calculus was used by Marek Rybaczuk (Wroclaw University of Technology in Poland) in his lecture "Fractal Models of Defects Evolution" at the 2004 South African Conference on Applied Mechanics. [258]

A perceptive discussion about non-Newtonian calculi and proper usage of the expression "multiplicative calculus" is included in the article "Multiplicative Hénon map" by Dorota Aniszewska and Marek Rybaczuk, both from Wroclaw University of Technology in Poland. (Please see the Multiplicative Calculus page.) The article's main topic is an application of geometric arithmetic [10, 11, 15], namely a multiplicative version of the Hénon map, which is a function that arose in the study of dynamical systems, fractals, and chaos theory. [288]

The bigeometric calculus is used in the article “New discrete chaotic multiplicative maps based on the logistic map“ by Dorota Aniszewska (Wroclaw University of Technology in Poland). [449] From the Abstract: “Chaos is a phenomenon which cannot be predicted if it manifests itself in a nonlinear system. ... The logistic map is the simplest mathematical model exhibiting chaotic behavior. ... In this paper, the properties of multiplicative [bigeometric] calculus were employed to transform the classical logistic map into multiplicative ones. The multiplicative logistic maps were tested for chaotic behavior.”

According to [21], "in dimensional spaces (in a similar way to physical quantities) you can multiply and divide quantities which have different dimensions but you cannot add and subtract quantities with different dimensions. This means that the classical additive derivative is undefined because the difference f(x+deltax)-f(x) has no value. However in dimensional spaces, the geometric derivative and the bigeometric derivative remain well-defined. Multiplicative dynamical systems can become chaotic even when the corresponding classical additive system does not because the additive and multiplicative derivatives become inequivalent if the variables involved also have a varying fractal dimension."

The geometric calculus and the bigeometric calculus were used by Bugce Eminaga (Girne American University in Cyprus), Hatice Aktore (Eastern Mediterranean University in North Cyprus), and Mustafa Riza (Eastern Mediterranean University in North Cyprus) in their article re dynamical systems called "A modified quadratic Lorenz attractor". [237] From that article: "Dynamical systems have also been discussed in the framework of various non-Newtonian calculi ... In Section 3, the modified quadratic Lorenz attractor is translated into geometric and bigeometric calculus, and the solutions of the the system are obtained using the corresponding multiplicative Runge-Kutta methods."

The geometric calculus was used by Mustafa Riza (Eastern Mediterranean University in North Cyprus), Hatice Aktore (Eastern Mediterranean University in North Cyprus), and Bugce Eminaga (Girne American University in Cyprus) in their lecture re dynamical systems called "A modified quadratic Lorenz attractor in geometric multiplicative calculus" at the 28th International Conference of the Jangjeon Mathematical Society. [233]

The geometric calculus was used by Luc Florack and Hans van Assen (both of the Eindhoven University of Technology in the Netherlands) in their work on biomedical image analysis and "complex imaging frameworks such as diffusion tensor imaging". In their article "Multiplicative calculus in biomedical image analysis" [88], they assert: "We advocate the use of an alternative calculus in biomedical image analysis, known as multiplicative (a.k.a. non-Newtonian) calculus. ... The purpose of this article is to provide a condensed review of multiplicative calculus and to illustrate its potential use in biomedical image analysis. ... Examples have been given in the context of cardiac strain analysis and diffusion tensor imaging to illustrate the relevance of multiplicative calculus in biomedical image analysis, and to support our recommendation for further investigation into practical as well as fundamental issues. ... After a condensed summary of multiplicative calculus collected from the literature, we will demonstrate its use by a multiplicative reformulation of two existing biomedical image analysis applications, viz. multi-scale representation (or spatial regularization) of diffusion tensor images in the framework of the log-Euclidean paradigm, and tensorial strain analysis in cardiac magnetic resonance imaging. These examples merely serve to illustrate the potential power of multiplicative calculus. In general, multiplicative calculus should come to mind as a potentially promising tool for addressing image analysis problems whenever some sort of multiplicative process lies beneath the surface. We shall point out what these processes are in our concrete examples. ... Phenomena driven by some (perhaps implicit) multiplicative mechanism may be more conveniently described in the context of multiplicative calculus than in the standard way. ... Examples have been given in the context of cardiac strain analysis and diffusion tensor imaging to illustrate the relevance of multiplicative calculus in biomedical image analysis, and to support our recommendation for further investigation into practical as well as fundamental issues.” (The expression "multiplicative calculus" refers here to the geometric calculus.)

In Professor Florack's article [96] he states: "Multiplicative calculus provides a natural framework in problems involving positive images and positivity preserving operators. In increasingly important, complex imaging frameworks, such as diffusion tensor imaging, it complements standard calculus in a nontrivial way. The purpose of this article is to illustrate the basics of multiplicative calculus and its application to the regularization of positive definite matrix fields." (The expression "multiplicative calculus" refers here to the geometric calculus.) [88, 96, 199, 111]

The geometric calculus was used in "Physically inspired depth-from-defocus", an article about image analysis and computer vision by four faculty-members of Saarland University in Germany: Nico Persch, Christopher Schroers, Simon Setzer, and Joachim Weickert (Gottfried Wilhelm Leibniz Prize winner). From the article: "We propose a novel variational approach to the depth–from–defocus problem. ... For the minimisation of our energy functional, we show the advantages of a multiplicative Euler-Lagrange formalism ... Our work is an example how one can benefit from physically refined modelling in conjunction with multiplicative calculi. It is our hope that both concepts will receive more popularity in future computer vision models." [316] (Article [316] is a revised version of articles [240] and [241].)

Non-Newtonian calculus was used in the 2018 doctoral dissertation of Martin Schmidt at Saarland University in Germany. The dissertation is entitled "Linear scale-spaces in image processing: drift-diffusion and connections to mathematical morphology". [414] From the dissertation: "The non-Newtonian calculi were introduced by Grossman and Katz [1972] and play an increasing role in current image processing applications. They allow to design problem-tailored algorithms. Instead of solving a given task directly, it is solved in an alternative domain where known or desired properties can be exploited. ... The Hopf-Cole transformation defined in (3.49) can be seen as a non-Newtonian calculus ... ."

According to Martin Schmidt's doctoral dissertation "Linear scale-spaces in image processing: drift-diffusion and connections to mathematical morphology" [414], the geometric calculus is used in the article "Gradient domain high dynamic range compression" [415] by Raanan Fattal, Dani Lischinski, and Michael Werman (all from The Hebrew University in Jerusalem). From Martin Schmidt's dissertation: "Whereas the classical or Newtonian calculus is recovered when α is the identity function, other choices of α lead to the so-called non-Newtonian calculi. The best known example is the geometric or multiplicative calculus where α is taken to be the exponential function. It is used by Fattal et al. [2002] to reflect the human’s visual perception as described by the Weber-Fechner law. This law states a logarithmic connection between the perceived colour values and their actual intensities."

The geometric calculus was used in an article on visual perception by Miguel Martinez-Garcia and Timothy Gordon, both from the University of Lincoln in the United Kingdom. The article is  called “A new model of human steering using far-point error perception and multiplicative control”. [457] From the article: “In addition, it can be verified that the log-normal distribution arises as a result of a multiplicative process ... The right hand side of (6b) can be interpreted as a multiplicative [geometric] derivative ...”

The geometric calculus was used by Gunnar Sparr (Lund Institute of Technology, in Sweden) in an article on computer vision. (The "multiplicative derivative" referred to in the article is the geometric derivative.) [155]

The geometric calculus was used in an article on cyber-physical systems and signal temporal logic by Noushin Mehdipour (Boston University), Cristian-Ioan Vasile (Boston University), and Calin Belta (Massachusetts Institute of Technology). [497]

The geometric calculus was used in three articles on medical-imaging science by Kiyoko Tateishi (Saint Marianna University School of Medicine in Japan and The University of Tokushima in Japan), Yusaku Yamaguchi (Shikoku Medical Center for Children and Adults in Japan), Omar M. A. Al-Ola (Tanta University in Egypt), Takeshi Kojima (The University of Tokushima in Japan), and Tetsuya Yoshinaga (The University of Tokushima in Japan). [271, 272, 359] The articles are entitled "Continuous analog of multiplicative algebraic reconstruction technique for computed tomography", "Noise reduction in computed tomography using a multiplicative continuous-time image reconstruction method", and "Continuous analog of accelerated OS-EM algorithm for computed tomography". The former two articles were presented at the SPIE conference Medical Imaging 2016: Physics of Medical Imaging in February/March of 2016. (SPIE, an affiliate of the American Institute of Physics, is an international society for optical engineering with more than 18,000 members.) From the latter article: "The integration of differential equations by discretization via the [geometric] calculus is derived as a multiplicative counterpart of classical Euler and RK [Runge-Kutta] methods."

The geometric calculus was used in an article about radiation therapy by T. Yoshinaga, Y. Tanaka, K. Fujimoto (all from the Institute of Health Biosciences at The University of Tokushima in Japan). [289] From the article: "In this paper, we propose an iterative method as discretization of the differential equation using the geometric multiplicative calculus, and show the effectiveness of our method. ... We derived the iterative method by using the first-order approximation based on the geometric multiplicative calculus applied to the differential equation."

The geometric calculus was used in the article "Tomographic image reconstruction based on minimization of symmetrized Kullback-Leibler divergence" by Ryosuke Kasai (Tokushima University in Japan), Yusaku Yamaguchi (Shikoku Medical Center for Children and Adults in Japan), Takeshi Kojima (Tokushima University), and Tetsuya Yoshinaga (Tokushima University). From the Abstract: "Iterative reconstruction (IR) algorithms based on the principle of optimization are known for producing better reconstructed images in computed tomography. ... We describe a hybrid Euler method combined with additive and multiplicative calculus for constructing an effective and robust discretization method, thereby enabling us to obtain an approximate solution to the differential equation. We performed experiments and found that the IR algorithm derived from the hybrid discretization achieved high performance." [437]

The geometric calculus was used in the article “Extended ordered-subsets expectation-maximization algorithm with power exponent for noise-robust image reconstruction in computed tomography” by Yusaku Yamaguchi (National Hospital Organization, Shikoku Medical Center for Children and Adults in Japan), Moe Kudo (Graduate School of Health Sciences, Tokushima University in Japan), Takeshi Kojima (Institute of Biomedical Sciences, Tokushima University), Omar Mohammad Abou Al-Ola (Tanta University in Egypt), and Tetsuya Yoshinaga (Institute of Biomedical Sciences, Tokushima University). From the Abstract: “In this paper, we propose an extended OS-EM algorithm with a power exponent. We theoretically prove the asymptotic stability of an equilibrium corresponding to the solution of the nonlinear hybrid dynamical system whose numerical discretization based on multiplicative calculus coincides with the extended OS-EM algorithm. We provide a numerical experiment to demonstrate the effectiveness of the proposed system and confirm the acceleration of the proposed method and the robustness against noise. The reconstruction of high-quality images made by the method even when the projection data is noisy allows patient dose reduction in clinical practice.”[512]

The geometric calculus is used in the article “Hybrid Euler method for discretizing continuous-time tomographic dynamical system” by Ryosuke Kasai (Graduate School of Health Sciences, Tokushima University in Japan), Yusaku Yamaguchi (Shikoku Medical Center for Children and Adults in Japan), Takeshita Kohima (Institute of Biomedical Sciences, Tokushima University),  and Tetsuya Yoshinage (Institute of Biomedical Sciences, Tokushima University). [517] (Tomography is imaging by sections or sectioning through the use of any kind of penetrating wave. The method is used in radiology, archaeology, biology, atmospheric science, geophysics, oceanography, plasma physics, materials science, astrophysics, quantum information, and other areas of science.) The article was a Selected Paper at NCSP'20 (The 2020 Research Institute of Signal Processing’s International Workshop on Nonlinear Circuits, Communications and Signal Processing). From the Conclusion: “We evaluated the hybrid Euler method constructed as a combination based on additive [classical] and multiplicative [geometric] calculus. We gave a simple vector field for which the hybrid Euler method is effective. Through numerical and physical experiments for a tomographic dynamical system, we found that the hybrid Euler method has an advantage over both the additive [classical] and multiplicative [geometric] Euler methods.”

The geometric calculus is used in the article on radiation therapy called “Intensity-modulated radiation therapy optimization for acceptable and remaining-one unacceptable dose-volume and mean-dose constraint planning" by Ryosei Nakada (Tokushima University in Japan), Omar M. Abou Al-Ola (Tanta University in Egypt), and Tetsuya Yoshinaga (Tokushima University). [521] From the article: “We give a novel approach for obtaining an intensity-modulated radiation therapy (IMRT) optimization solution based on the idea of continuous dynamical methods.  ... The system of nonlinear differential equations defined in this paper has a different vector field from that of the previously proposed system. The previous method for obtaining an acceptable solution has a disadvantage in that it requires a long calculation time because of the high numerical cost of integrating piecewise differential equations describing the hybrid dynamical system. To resolve the disadvantage, we conduct a numerical discretization with multiplicative calculus. ...  By using geometric multiplicative Euler discretization we obtain the iterative algorithm of the variable for radiation beam weight."

The geometric calculus was used by Luc Florack (Eindhoven University of Technology in the Netherlands) in his presentation "Neuro and cardio imaging" at the 2011 BIRS Workshop (Banff International Research Station for Mathematical Innovation and Discovery). [195]

Application of the geometric calculus to image analysis is discussed in the article "Direction-controlled DTI [Diffusion Tensor Imaging] interpolation" by Luc Florack, Tom Dela Haije, and Andrea Fuster, all from Eindhoven University of Technology in the Netherlands. [231] From that article: "The methodology we propose in the next section can be seen as an application of Riemann-Finsler geometry and exploits [geometric] calculus to implement positivity preserving 'linear' operations."

The geometric calculus is used in the chapter “Geodesic Tubes for Uncertainty Quantification in Diffusion MRI” in the 2021 book Information Processing in Medical Imaging. [556] The chapter was written by Rick Sengers, Luc Florack, and Andrea Fuster, all from Eindhoven University of Technology in The Netherlands.

Non-Newtonian calculus plays a prominent role in the article "Ultrasound image edge detection based on a novel multiplicative gradient and Canny operator" by Xiaohong Gong, Yali Zhou, Hao Zhou, and Yinfei Zheng (all from Zhejiang University in Hangzhou, China). The authors used non-Newtonian calculus to devise a technique for medical ultrasound-imaging that has "great research and application value". [211] From that article: "The proposed technique combines a new multiplicative gradient operator of non-Newtonian type with the traditional Canny operator to generate the initial edge map ....  Thus, the proposed method is very suitable for fast and accurate edge detection of medical ultrasound images. ... A calculus system based on non-additive perspective is named as non-Newtonian calculus, which has been used in the field of Bayesian analysis and image processing. ... In this section, experiments are performed on five clinical ultrasound images containing ovary, carotid artery, left ventricular, lymph cancer, and the abdominal aneurysm obtained from the Second Affiliated Hospital of Zhejiang University."

Non-Newtonian calculus was used by X. H. Gong, Y. F. Zheng, J. L. Qin, and H. Zhou (all from Zhejiang University in Hangzhou, China) in an article on medical ultrasound imaging called “Multiplicative gradient based edge detection method for medical ultrasound image”. [323]

Non-Newtonian calculus was used in a study of contour detection in images with multiplicative noise by Marco Mora, Fernando Córdova-Lepe, and Rodrigo Del-Valle (all of Universidad Católica del Maule in Chile). [99]  From the Abstract: "In this paper, a new operator for contour detection in images with multiplicative noise is presented. Traditional methods of edge detection, as those based in gradient operator or measures of variance, follow a logic and a math formulation in correspondence with the Differential and Integral Calculus of Newton. This work presents a new operator of non-Newtonian type which had shown be more efficient in contour detection than the traditional operators. Like the regular gradient, a non-Newtonian gradient can be used in a number of more complex methods, which shows its potential in the contours detection in images affected by multiplicative noise."

The bigeometric calculus was used in the article “A multi-directional gradient with bi-geometric calculus to detect contours in images with multiplicative noise” by M. Acevedo-Letelier, K. Vilches, and M. Mora (all from Universidad Católica del Maule in Chile). [459] The Abstract: In this paper a new operator is presented for the detection of contours in images with multiplicative noise, by using the operations introduced in the bi-geometric calculus, since recent results in the literature show that multiplicative operators tend to make more accurate approximations of the reality in images with multiplicative noise. The operator introduced corresponds to a multiplicative multi-directional gradient. The Global Efficiency was used as performance function to make a comparison about the effectiveness in the detection of contours, between the multi-gradient and its multiplicative version. ... According to the results obtained from the objective comparison, the multiplicative multi-directional gradient operator presents improved efficiency in obtaining contours versus its classical version.”

The bigeometric calculus was used in the article “Contour detection in images with speckle by using the bi-geometric structure” by M. Acevedo-Letelier, K. Vilches, and M. Mora (all from Universidad Católica del Maule in Chile). [476] Abstract: “The speckle produces severe degradations in digital images, which could be introduced by technological devices during the acquisition, capture or transmission of the images. This perturbation affects, for instance, medical and satellite images. Recently, some methods defined in the bi-geometric structure have presented significant improvements in the processing of image with speckle. In this work, we review some aspects of the bi-geometric structure including a discussion about some of the applications found in the literature focused in the detection of contours in grey scale images with speckle.” From the article: “Our main objective is to review some of the improvements obtained as consequence of applying the bi-geometric structure in the treatment of images with speckle, including the recall of some principles involved in the multiplicative noise production, in ultrasound images.”

Non-Newtonian calculus was used in the article "A multiplicative gradient-based anisotropic diffusion approach for speckle noise removal" by Romulus Terebes (Technical University of Cluj-Napoca [TUCN], Romania), Monica Borda (TUCN, Romania), Christian Germain (IMS Laboratory, Bordeaux, France), Raul Malutan (TUCN, Romania), and Ioana Ilea (TUCN, Romania). [268] From the Abstract: "We propose a novel directional diffusion method for speckle noise removal that uses the multiplicative gradient as an edge detector and operates on a moving orthonormal basis issued by a structure tensor based-approach and stochastic modelling. The method has good speckle removal and edge preservation properties and it can be used for filtering ultrasound, optical coherence tomography medical images or other types of images degraded by speckle, such as those acquired in Synthetic Aperture Radar (SAR) imaging systems."

Non-Newtonian calculus was used in the article "Polsar image denoising using directional diffusion" by Romulus Terebes (Technical University of Cluj-Napoca, Romania), Monica Borda (Technical University of Cluj-Napoca), Raul Malutan (Technical University of Cluj-Napoca), Christian Germain (University of Bordeaux, France), Lionel Bombrun (University of Bordeaux), and Ioana Ilea (University of Bordeaux). [407] From the article: "We employ the multiplicative gradient [99] for detecting edges, junction and interest points on PolSAR data."

A course on non-Newtonian calculus was conducted in the summer-term of 2012 by Joachim Weickert (Gottfried Wilhelm Leibniz Prize winner), Laurent Hoeltgen, and other faculty from the Mathematical Image Analysis Group of Saarland University in Germany. Among the topics covered were applications of non-Newtonian calculus to digital image processing, rates of return, and other growth processes. [106]

The geometric calculus was used in a course given by Joachim Weickert (Gottfried Wilhelm Leibniz Prize winner) at Saarland University in Germany in the summer-term of 2012: "Differential Equations in Image Processing and Computer Vision, CS 101". [173]

The geometric calculus was one of the topics covered in Professor Ross Hatton's graduate-course Geometric Mechanics (ROB 541) at Oregon State University in 2015.  [400] From the Course Overview: "This  course  serves  as  an  introduction  to  geometric  methods  in  the  analysis  of  dynamic systems."

In their article "A non-Newtonian examination of the theory of exogenous economic growth", Diana Andrada Filip (Babes-Bolyai University of Cluj-Napoca, Romania) and Cyrille Piatecki (LEO, Orléans University, France) assert: "In this paper, we have tried to present how a non-Newtonian calculus could be applied to repostulate and analyse the neoclassical [Solow-Swan] exogenous growth model [in economics]. ... In fact, one must acknowledge that it’s only under the effort of Grossman & Katz (1972) ... that such a non-Newtonian calculus emerged to give a natural answer to many growth phenomena. ... We must underscore that to discover that there was a non-Newtonian way to look to differential equations has been a great surprise for us. It opens the question to know if there are major fields of economic analysis which can be profoundly re-thought in the light of this discovery." [82, 34, 121]

Applications of non-Newtonian calculus to economics and statistics are discussed in the article "An overview on non-Newtonian calculus and its potential applications to economics" by Diana Andrada Filip (Babes-Bolyai University of Cluj-Napoca, Romania) and Cyrille Piatecki (LEO, Orléans University, France). [181] From that article: "In this paper, after a brief presentation of [the geometric] calculus, we try to show how it could be used to re-explore from another perspective classical economic theory, more particularly economic growth and the maximum-likelihood method from statistics."

A strong argument for "a non-Newtonian economic analysis" based on a new multiplicative accounting system and non-Newtonian calculus is presented in the article "In defense of a non-Newtonian economic analysis through an accounting paradigm" by Diana Andrada Filip (Babes-Bolyai University of Cluj-Napoca in Romania) and Cyrille Piatecki (Orléans University in France). [216] In that article they state: In that article they state: "The double-entry bookkeeping promoted by Luca Pacioli in the fifteenth century could be considered a strong argument in behalf of the multiplicative calculus which can be developed from the Grossman and Katz non-Newtonian calculus concept provided that one goes from an additive bookkeeping system to a multiplicative one. ... If for instance a non-Newtonian analysis of economic growth should be implemented, ... the involved [accounting system] must also be non-Newtonian, that is non-additive."

Non-Newtonian calculus is used in the 2019 book Multiplikative Euklidische Vektorräume als Grundlage für das Rechnen mit positiv-reellen Größen (Multiplicative Euclidean Vector Spaces as the Basis for Calculating with Positive-Real Quantities) by Björn Friedrich (Leibniz Institute for Neurobiology and Department of Experimental Audiology at the Otto von Guericke University in Magdeburg, Germany). Björn Friedrich is a researcher in the field of auditory neuroscience; he has a theoretical and computational background. The book includes application of NNC to psychophysics. (Psychophysics is the subfield of psychology devoted to the study of physical stimuli and their interaction with sensory systems. Psychophysical tasks have been extensively used to draw conclusions on how information is processed by the visual and other sensory systems.) Section 3.2.1 is called “Multiplikative Calculi Nach Grossman and Katz (1972)”, and includes discussion of the geometric calculus, the anageometric calculus, and the bigeometric calculus. From a translation of the Abstract: “Altogether, the multiplicative perspective has far-reaching consequences for many areas, in which positive-real quantities occur. Hence, there are numerous starting points for future research, including the expansion of the ideas presented herein to other non-additive structures.” From the publisher’s synopsis:

    “Vector spaces serve as a visual space for many aspects of our reality. While additive forms of abstract vector spaces belong to the curriculum of mathematical education, non-additive forms are rather unknown. In certain cases, this could create more appropriate visual spaces.

    ‘The present work is based on the guiding question, what are the consequences if one considers the positive-real numbers as an independent multiplicative Euclidean vector space (multiplicative perspective) instead of just a subset of the positive elements of the additive Euclidean vector space of the real numbers (additive Perspective).

    “The author first uses various examples and well-known statistical paradoxes to demonstrate that a change of perspective can lead to far-reaching theoretical and practical consequences. Then he devotes himself to positive, real variables that appear as stimulus characteristics in psychophysics. The focus is on a thought experiment, the findings of which question many well-known perceptive illusions that have been characterized with conventional psychophysical measures. New measures based on multiplicative Euclidean vector spaces could lead to different results. This is supported by empirical data on the adjustment of onset durations of acoustic stimuli: With the help of the new measures, influences of stimulus parameters can be determined, which would remain undetected with conventional measures.”

Discussions concerning a new multiplicative accounting system and the advantages of using the geometric calculus in economic analysis are included in an article by Diana Andrada Filip (Babes-Bolyai University of Cluj-Napoca in Romania) and Cyrille Piatecki (Orléans University in France). [149]

The geometric calculus was used by Hasan Özyapıcı (Eastern Mediterranean University in Cyprus), İlhan Dalcı (Eastern Mediterranean University in Cyprus), and Ali Özyapıcı (Cyprus International University) in their article "Integrating accounting and multiplicative calculus: an effective estimation of learning curve". [290] From the Abstract: "The results of this study are also expected to help researchers, practitioners, economists, business managers, and cost and managerial accountants to understand how to construct a multiplicative based learning curve to improve such decisions as pricing, profit planning, capacity management, and budgeting." (The expression "multiplicative calculus" refers here to the geometric calculus.)

The non-Newtonian approach to accounting [82, 121, 149, 181, 216]  was advocated by Amelia Correa (St. Andrews College in India) and Romar Correa (University of Mumbai in India) in their article "Accounting for Financialization: Stock-Flow-Consistent Political Economy". [259]

Many applications of non-Newtonian calculus have been made by Agamirza E. Bashirov, Mustafa Riza, and Yucel Tandogdu (all of Eastern Mediterranean University in North Cyprus); Emine Misirli Kurpinar and Yusuf Gurefe (both of Ege University in Turkey); and Ali Ozyapici (Lefke European University in Turkey). Their work on non-Newtonian calculus has application to differential equations, calculus of variations, finite-difference methods, approximation theory, multivariable calculus, complex analysis, actuarial science, finance, economics, biology, and demographics. [2, 24, 27, 33, 84, 87, 94, 95, 123, 140, 145, 157, 200] The article [2] was "submitted by Steven G. Krantz [Chauvenet-Prize winner]",  published in 2008 by the Journal of Mathematical Analysis and Applications, and includes discussion of multiplicative metric-spaces. From [94]: "This work is aimed to show that various problems from different fields can be modeled more efficiently using multiplicative calculus, in place of Newtonian calculus. ... Examples from finance, actuarial science, and economics are presented with solutions using both Newtonian and multiplicative calculus concepts. Based on the encouraging results obtained it is recommended that further research into this field be vested to exploit the applicability of multiplicative calculus in different fields as well as the development of multiplicative calculus concepts. ... In this study it becomes evident that the multiplicative calculus methodology has some advantages over additive [classical] calculus in modeling some processes in the areas such as actuarial, financial, economical, biological, demographic etc. It is recommended to take into account multiplicative differential equations in modeling different problems, where the process easily relates to its growth or decay factor. Rehandling of existing models in the multiplicative form will be beneficial." (The expression "multiplicative calculus" refers here to the geometric calculus.)

The geometric calculus was used by Bulent Bilgehan (Girne American University in Cyprus/Turkey) in his article about signal processing called "Efficient approximation for linear and non-linear signal representation". [222] The Abstract: "This paper focuses on optimum representation for both linear and non-linear type signals which have a wide range of applications in the analysis and processing of real-world signals, that is, noise, filtering, audio, image etc. Accurate representation of signals, usually is not an easy process. The optimum representation is achieved by introducing exponential bases within multiplicative calculus which enables direct processing to reveal the unknown fitting parameters. Simulation tests confirm that the newly introduced models produce accurate results while using substantially less computation and provide support for applying the new model in the field of parametric linear, non-linear signal representation for processing." (The expression "multiplicative calculus" refers here to the geometric calculus.)

The geometric calculus was used in the 2015 article "Finite product representation via multiplicative calculus and its applications to exponential signal processing" by Ali Ozyapici (Cyprus International University in Cyprus/Turkey) and Bülent Bilgehan (Girne American University in Cyprus/Turkey). (The expression "multiplicative calculus" refers here to the geometric calculus.) [225] The Abstract: "In this paper, the multiplicative least square method is introduced and is applied to integrals for the finite product representation of the positive functions. Hence, many nonlinear functions can be represented by well-behaved exponential functions. Product representation produces an accurate representation of signals, especially where exponentials occur. Some real applications of nonlinear exponential signals will be selected to demonstrate the applicability and efficiency of the proposed representation."

The geometric calculus and the multiplicative least-squares method are used in the article "Multiplicative based path loss model" by Bülent Bilgehan (Near East University in Cyprus) and Stephen Ojo (Girne American University in Cyprus). [446] From the article: "We present a newly introduced multiplicative based path loss model for the wireless channel. [Path loss is the reduction in power density of an electromagnetic wave as it propagates through space.] ... The new method uses the multiplicative least square fitting model that relates the decibel path loss to the distance with parameterized exponential‐type basis. ... The newly generated multiplicative base path loss model tested and compared with the best existing suitable models. ... The new model can be used to determine accurately the path loss for microcells in the wireless coverage area as well as in the applications of data analysis. [A microcell is a cell in a mobile phone network served by a low power cellular base station (tower), covering a limited area such as a mall, a hotel, or a transportation hub.] ... The model is based on developing formulas in the non‐Newtonian calculus to represent the regression fits to measurements given in the single‐cell signal strength for rural, urban, and suburban areas. ... The model uses 2 powerful methods for the derivation. Firstly, the model is derived from geometric multiplicative calculus, and secondly, the parameter values are extracted from the data." 

The geometric calculus was used by Ali Ozyapici and Bulent Bilgehan (both of Girne American University in Cyprus/Turkey) in their lecture "Applications of multiplicative calculus to exponential signal processing" at the 2013 Algerian Turkish International Days on Mathematics. [162] In the abstract to that lecture they state: "In this work, we study the representation of signals based on multiplicative calculus. ... Consequently, we demonstrate that multiplicative calculus representation results in a highly efficient model for the representation of exponential type signals." (The expression "multiplicative calculus" refers here to the geometric calculus.)

The geometric calculus was used by Bulent Bilgehan (Girne American University in Cyprus/Turkey) in his lecture "Finite product representation via multiplicative calculus in signal processing" at the First International Symposium on Engineering, Artificial Intelligence & Applications at Girne American University in Cyprus/Turkey. (The expression "multiplicative calculus" refers here to the geometric calculus.) [171]

Geometric arithmetic  [15] was used in an article about signal processing by Norman Zacharias (Leibniz Institute for Neurobiology, Germany), Cezary Sieluzycki (Leibniz Institute for Neurobiology, Germany), Wojciech Kordecki (University of Business in Wroczaw, Poland) , Reinhard Konig (Leibniz Institute for Neurobiology, Germany), and Peter Heil (Leibniz Institute for Neurobiology, Germany). From that article: "We therefore propose geometric, rather than arithmetic, averaging of the M100 component across subjects and suggest a novel and superior normalization procedure.  ... According to our knowledge, it has never been investigated whether ERFs and ERPs follow the additive or the multiplicative model, but this knowledge is crucial, ... ." [266]

Non-Newtonian calculus was used by Ugur Kadak (Gazi University in Turkey) and Muharrem Ozluk (Batman University in Turkey) in their article "Generalized Runge-Kutta method with respect to non-Newtonian calculus". [210] From the Abstract: In this paper, the well-known Runge-Kutta method for ordinary differential equations is developed in the frameworks of non-Newtonian calculus given in generalized form and then tested for different generating functions. The efficiency of the proposed non-Newtonian Euler and Runge-Kutta methods is exposed by examples, and the results are compared with the exact solutions.

The geometric calculus was used in the article "The Runge-Kutta Method in geometric multiplicative calculus" by Mustafa Riza and Hatice Aktore (both from Eastern Mediterranean University in North Cyprus). The article includes applications to selected well-known topics in biology, physics, and mathematics, including the Baranyi model for bacterial growth and the Rössler attractor problem. [169] From the article: "This paper illuminates the derivation, applicability, and efficiency of the multiplicative Runge-Kutta method, derived in the framework of geometric multiplicative calculus. ... We  saw  that  in  these  examples  the  multiplicative Runge-Kutta method produces signicantly better results for the same step width than the ordinary Runge-Kutta method. Furthermore, the performance of both methods was compared explicitly for one example. We observed that the multiplicative Runge-Kutta method produced smaller errors for the same computation time than the ordinary Runge-Kutta method, demonstrating the universal applicability of the proposed method. The multiplicative Runge-Kutta method was also applied to the solution of a bacterial growth model proposed by Baranyi and compared to the ordinary Runge-Kutta method, with similar results."

The bigeometric calculus was used in the article "Bigeometric calculus - a modelling tool" by Mustafa Riza (Eastern Mediterranean University in North Cyprus) and Bugce Eminaga (Girne American University in Cyprus). The article includes a new mathematical model for studying tumor therapy with oncolytic virus. In the article, Professors Riza and Eminaga state: "The results show that the Bigeometric Runge-Kutta method is superior to the ordinary Runge-Kutta method for a certain family of problems." [178]

The geometric and bigeometric calculi were used in the article "Bigeometric Calculus and Runge Kutta Method" by Mustafa Riza (Eastern Mediterranean University in North Cyprus) and Bugce Eminaga (Girne American University in Cyprus). The article (a revision of article [178]), includes new mathematical models (of the growth of cells, genes, bacteria, and viruses) for studying such things as tumor therapy with oncolytic virus and cell-cycle-specific cancer-chemotherapy. In the article, Professors Riza and Eminaga state: "Bigeometric Runge-Kutta method is, at least for a particular set of initial value problems, superior with respect to accuracy and computation-time to the ordinary Runge-Kutta method." [215]

The bigeometric calculus was used in "New solution method for electrical systems represented by ordinary differential equation", an article recommended by Piero Malcovati (University of Pavia in Italy) and written by Bülent Bilgehan (Girne American University in Cyprus), Buğçe Eminağa (Girne American University in Cyprus), and Mustafa Riza (Eastern Mediterranean University in Cyprus). [250] From the article's abstract: "The aim is to examine the existing models from bigeometric calculus point of view to obtain accuracy on the results. This work is an application of bigeometric Runge–Kutta (BRK4) method ... . This type of work arises from applications where the systems are defined by ordinary differential equations such as noise, filter, audio, chaotic circuits, etc. ... The improvement in this work is obtained by introducing bigeometric calculus in the process of seeking a solution to differential equations. ... The applicability is tested against the classical method called Runge–Kutta (RK4). Simulation results confirm the application of BRK4 method in electrical circuit analysis. The new method also provides better results for all types of input signals, i.e., linear, nonlinear, constant or Gaussian." From the article: "There has been great research in geometric (multiplicative) and bigeometric calculus within the recent years."

The geometric calculus and the bigeometric calculus were used by Hatice Aktore (Eastern Mediterranean University in North Cyprus) in an article on multiplicative Runge-Kutta methods. [133]

A lecture about a complex multiplicative Runge-Kutta method was presented by Hatice Aktore and Mustafa Riza (both of Eastern Mediterranean University in North Cyprus) at the 2012 International Conference on Applied Analysis and Algebra at Yıldız Technical University in Istanbul, Turkey. [148]

The geometric calculus was used by Yusuf Gurefe (Usak University in Turkey) and Emine Misirli (Ege University in Turkey) in their lecture "New Runge-Kutta methods for numerical solutions of multiplicative initial value problems" at the 2014 International Conference on Recent Advances in Pure and Applied Mathematics at Antalta, Turkey. In the abstract to that lecture they state: "[Geometric] calculus, which is defined in a manner analogous to the concepts in the classical calculus, has become important in recent years." [207]

The bigeometric calculus was used in a presentation by Bugce Eminaga (Girne American University in Cyprus) and Mustafa Riza (Eastern Mediterranean University in North Cyprus) at The Third International Symposium on Engineering, Artificial Intelligence and Applications (ISEAIA 2015). [404] From the Abstract: "In this study, we have ... derived the bigeometric Taylor theorem on the basis of the geometric  multiplicative Taylor theorem exploiting the relation between the geometric and bigeometric multiplicative derivative.  As  an  application  of  the  bigeometric  Taylor expansion, we derived the bigeometric Runge Kutta method. The bigeometric Runge-Kutta  method  is  applied  to  problems  with  known  closed  form  solutions  to  show  the  superiority of this method for a certain family of problems compared to the one in Newtonian calculus. Furthermore, the bigeometric Runge-Kutta method is tested on for the mathematical model of Agarwal for tumor therapy with oncologic virus and the bigeometric Rössler attractor, showing the general applicability of the method."

The bigeometric calculus was used in a presentation by Bulent Bilgehan (Girne American University in Cyprus/Turkey) and Bugce Eminaga (Girne American University in Cyprus) at The Third International Symposium on Engineering, Artificial Intelligence and Applications (ISEAIA 2015). [405] From the Abstract: "This work is an application of bigeometric Runge Kutta method aiming to solve differential  equations with nonzero  initial  condition. This type  of  work  arises  from applications  where  the  system is represented  by  ordinary differential equations such as noise, filter, audio,  caotic  circuits  e.t.c.  Solutions  to  these  types  of  equations  are  not  always easy. The improvement in this work is obtained by introducing bigeometric calculus in the process of seeking a solution to differential equations. ... Simulation results confirm the application of bigeometric Runge-Kutta method in electrical circuit analysis. The new method also provides better results for all type of input signals, i.e., linear, nonlinear, constant or Gaussian."

The bigeometric calculus was used in a presentation by Hatice Aktore and Bugce Eminaga (both from Girne American University in Cyprus) at The Third International Symposium on Engineering, Artificial Intelligence and Applications (ISEAIA 2015). [406] From the Abstract: "In this study a new modified quadratic Lorenz attractor is introduced. ... Then the multiplicative counterpart of the new system is also presented and the simulations are conducted using the multiplicative Runge-Kutta methods."

Non-Newtonian calculus is used in the article “Lorenz-generated bivariate Archimedean copulas” by Andrea Fontanari, Pasquale Cirillo, and Cornelis W. Oosterlee, all from Delft University of Technology in the Netherlands. [514]

The geometric calculus was used in the article "New 2-point implicit block multistep method for multiplicative initial value problems" by Yusuf Gurefe (Usak University in Turkey) and Emine Misirli (Ege University in Turkey). [291]

The geometric calculus was used by Muhammet Yazıcı (Karadeniz Technical University in Turkey) and Harun Selvitopi (Erzurum Technical University in Turkey) in their article "Numerical methods for the multiplicative partial differential equations". [381] From the Conclusion: "In this article, explicit Euler method, implicit Euler method and Crank-Nicolson method based on multiplicative calculus have been developed for numerical solutions of the multiplicative heat equation with the initial and boundary conditions. The algorithms are tested and the numerical results compared with the exact solution are quite satisfactory. The present methods with some modifications might be applied to many multiplicative partial differential equations arising in engineering and sciences."

The geometric calculus is among the topics presented in the mathematics textbook Mathematical Analysis: Fundamentals by Agamirza Bashirov. [179] Included is application of the geometric calculus to differential equations. From the Abstract to Chapter 11: "An interesting feature of this chapter is an introduction to multiplicative calculus, which is an alternative to the [classical] calculus of Newton and Leibnitz. By use of methods of multiplicative calculus it is proved that an infinitely-many times differentiable function may not be analytic." From a list of items in the publisher's (Academic Press) Description: "Elements of multiplicative calculus aiming to demonstrate the non-absoluteness of Newtonian calculus." (The expression "multiplicative calculus" refers here to the geometric calculus.)

Non-Newtonian calculus is used in the lecture "A new analysis of non-smooth convex optimisation problems by using non-Newtonian calculus" by Ali Hakan Tor (Abdullah Gül University in Turkey), at the International Congress of Mathematicians 2018 (ICM 2018) in Rio De Janeiro, Brazil. [422] The International Congress of Mathematicians (ICM) is the world's largest mathematics conference. Hosted by the International Mathematical Union, the ICM meets once every four years and awards four prizes for excellence in mathematics research: the Fields Medals, the Rolf Nevanlinna Prize, the Carl Friedrich Gauss Prize, and the Chern Medal Award. The Fields Medal is widely regarded as the highest honor a mathematician can receive, and is considered by some to be the mathematician's "Nobel Prize".

Multiplicative calculus was included among the topics proposed for "Summability theory and its applications", a session at the 11th ISAAC congress (The International Society for Analysis, its Applications and Computation) at Linnaeus University in Sweden, 14-18 August 2017. From the conference-announcement in February of 2017: "The aim of this session [Summability theory and its applications] is to discuss the recent progress on the summability theory, including but not limited to special summability methods, Tauberian theorems and conditions, fuzzy Tauberian theorems, statistical convergence, Tauberian theorems in quantum calculus, Tauberian theorems in multiplicative calculus and related topics and applications." [332]

Application of the bigeometric derivative to the theory of elasticity in economics was made by Fernando Córdova-Lepe (Universidad Católica del Maule in Chile) . (He referred to the bigeometric derivative as the "multiplicative derivative.") [3, 4, 105] Elasticity is also discussed in Non-Newtonian Calculus [15], Bigeometric Calculus: A System with a Scale-Free Derivative [10], and The First Systems of Weighted Differential and Integral Calculus [9].

The geometric calculus was used in the article "A q-analogue of the multiplicative calculus: q-multiplicative calculus" by Gokhan Yener and Ibrahim Emiroglu (both of Yildiz Technical University in Turkey). [267] From that article: "According to our research, we strongly believe that q-multiplicative calculus, which is the extended version of multiplicative calculus [i.e., the geometric calculus], can show the way for further research fields with new definitions, theories and applications."

The geometric calculus was used in the article "Some of the methods used for the approximate account of the roots of univariate equations and their applications in q-multiplicative calculus" by Gokhan Yener (Yildiz Technical University in Turkey). [328]

The geometric calculus was used by Paolo Perrone (University of Milan in Italy and Max Planck Institute in Germany/Italy) in his article "A gauge theoretic approach to quantum physics". [194]

Application of non-Newtonian calculus to information technology was made by S. L. Blyumin of the Lipetsk State Technical University in Russia. [23]

An article concerning minimization methods based on the geometric and bigeometric calculi was written by Ali Ozyapici (Girne American University in Cyprus/Turkey), Mustafa Riza (Eastern Mediterranean University in North Cyprus), Bulent Bilgehan (Girne American University), and Agamirza E. Bashirov (Eastern Mediterranean University). [176] From the Abstract to the article: "Theory and applications of [geometric] and [bigeometric] calculi have been evolving rapidly over the recent years. As numerical minimization methods have a wide range of applications in science and engineering, the idea of the design of minimization methods based on [geometric] and [bigeometric] calculi is self-evident. In this paper, the well-known Newton minimization method for one and two variables is developed in the framework of [geometric] and [bigeometric] calculi. The efficiency of these proposed minimization methods is demonstrated by examples, ... . One of the striking results of the proposed method is that the rate of convergence and the range of initial values are considerably larger compared to the original method."

The geometric calculus was used in "Experimentally approved generalized model for circuit applications", an article about electrical system analysis by Bülent Bilgehan (Near East University, in Turkey), Ali Özyapıcı (Cyprus International University, in Turkey), and Zehra B.Sensoy (Cyprus International University, in Turkey). [373] From the Summary: "In this paper, a generalized model based on the multiplicative least square method is presented ... The advantage of the method is due to exponential derivation process within multiplicative [geometric] calculus and has the flexibility to represent used functions such as Gaussian and exponentials."

The bigeometric calculus was used in the article "A new approach to the concept of linearity. Some elements for a multiplicative linear algebra" by Fernando Córdova-Lepe, Rodrigo del Valle, and Karina Vilches Ponce (all from Universidad Católica del Maule in Chile). [416] From the Abstract: "An arithmetic based on the multiplication is the base of the Bigeometric (and Multiplicative) Calculus, a type of non-Newtonian Calculus which has had uses in the digital processing of images. In this work, we present the fundamental elements to construct a proposal of Multiplicative Linear Algebra with view of being a support for a Multiplicative Calculus in Several Variables and its applications. Some important concepts such as vector space, linear transformation and matrix representation are reviewed. In particular, we give a meaning and context to the operation: matrix raised to matrix."

Non-Newtonian calculus was used in the article "On line and double integrals in the non-Newtonian sense" by Ahmet Faruk Çakmak (Yıldız Technical University in Turkey) and Feyzi Başar (Fatih University in Turkey). [201] From the Abstract: "This paper is devoted to line and double integrals in the sense of non-Newtonian calculus (*-calculus). Moreover, in the sense of *-calculus, the fundamental theorem of calculus for line integrals and double integrals are stated and proved, and some applications are presented."

Application of non-Newtonian calculus to functional analysis was made by Cengiz Türkmen and  Feyzi Basar, both of Fatih University in Turkey. Their work was presented at the First International Conference on Analysis and Applied Mathematics, whose purpose was "to bring together mathematicians working in the area of analysis and applied mathematics to share new trends of applications of math". [112]

The geometric calculus was used in the article "On multiplicative fractional calculus" by Thabet Abdeljawad (Prince Sultan University in Saudi Arabia). [251]  From that article: "In this work, we bring together [the geometric] calculus and fractional calculus."

The geometric calculus was used in the article "On geometric fractional calculus" by Thabet Abdeljawad (Prince Sultan University in Saudi Arabia) and Michael Grossman (United States). [270] From that article: "In this work, we bring together [the geometric] calculus and fractional calculus."

Non-Newtonian calculus is used in the article “Some applications of extended calculus to non-Newtonian flow in pipes” by M. Letelier (University of Santiago in Chile) and J. Stockle (Diego Portales University in Chile). [531] The article includes an application of weighted non-Newtonian calculus. The Abstract: “Fractional and non-Newtonian calculus are an extension of classical calculus, usually known for providing new mathematical tools useful in science, developed from alternative approaches. Among fractional calculus, Riemann-Liouville and Caputo fractional derivatives have been the most popular operators employed in spite of their complexity. In this work, two novel and compact methods are presented as an alternative to the fractional calculation options. To test the feasibility of proposed methods, three classical fluid mechanic problems are studied: the flow through circular pipe, parallel plates and annulus, by modifying the constitutive equations into their fractional equivalent. On the other hand, a new weighted non-Newtonian derivative is proposed to extend the possibilities to model fluid viscosity based on the influence of nonadjacent layers, using the pipe flow as an example. Results show that proposed fractional models can describe shear-thinning and shear-thickening behaviors depending on the fractional order of the derivative, while the weighted derivative allows to expand the way viscosity is modeled, demonstrating the suitability of these approaches to describe physical problems.”

The geometric calculus was used in the article "Generalization of Special Functions and its Applications to Multiplicative and Ordinary Fractional Derivatives" by Ali Ozyapıcı (Cyprus International University in Cyprus), Yusuf Gurefe (Usak University in Turkey), and Emine Mısırlı (Ege University in Turkey). [334] From the article: "The goal of this paper is to extend the classical and multiplicative fractional derivatives. For this purpose, it is introduced the new extended modified Bessel function and also give an important relation between this new function and the confluent hypergeometric function. ...  In [the past 10 years], superiority of the multiplicative calculus over ordinary calculus was proved by many studies. The most significant among these studies are in biomedical image analysis, complex analysis, growth phenomena, numerical analysis, actuarial science, finance, demography, biology, and recently in accounting. In order for multiplicative calculus to be used efficiently in all respects more studies needs to be done in various fields. ... First application of the multiplicative calculus to fractional derivative is executed by Abdeljawad and Grossman in "On geometric fractional calculus". ... It can be easily seen that the results obtained in this paper are new and effective mathematical tools and, also extensions of many results in the literature".

The geometric calculus was used in the article "On multiplicative complex integral" by Agamirza E. Bashirov and Sajedeh Norozpour (both from Eastern Mediterranean University in North Cyprus) at the Third International Conference on Recent Advances in Pure and Applied Mathematics. [293] From the Abstract: "In 1972 Grossman and Katz [Non-Newtonian Calculus] proposed alternative calculi to the calculus of Newton and Leibnitz. ... This pioneering work initiated numerous studies."

Non-Newtonian calculus was used in the article "On multivalued complex functions" by Agamirza E. Bashirov and Sajedeh Norozpour (both from Eastern Mediterranean University in North Cyprus). From the article: " Is it possible to present complex calculus where log-function and other related functions are single-valued? In this paper we give a positive answer to this question. ... Michael Grossman and Robert Katz [Non-Newtonian Calculus] pointed out the realizations of CALCULUS which are different from Newtonian one, calling them as non-Newtonian calculi. ... We expect many items of complex calculus to be changed, mainly being simplified." [303]

Non-Newtonian calculus was used in a novel approach to complex analysis, developed by Agamirza E. Bashirov and Sajedeh Norozpour (both from Eastern Mediterranean University in North Cyprus) and explained in their article "Riemann surface of complex logarithm and multiplicative calculus". [309] From that article (which is a revision of [303]): "Could elementary complex analysis ... be made more elementary? In this paper we demonstrate that a little reorientation of existing elementary complex analysis brings a lot of benefits, including operating with single-valued logarithmic and power functions ... But instead of advanced mathematical concepts such as manifolds, differential forms, integration on manifolds, etc, which are necessary for introducing complex analysis in Riemann surfaces, we use rather elementary methods of multiplicative calculus [geometric calculus]. We think that such a reoriented elementary complex analysis could be especially successful as a first course in complex analysis for students of engineering and physics, and even for applied mathematics students who indeed do not see a second and more advanced course in complex analysis. It would be beneficial for the students of pure mathematics programs as well, because it is a more appropriate introduction to complex analysis on Riemann surfaces than the existing one. ... We think that an appropriate calculus for Fourier series is complex [geometric] calculus ... This paper just demonstrates that the Riemann surface of complex logarithm can be replaced by methods of multiplicative calculus. This raises the following challenging question: Can complex analysis on Riemann surfaces be easily covered by considering different non-Newtonian calculi? This requires a wide research of interested experts in the field.  We expect many items of complex analysis to be changed, mainly being simplified."

Non-Newtonian calculus was used in the article "On an alternative view to complex calculus" by Agamirza E. Bashirov and Sajedeh Norozpour (both from Eastern Mediterranean University in North Cyprus). [412] From the Abstract: "In most (if not all) textbooks on complex calculus, the differentiation and integration of complex functions are presented by using the algebraic form of complex variables because the respective formulae in terms of the polar form are inappropriate. In this paper, we demonstrate that by transferring the field structure of the system of complex numbers to the Riemann surface of complex logarithm and changing the sense of derivative and integral [to bigeometric], complex calculus can be delivered in terms of the polar form of complex variables identically to the presentation in terms of algebraic form."

The geometric calculus was used in the article "Multiplicative Laplace transform and its applications" by Numan Yalcın (Gumushane University in Turkey) , Ercan Celik (Ataturk University in Turkey), and Ahmet Gokdogan (Gumushane University in Turkey). [295]  From the Abstract: "In this work, taking definitions and properties of Laplace transform in classical analysis as a basis, we give some basic definitions and properties of the multiplicative Laplace transform. In addition, solutions of some multiplicative differential equations are obtained by the help of this transform." From the article: "Non-Newtonian calculus allowed scientists to look from a different point of view to the problems encountered in science and engineering."

The geometric calculus was used in the article “Multiplicative Fourier transform and its applications to multiplicative differential equations” by Aarif Hussain Bhat, Javid Majid, Tafazul Rehman Shah, Imtiyaz Ahmad Wani, and Renu Jain (all from Jiwaji University Gwalior in India). [482] From the Conclusion: “In this paper, we have defined multiplicative Fourier transform for positive definite functions. ... Further, the relation between multiplicative Laplace transform and multiplicative Fourier transform has been established. Moreover, it has been shown that this transform can be employed as an alternative method to find solutions of multiplicative differential equations.”

Non-Newtonian calculus was used in an article on Markov processes and multiplicative (geometric) Bernstein polynomials. The article is entitled “Modifying an approximation process using non-Newtonian calculus” and was written by Octavian Agratini (Babes-Bolyai University in Romania) and Harun Karsli (Bolu Abant Izzet Baysal University in Turkey). [513] From the article: “In the present note we modify a linear positive Markov process of discrete type by using so called multiplicative [geometric] calculus. In this framework, a convergence property and the error of approximation are established. In the final part some numerical examples are delivered. ... At this point we refer to non-Newtonian calculus also called as multiplicative calculus. In the 1970s, Michael Grossman and Robert Katz developed this type of calculation moving the roles of subtraction and addition to division and multiplication. ... This type of calculus was also called geometric calculus in order to emphasize that changes in function arguments are measured by differences, while changes in values are measured in ratios. ... In the present paper our aim is to bring up multiplicative calculus to the attention of researchers in the branch of positive approximation processes. ... In the next two sections we present basic elements of multiplicative calculus and new results as regards the positive approximation processes.”

The geometric calculus was used in the article “Multiplicative Sumudu transform and its Applications” by Aarif Hussain Bhat, Javid Majid, and Imtiyaz Ahmad Wani, all from Jiwaji University in India. [471] From the article: “With the passage of time some researchers have proved that multiplicative calculus is very helpful in solving problems related to science and engineering fields.”

Non-Newtonian analysis was used by Ugur Kadak (Gazi University in Turkey) and Yusuf Gurefe (Usak University in Turkey) in their article "A generalization on weighted means and convex functions with respect to the non-Newtonian calculus". [301]

The concept of ‘anageometric convex function’ is introduced by Ugur Kadak (Gazi University in Turkey) and Yusuf Gurefe (Usak University in Turkey) in their article "A generalization on weighted means and convex functions with respect to the non-Newtonian calculus". [301]

Non-Newtonian calculus was used in the article “Non-Newtonian fuzzy numbers” by Ugur Kadak (Gazi University in Turkey). [230]

Non-Newtonian calculus was used in the article “Non-Newtonian fuzzy numbers and related applications" by Ugur Kadak (Bozok University in Turkey). [399]

Non-Newtonian Calculus was used by Emine Misirli (Ege University in Turkey) and Yusuf Gurefe (Bozok University in Turkey) in their article "Some numerical methods on multiplicative calculus". [302] From the article: "In this study, we presented multiplicative methods to solve the first order multiplicative differential equations. ... Comparing the obtained approximate numerical solutions with the other methods, we observe that the presented algorithms give more approximate results than the others.  ... Consequently it can be said that many problems in engineering and sciences might be solved by these developed methods."

The geometric calculus is used in the article "Multiplicative Newton’s methods with cubic convergence" by Emrah Unal (Artvin Coruh University, in Turkey), Ishak Cumhur (Recep Tayyip Erdogan University, in Turkey), and Ahmet Gokdogan (Gumushane University, in Turkey). [374] From the article: "In this work, since we work in multiplicative analysis ... ."

The geometric calculus was used in the article "Some variants of multiplicative Newton method with third-order convergence" by Emrah Unal (Artvin Coruh University in Turkey), Ishak Cumhur (Recep Tayyip Erdogan University in Turkey), and Ahmet Gökdoğan (Gumushane University in Turkey). [397]

In their article "On relativistic velocity addition", Agamirza Bashirov and Sinem Genc (both from Eastern Mediterranean University in North Cyprus) discuss the "tanh-calculus", a non-Newtonian calculus related to Einstein's relativistic sum of two speeds. ("Tanh" denotes the well-known hyperbolic-tangent function.) [444]

Non-Newtonian calculus was used in the article "Arithmetic summable sequence space over non-Newtonian field" by Taja Yaying (Dera Natung Government College in India) and Bipan Hazarika (Guwahati University in India). [401] From the article: "We expect that our given notions, definitions, results and the related investigations might be useful for others in modelling various problems in the fields of economics, engineering, numerical analysis, mathematical physics, fuzzy theory, etc."

Lectures concerning non-Newtonian calculus were presented at the International Conference on Computational and Statistical Methods in Applied Sciences (COSTAS) at Ondokuz Mayıs University in Samsun, Turkey, on 9-11 November 2017: "Non-Newtonian improper integrals" by Murat Erdoğan and Cenap Duyar, "Some geometric properties of the non-Newtonian sequence space" by Nihan Güngör, and "On the function sequences and series in the non-Newtonian calculus" by Fatmanur Erdoğan and Birsen Sağır Duyar. [378]

Non-Newtonian calculus was used in a presentation at the 2018 International Conference on Analysis and Its Applications (ICAA-2018): “Some new inequalities and Hermite-Hadamard-Fejer inequality via non-Newtonian calculus” by Erdal Unluyol and Yeter Erdas (both from Ordu University in Turkey). [450]

Non-Newtonian calculus was used in a presentation at the 2019 International Conference on Applied Analysis and Mathematical Modeling (ICAAMM19): “*-Hermite-Hadamard-Fejer inequality and some new inequalities via *-calculus” by Erdal Unluyol, Yeter Erdas, and Seren Salas (all from Ordu University in Turkey). [478]

The geometric calculus was used in the article “A new look at the classical sequence spaces by using multiplicative calculus” by Yusuf Gurefe (Usak University in Turkey), Ugur Kadak (Bozok University in Turkey), Emine Misirli (Ege University in Turkey), and Alia Kurdi (University Politehnica of Bucharest in Romania). (The expression "multiplicative calculus" refers here to the geometric calculus.) [232]

Non-Newtonian calculus was used in the article "On the classical paranormed sequence spaces and related duals over the non-Newtonian complex field" by Ugur Kadak (Bozok University in Turkey), Murat Kirisci (Istanbul University in Turkey), and Ahmet Faruk Cakmak (Yıldız Technical University in Turkey). [226]

Geometric arithmetic [15] is used, and non-Newtonian calculus is discussed, in the article "On multiplicative difference sequence spaces and related dual properties" by Ugur Kadak (Gazi University in Turkey). [307]

Nonclassical arithmetrics (i.e., complete ordered fields distinct from the real number system) [10, 11, 15] are used and non-Newtonian calculus is discussed in the article "Convex functions and some inequalities in terms of the non-Newtonian calculus" by Erdal Unluyol (Ordu University in Turkey), Seren Salas (Ordu University), and Imdat Iscan (Giresun University in Turkey). [347]

Nonclassical arithmetrics (i.e., complete ordered fields distinct from the real number system) [10, 11, 15] are used, and non-Newtonian calculus is discussed in the article "Convexity and Hermite-Hadamard type inequality via non-Newtonian calculus" by Erdal Unluyol (Ordu University, in Turkey), Seren Salas (Ordu University), and Imdat Iscan (Giresun University in Turkey). [376]

In their article "A new view of some operators and their properties in terms of the non-Newtonian calculus", Erdal Unluyol (Ordu University, in Turkey), Seren Salas (Ordu University), and Imdat Iscan (Giresun University in Turkey) discuss nonclassical arithmetic [10, 11, 15], non-Newtonian calculus, non-Newtonian Hilbert spaces, and non-Newtonian normed spaces. [377]

Geometric arithmetic and geometric calculus are featured topics in the article "Generalized Köthe-Toeplitz dual of some geometric difference sequence spaces" by Shadab Ahmad Khan and Ashfaque A. Ansari (both from Gorakhpur University in India). [319]

Geometric arithmetic and geometric calculus were used in the article “Geometric difference sequence spaces in numerical analysis” by Shadab Ahmad Khan, from Mahamaya Government Degree College and Lucknow University in India. [470]

Geometric calculus, geometric arithmetic, and geometric complex-numbers are featured topics in the article "Applications of infinite matrices in non-Newtonian calculus for paranormed spaces and their Toeplitz duals" by Kuldip Raj and Charu Sharma (both from Shri Mata Vaishno Devi University in India. [345]

Non-Newtonian calculus was used in the article "Cesaro summable sequence spaces over the non-Newtonian complex field" by Ugur Kadak (Bozok University in Turkey). [253]

The bigeometric calculus was used in the article “Bigeometric Cesaro difference sequence spaces and Hermite interpolation” by Sanjay Kumar Mahto, Atanu Manna, and P. D. Srivastava (all from the Indian Institute of Technology). [456] From the article: “The important applications of bigeometric calculus are seen in fractal dynamics of materials, fractal dynamics of biological systems, etc. Moreover [geometric] calculus is used to establish non-Newtonian Runge-Kutta methods, Lorenz systems, and some finite difference methods.”

The geometric calculus was used in the article “On integral inequalities for product and quotient of two multiplicatively convex functions” by M. A. Ali (Government College University in Pakistan), M. Abbas (University of Pretoria in South Africa), Z. Zhang (Nanjing Normal University in China), I. B. Sial (Government C. U.), and R. Raif (Government U. C.). [475] Abstract: “In this paper, we derived integral inequalities of Hermite-Hadamard type in the setting of multiplicative calculus for multiplicatively convex and convex functions. We also derived integral inequalities of Hermite-Hadamard type for product and quotient of multiplicatively convex and convex functions in multiplicative calculus.”

Non-Newtonian calculus was used in an article by S. L. Blyumin (Lipetsk State Technical University in Russia) involving binary arithmetic-operations and functional equations. [180]

The geometric calculus was used in the article ‘“An introduction to non-smooth convex analysis via multiplicative derivative” by Ali Havana Tor (Abdullah Gul University in Turkey). [474] The Abstract: “In this study, *-directional derivative and *-subgradient are defined using the multiplicative [geometric] 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.”

The geometric calculus was used in an article on marketing by David Godes (University of Maryland). [192]

Several of the books and articles by Jane Grossman, Michael Grossman, and Robert Katz contain discussion about applications of non-Newtonian calculus to subjects such as growth/decay analysis, analytic geometry, vectors, least-squares methods, centroids, complex numbers, sigmoidal functions, relativistic composition of speeds, measurement (physics), psychophysics, weighted calculus, meta-calculus, averages (of functions), and means (of two positive numbers). [15, 11, 9, 7, 10, 8, 12, 14, 34, 35]

Non-Newtonian calculus may have application to neuroscience. According to Roberto Sotero Diaz (Hotchkiss Brain Institute of the University of Calgary in Canada): "... I’m very interested in the application of non-Newtonian calculus to computational neuroscience, specifically for solving biophysical models of the generation of neuronal activity. The sigmoidal calculus, as introduced in your book Non-Newtonian Calculus has the potential to be a very useful approach to the problems I want to solve ... ." [15]

The non-Newtonian averages (of functions) were used to construct a family of means (of two positive numbers). [8, 14] Included among those means are some well-known ones such as the arithmetic mean, the geometric mean, the harmonic mean, the power means, the logarithmic mean, the identric mean, and the Stolarsky mean.  The family of means was used to yield simple proofs of some familiar inequalities. [14] Publications [8, 14] about that family are cited in articles [29-32, 118, 153, 331].

Non-Newtonian calculus was used in the famous 2006 report "Stern Review on the Economics of Climate Change", according to a 2012 critique of that report (called "What is Wrong with Stern?") by former UK Cabinet Minister Peter Lilley and economist Richard Tol. "Stern Review on the Economics of Climate Change", which is over 700-pages long, was commissioned by the UK government, was written by a team led by Nicholas Stern (former Chief Economist at the World Bank), and has drawn worldwide attention. [116, 165]

The geometric calculus was used in an article on "statistics of acoustically induced bubble-nucleation events in in-vitro blood" by Jérôme Gateau, Nicolas Taccoen, Mickaël Tanter, and Jean-François Aubry (their affiliations: Institut Langevin; ESPCI ParisTech; CNRS UMR 7587, INSERM U979; Université Paris Diderot, Paris 7). [166]

Non-Newtonian calculus was used in the article "A new theoretical discrete growth distribution with verification for microbial counts in water" by James Englehardt (University of Miami), Jeff Swartout (US EPA Facilities, Cincinnati, OH, USA), and Chad Loewenstine (BLDG 27958-A, Quantico, VA, USA). [287]

Non-Newtonian calculus was used in the article "The discrete Weibull distribution: an alternative for correlated counts with confirmation for microbial counts in water" by James D. Englehardt (University of Miami) and Ruochen Li (Shenzhen, China). [85]

The geometric calculus was used in the article "Distributions of autocorrelated first-order kinetic outcomes: illness severity" by James D. Englehardt (University of Miami). (The expression "product integral" used in the article refers to the geometric integral.) [227] From that article: "Process increments represent multiplicative causes. In particular, illness severities are modeled as such, occurring in proportion to products of, e.g., chronic toxicant fractions passed by organs along a pathway, or rates of interacting oncogenic mutations. ... [the geometric integral] is fundamental to developing the continuously multiplicative nature of the continuous first-order kinetic rate law, not otherwise obvious."

The geometric calculus was used in the article "Distributions of autocorrelated first-order kinetic outcomes: illness severity [Revised author manuscript incorporating corrections published March 21, 2017 http://dx.doi.org/10.1371/journal.pone.0174526]" by James D. Englehardt (University of Miami). [382] From the article: "Process increments represent multiplicative causes. In particular, illness severities are modeled as such, occurring in proportion to products of, e.g., chronic toxicant fractions passed by organs along a pathway, or rates of interacting oncogenic mutations. ... Although use of the [geometric] integral in Equations 4 – 5 represents an unfamiliar representation of a known result, it is fundamental to developing the continuously multiplicative nature of the continuous first-order kinetic rate law, not otherwise obvious. ... Thus, Equations 4 and 5 provide the required basis for simulation of such “continuous” first-order processes in the Results section. Further, they clarify the definition of a population geometric mean [16]."

"Application of geometric calculus in numerical analysis and difference sequence spaces" is the title of an article by Khirod Boruah and Bipan Hazarika, both from Rajiv Gandhi University in India. Included in the article is discussion of the "advantages of geometric interpolation formulae over ordinary interpolation formulae". (In that article the word "geometric" refers to the geometric calculus, not to the subject of geometry.) From the article: "The main aim of this paper is ... and to obtain the Geometric Newton-Gregory interpolation formulae which are more useful than Newton-Gregory interpolation formulae. ... In this paper, we have defined geometric difference sequence space and obtained the Geometric Newton-Gregory interpolation formulae. Our main aim is to bring up geometric calculus to the attention of researchers in the branch of numerical analysis and to demonstrate its usefulness. We think that geometric calculus may especially be useful as a mathematical tool for economics, management and finance. ... The authors would like to thank [Chauvenet-Prize winner] Prof. Steven Krantz, Editor-in-Chief [Journal of Mathematical Analysis and Applications] for his comments.”[276]

Geometric arithmetic [15] and geometric complex-numbers [11] were used in the article "Generalized geometric difference sequence spaces and its duals" by Khirod Boruah (Rajiv Gandhi University in India), Bipan Hazarika (Rajiv Gandhi University in India), and Mikail Et (Firat University in Turkey). From that article: "Generally speaking multiplicative calculus is a methodology that allows one to have a different look at problems which can be investigated via calculus. In some cases, for example for growth related problems, the use of multiplicative calculus is advocated instead of a traditional Newtonian one." (The expression "multiplicative calculus" refers here to the geometric calculus.) [277]

Geometric arithmetic [15] and geometric complex-numbers [11] were used in the article "On some generalized geometric difference sequence spaces" by Khirod Boruah and Bipan Hazarika (both from Rajiv Gandhi University in India). [380]

"Bigeometric calculus and its applications" is the title of an article by Khirod Boruah and Bipan Hazarika, both from Rajiv Gandhi University in India. [298] From the Abstract: "Based on M. Grossman in Bigeometric Calculus: A System with a Scale-Free Derivative [10]  and Grossman and Katz in Non-Newtonian Calculus [15], in this paper we discuss applications of bigeometric calculus in different branches of mathematics and economics."

Non-Newtonian calculus was used in the article “Applications of non-Newtonian calculus for classical spaces and Orlicz functions” by Kuldip Raj1 and Charu Sharma1, both from Shri Mata Vaishno Devi University in India. [463] From the Abstract: “The objective of this paper is to introduce and study some sequence spaces over the geometric complex numbers by means of Museilak–Orlicz function. ... Moreover, by using the concept of non-Newtonian calculus we prove the completeness of the spaces.”

The bigeometric calculus and its applications are discussed in the article "Some basic properties of G-Calculus and its applications in numerical analysis" by Khirod Boruah and Bipan Hazarika, both from Rajiv Gandhi University in India. [294] (The authors used the expression "G-calculus" instead of "bigeometric calculus".)

The bigeometric calculus was used in the article "G-Calculus" by Khirod Boruah and Bipan Hazarika, both from Rajiv Gandhi University in India. [436] From the article: "Generally, in growth related problems, price elasticity, and numerical approximations problems, bigeometric-calculus can be advocated instead of the traditional Newtonian one."

The bigeometric calculus was presented in the article “Bigeometric Integral Calculus” by Khirod Boruah and Bipan Hazarika, both from Rajiv Gandhi University in India. [477]

The bigeometric calculus was used in the article “Solvability of bigeometric differential equations by numerical methods” by Khirod Boruah (Rajiv Gandhi University in India), Bipan Hazarika (Rajiv Gandhi University) and A. E. Bashirov (Eastern Mediterranean University in North Cyprus). [469] The article includes applications of the following methods: bigeometric Euler, bigeometric Taylor-series, and bigeometric Runge-Kutta.

The bigeometric calculus was used by William Campillay and Manuel Pinto (both of the Universidad de Santiago de Chile) in a lecture on bigeometric differential-equations at the VIII Congreso de Análisis Funcional y Ecuaciones de Evolución at Universidad de Santiago de Chile. [172]

The geometric calculus was used in an article on probability and statistics called "Estimating the maximum expected value in continuous reinforcement learning problems" by Carlo D’Eramo, Alessandro Nuara, Matteo Pirotta, and Marcello Restelli (all from Politecnico di Milano in Italy). (Reinforcement learning is an area of machine learning that is studied in disciplines such as game theory, control theory, operations research, information theory, simulation-based optimization, multi-agent systems, swarm intelligence, statistics, and genetic algorithms.) [349]

A new Lebesgue-type theory extending the notion of geometric-integration was set forth in the Wikipedia article "Product integral" on 2 April 2018. [413]

Non-Newtonian calculus was used in the article "Modeling quantum information dynamics achieved with time-dependent driven fields in the context of universal quantum processing" by Francisco Delgado and Suset Rodriguez (both from Tecnologico de Monterrey: School of Engineering and Sciences, in Mexico). [423]

Non-Newtonian calculus was used in an article on on quantum information and quantum mechanics by Francisco Javier Delgado Cepeda, from Tecnológico de Monterrey in Mexico. The article is called "Assembling Large entangled states in the Rényi-Ingarden-Urbanik entropy measure under the SU(2)-dynamics decomposition for systems built from two-level subsystems". [479] From the article: “Quantum Information is a discipline derived from Quantum Mechanics which uses quantum systems to exploit their states as information recipients. ... Quantum information is pursuing novel approaches to set information and processing on physical systems exhibiting quantum properties, such as superposition and entanglement. ... The aim of this paper is focused in the complexity to get general quantum states departing from the most simple ones, particularly those possibly exhibiting several types of entanglement (really, from separable to the genuine entangled states as a spectrum). ... Thus, because the evolution operator can be calculated from the Hamiltonian as the time-ordered integral (Grossman and Katz 1972), basically containing time-ordered products preserving that structure, it inherits the same block structure.”

The geometric integral is discussed in the article "Product integrals and sum integrals" by Raymond A. Guenther (University of Nebraska at Omaha). [236]

Weighted non-Newtonian calculus [9] was used by Ziyue Liu and Wensheng Guo (both of the University of Pennsylvania) in an article on spline smoothing. [119]

Weighted non-Newtonian calculus [9] was used by David Baqaee (Harvard University) in an article on an axiomatic foundation for intertemporal decision making. [86]

The First Systems of Weighted Differential and Integral Calculus [9] was used in the book Minimization of Climatic Vulnerabilities on Mini-hydro Power Plants: Fuzzy AHP, Fuzzy ANP Techniques and Neuro-Genetic Model Approach by Mrinmoy Majumder (National Institute of Technology Agartala in India). [274] From page 26: "The weight function was proposed by Grossman et al. (1980) [The First Systems of Weighted Differential and Integral Calculus ] which represents the weighted impact of a set of parameters based on their priority values and magnitude."

The First Systems of Weighted Differential and Integral Calculus [9] is used in the article "Diversification into bioenergy" by Fredrik Hansson and Staffan Marklund (both from Chalmers University of Technology in Göteborg, Sweden). [297]

Averages: A New Approach [8] is cited in an article on radiation-oncology physics entitled “Using weighted power mean for equivalent square estimation”. [447] (Power means are discussed in Averages: A New Approach.) The authors: Sumin Zhou, Qiuwen Wu, Xiaobo Li, Rongtao Ma, Dandan Zheng, Shuo Wang, Mutian Zhang, Sicong Li, Yu Lei, Qiyong Fan, Megan Hyundai, Tye Diener, and Charles Enke (University of Nebraska Medical Center in USA, Duke University Health System in USA, Fujian Medical University in China). From the article: “However, tests using the AIC criteria presented in this work unequivocally established the superiority of the new formula, especially in clinical scenarios enabled by modern technological revolutions. ... The WPM  [Weighted Power Mean] formula outperformed the traditional formulae on three testing datasets. With increasing utilization of very elongated, small rectangular fields in modern radiotherapy, improved photon output factor estimation is expected by adopting the WPM formula in treatment planning and secondary MU check.” (In radiotherapy, radiation treatment planning is the process in which a team consisting of radiation oncologists, radiation therapists, medical physicists, and medical dosimetrists plan the appropriate external beam radiotherapy or internal brachytherapy treatment technique for a patient with cancer. MU check is an independent second-check verification software package designed to validate monitor unit (MU) calculations performed by a primary radiation treatment planning.)

Non-Newtonian calculus was used by Stanley Paul Palasek (Sonoran Science Academy in Tucson, Arizona) in a biology project on opioid peptide delivery at an Intel® International Science and Engineering Fair. [97]

Non-Newtonian calculus was used by J. I. King in a study on atmospheric temperature (optical measure theory and inverse transfer theory). [102] (This study is used in [89].)

A lecture about the bigeometric calculus was presented by Ahmet Faruk Çakmak at the 2011 International Conference on Applied Analysis and Algebra at Yıldız Technical University in Istanbul, Turkey. [107]

Non-Newtonian calculus, and related matters are topics in the article "Matrix transformations between certain sequence spaces over the non-Newtonian complex field" by Ugur Kadak and Hakan Efe (both of Gazi University in Turkey). [189]

Non-Newtonian calculus and related matters are topics in the article "The construction of Hilbert spaces over the non-Newtonian field" by Ugur Kadak and Hakan Efe (both of Gazi University in Turkey). [190]

The geometric calculus was used by Uğur Kadak (Gazi University in Turkey) and Yusuf Gurefe (Bozok University in Turkey) in their presentation at the 2012 Analysis and Applied Mathematics Seminar Series of Fatih University in Istanbul, Turkey. [117]

Application of non-Newtonian analysis to function spaces was made by Ahmet Faruk Cakmak (Yıldız Technical University in Turkey) and Feyzi Basar (Fatih University in Turkey) in their lecture at the 2012 conference The Algerian-Turkish International Days on Mathematics, at University of Badji Mokhtar at Annaba, in Algeria. [127]

Non-Newtonian analysis was used by Ahmet Faruk Cakmak (Yıldız Technical University in Turkey) and Feyzi Basar (Fatih University in Turkey) in their 2015 article “Some sequence spaces and matrix transformations in multiplicative sense”, and in their 2014 lecture with the same name at the Çankırı Karatekin University Mathematics Days, Çankırı, Turkey. [217, 204] In that article the authors state: "As an alternative to the classical calculus of Newton and Leibniz, [the geometric] calculus gives more convenient results in some specific problems."

Application of non-Newtonian analysis to "continuous and bounded functions over the field of non-Newtonian/geometric complex numbers" was made by Zafer Cakir (Gumushane University, Turkey). [137, 159]

Non-Newtonian calculus was one of the topics of discussion at the 2013 Algerian-Turkish International Days on Mathematics conference at Fatih University in Istanbul, Turkey. [138]

Non-Newtonian calculus is listed among the topics covered in the International Journal on Recent Trends in Life Science and Mathematics (IJLSM). [167]

Non-Newtonian calculus and related matters are used in the article "Certain sequence spaces over the non-Newtonian complex field" by Sebiha Tekin and Feyzi Basar, both of Fatih University in Turkey. [144]

The geometric calculus was used in the article "Matrix transformation between geometric difference sequence spaces" by Khirod Boruah (Rajiv Gandhi University in India). [403] From the article: "In the area of non-Newtonian calculus pioneering work was carried out by Grossman and Katz [Non-Newtonian Calculus] ... Nowadays geometric calculus is an alternative to the usual calculus of Newton and Leibniz. ... In  some  cases, mainly  problems of price elasticity, resiliency, multiplicative growth etc. the use of multiplicative calculus is advocated instead of a traditional Newtonian calculus."

The geometric integral is useful in stochastics. [22]

The geometric calculus is the topic of an article by Dick Stanley of the University of California at Berkeley. [19]

Various student projects regarding the geometric calculus are discussed in an article by Duff Campbell of the University of California at Berkeley. [193]

The geometric calculus was used in statistics and data analysis by Jarno van Roosmalen (Eindhoven University of Technology in the Netherlands) in his bachelor project on "multiplicative principal component analysis". [120]

Non-Newtonian calculus may have application in situations involving discontinuous phenomena. [35]

The geometric calculus was the subject of Christopher Olah's lecture at the Singularity Summit on 13 October 2012. [134] Singularity University's Singularity Summit is a conference on robotics, artificial intelligence, brain-computer interfacing, and other emerging technologies including genomics and regenerative medicine. [135]  Christopher Olah is a Thiel Fellow.

A graduate-seminar involving non-Newtonian calculus was conducted by Jared Burns at the University of Pittsburgh on 13 December 2012. [154]

In 2018, “Non-Newtonian calculus” was included among the research topics and course offerings of Eastern Mediterranean University in Northern Cyprus. [458]

Non-Newtonian calculus is among the subjects studied in the course “Application of Calculus in Commerce and Economics - Introduction” at Delhi University in India. [529]

The geometric and bigeometric calculi were used in a seminar (December of 2013) given by Ali Ozyapici and held by the Faculty of Engineering at Girne American University in Cyprus: "Applications of Multiplicative Calculi to Economical and Numerical Problems". [175]

Each of the topics Non-Newtonian calculus and non-Newtonian metric spaces is the subject of a presentation given at the International Conference on Mathematics and Engineering (ICOME-2017) in Istanbul, Turkey, 10-12 May 2017. [343, 344]

Non-Newtonian calculus was used by Ugur Kadak (Gazi University in Turkey), Feyzi Basar (Fatih University in Turkey), and Hakan Efe (Gazi University in Turkey) in an article on sequence spaces. [160]

Non-Newtonian calculus was used by Ugur Kadak (Gazi University in Turkey), Feyzi Basar (Fatih University in Turkey), and Hakan Efe (Gazi University in Turkey) in their lecture “Construction of the duals of classical sets of sequences and related matrix transformations with non-Newtonian calculus” at the 2013 Algerian Turkish International Days on Mathematics. [186]

Non-Newtonian calculus and related matters are topics in the article "Determination of the Kothe-Toeplitz duals over the non-Newtonian complex field" by Ugur Kadak (Gazi University in Turkey). [183]

Non-Newtonian calculus was used by Ahmet Faruk Çakmak (Yıldız Technical University in Turkey) and Feyzi Basar (Fatih University in Turkey) in an article on function spaces. [161]

A lecture entitled "Difference sequence spaces and non-Newtonian calculus" was presented by Khiord Boruah (Rajiv Gandhi University in India) at the National Conference on Recent Trends of Mathematics and its Applications, in May of 2014. [188]

Several specific non-Newtonian calculi are discussed in an article by H. Vic Dannon (Gauge Institute in USA). [196]

Several specific non-Newtonian calculi are discussed in a book by H. Vic Dannon (Gauge Institute in USA). [197]

The geometric calculus was used by Muhammad Waseem, Muhammad Aslam Noor, Farooq Ahmed Shah, and Khalida Inayat Noor (all from COMSATS Institute of Information Technology in Pakistan) in their article "An efficient technique to solve nonlinear equations using multiplicative calculus". [358] From the article: "In this paper, we develop an efficient technique in the framework of multiplicative calculus and suggest a new class of numerical methods for solving multiplicative nonlinear equations g(x) = 1. ... We solve the population growth model and minimization problem which demonstrate the implementation and efficiency of the new techniques. We also show that these techniques perform much better as compared to the similar ordinary methods for solving ordinary nonlinear equations f(x) = 0. ... There is a broad range of applications of multiplicative nonlinear equations g(x) = 1, almost in every field of life, especially in engineering and science."

The geometric calculus was used by Tolgay Karanfiller (Cyprus International University) in his lecture "Numerical solution of non-linear equations via multiplicative calculus" at the 2014 International Conference on Recent Advances in Pure and Applied Mathematics at Antalta, Turkey. [209]

The bigeometric calculus was used by Ali Ozyapici, Zehra B. Sensoy, and Tolgay Karanfiller (all from Cyprus International University in Turkey) in their article "Effective root-finding methods for nonlinear equations based on multiplicative calculi". [300]

The mathematics department of Eastern Mediterranean University in North Cyprus has established a research group for the purpose of studying and applying the geometric calculus and the bigeometric calculus. [110]

The geometric integral and geometric average were used in the doctoral dissertation "Evolutionary dynamics in changing environments” by Frank Stollmeier at Georg-August-Universität Göttingen in Germany, 2018. From the article: “The most convenient way to express this is the product-integral [i.e., geometric integral].”[464]

The geometric calculus and some of its applications are the topics of the 2009 doctoral dissertation of Ali Ozyapici at Ege University in Turkey. The dissertation is entitled "Multiplicative calculus and its applications". [191] 

Non-Newtonian calculus and some of its applications are the topics of the 2011 doctoral dissertation of Ugur Kadak at Gazi University in Turkey. The dissertation is entitled "Non-Newtonian analysis and its applications". [187]

Non-Newtonian calculus is used in the article "Stability of linear Volterra-Stieltjes differential equations" by Pham Huu Anh Ngoc from Ho Chi Minh City-International University in Vietnam. [421]

The bigeometric calculus is used in the article “BG-Volterra integral equations and relationship with BG-differential equations” by Nihan Gungor from Gümüşhane Üniversitesi in Turkey. [516] The Abstract: “In this study, the Volterra integral equations are defined in the sense of bigeometric calculus by the aid of bigeometric integral. The main aim of the study is to research the relationship between bigeometric Volterra integral equations and bigeometric differential equations.”

The geometric calculus and some of its applications are the topics of the 2113 doctoral dissertation of Yusuf Gurefe at Ege University in Turkey. The dissertation is entitled "Multiplicative differential equations and applications". [208]

Non-Newtonian analysis was used in the 2014 doctoral dissertation of Ahmet Faruk Cakmak at Yıldız Technical University in Turkey. The dissertation is entitled "Some new sequence spaces over a new field". [205]

The bigeometric calculus is used in the 2015 doctoral dissertation of Bugce Eminaga at Eastern Mediterranean University in North Cyprus. The dissertation is entitled "Bigeometric Taylor Theorem and its application to the numerical solution of bigeometric differential equations". [392]

The bigeometric calculus is used in the 2015 doctoral dissertation of Hatice Aktore at Eastern Mediterranean University in North Cyprus. The dissertation is entitled "Numerical approximation methods using multiplicative calculus". [393]

The bigeometric calculus was used in the 2017 doctoral thesis "Study on sequence spaces over non-Newtonian calculus" by Khirod Boruah at Rajiv Gandhi University in India. [441]

The concept of non-Newtonian convexity and related concepts were introduced in the article "Some geometric properties of the non-Newtonian sequence spaces lp(N)" by Nihan GÜNGÖR (Gümüşhane University in Turkey). [388]

Non-Newtonian calculus was used in the 2017 doctoral dissertation of Kumari Jyoti at SRM University Delhi-NCR (Haryana) in India. The dissertation is entitled "Digital images and non-Newtonian calculus endowed with fixed point theory". [367]

Non-Newtonian calculus is a featured topic of the 2016 doctoral dissertation of Zakaria Adnan at Kwame Nkrumah University of Science and Technology in Ghana. The dissertation is entitled "An analysis of Runge-Kutta method in non-Newtonian calculus". [278]

The geometric calculus was used in “Evolutionary dynamics in changing environments”, the 2018 doctoral dissertation of Frank Stollmeier at Georg-August-Universität Göttingen, in Germany. [453]

Knowledge of the geometric calculus ("multiplicative calculus") is a requirement for the master's degree in computer-engineering at Inonu University (Malatya, Turkey). [136]

Non-Newtonian calculus and the geometric calculus are discussed in the article "Herramientas matemáticas para la enseñanza de un nuevo cálculo basado en la multiplicación y no en la adición, con aplicacio" ("Mathematical tools for teaching a new calculus based on multiplication and not on addition, with application") by Sixta Vivanco, Jaider Blanco, and Gabriel Vergara (all from Universidad del Atlántico in Columbia). [445] From the Abstract: "The present work aims to provide undergraduate calculus students with new mathematical tools from non-Newtonian calculus, for use in science, engineering and mathematics. ... It is sought through this work to enable students to learn a new calculus based on multiplication and division and not on addition and subtraction (and based on the exponential function and its properties) in order to have more and better tools with which they can apply calculus in their careers."

The geometric calculus ("multiplicative calculus") is included on the list "Proposed Topics for the Master's Degree" of the Institute of Mathematics of the Jagiellonian University in Poland. [174]

Non-Newtonian calculus was used in the 2005 master's thesis of Ali Ozyapici at Eastern Mediterranean University in North Cyprus. [220]

Non-Newtonian calculus was used in the 2011 master's thesis of Hatice Aktore at Eastern Mediterranean University in North Cyprus. [221]

The geometric calculus was used in the 2014 master's thesis of Jaafar Anwar H. Ameen at Eastern Mediterranean University in North Cyprus. [219]

Weighted non-Newtonian calculus [9] was used in the 2017 master's thesis "The study of clinical prognostic factors of stranded sea turtles in northern Taiwan" ("北台灣救傷海龜之臨床預後因子研究") by Chen Yilin (National Taiwan University / College of Veterinary Medicine). [390]

The geometric calculus was used in the 2018 master’s thesis “Vehicle speed measurement using image processing” by Abdulgader Ramadan Gadah at Near East University in North Cyprus. [455]

The First Systems of Weighted Differential and Integral Calculus [9] is used by Riswan Efendi (Universiti Teknologi Malaysia), Zuhaimy Ismail (Universiti Teknologi Malaysia), Nor Haniza Sarmin (Universiti Teknologi Malaysia), and Mustafa Mat Deris (Universiti Tun Hussein Onn Malaysia) in their article "A reversal model of fuzzy time series in regional load forecasting". [245]

The First Systems of Weighted Differential and Integral Calculus [9] is used by Ping Jiang, Qingli Dong, Peizhi Li, and Lanlan Lian (all from Dongbei University of Finance and Economics, in Dalian, China) in their article "A novel high-order weighted fuzzy time series model and its application in nonlinear time series prediction". [329]

The First Systems of Weighted Differential and Integral Calculus [9] is discussed in the article "Weighting technique on multi-timeline for machine learning-based anomaly detection system" by Kriangkrai Limthong, Kensuke Fukuda, Yusheng Ji, and Shigeki Yamada (the former from Bangkok University in Thailand, and the latter three from the National Institute of Informatics in Tokyo-Japan). From the article: "Weighting is our proposed process for the multi-timeline detection module during the learning process. This module guides the learning algorithm to generate a decision function which relies on particular timelines. ... There are various weighting techniques for different purposes that could be plugged into this module, such as by performing a sum, integral, average or even calculus [The First Systems of Weighted Differential and Integral Calculus]." [330]

The geometric calculus was presented in the book Alternative Picture of the World, Volume 1 by Leonid G. Kreidik and George Shpenkov (both from the Institute of Mathematics & Physics at the University of Technology & Agriculture in Poland). From the book: "The multiplicative calculus allows us to see a great many facts which would be impossible to find by the classical additive calculus." (The expression "multiplicative calculus" refers here to the geometric calculus.) [285]

The geometric calculus was presented in the article "Additive and multiplicative judgements of dialectical logic. Additive and multiplicative differentials and integrals of dialectical judgements" by Leonid G. Kreidik of the Dialectical Academy in Russia-Belarus. From the article: "The multiplicative calculus allows us to see a great many facts which would be impossible to find by the classical additive calculus." (The expression "multiplicative calculus" refers here to the geometric calculus.) [256]

The geometric calculus is used in the article "Invariant functions for solving multiplicative discrete and continuous ordinary differential equations" by Reza Hosseini Komlaei and Mohammad Jahanshahi (both from Azarbaijan Shahid Madani University, in Iran). [370] From the Abstract: "...  a vast class of difference equations with variable coefficients and nonlinear difference and differential equations are investigated and solved by making use of multiplicative difference and differential equations." [370] From the Abstract: "Finally a vast class of difference equations with variable coefficients and nonlinear difference and differential equations are investigated and solved by making use of multiplicative difference and differential equations."

The geometric calculus was used in the article "Solution of multiplicative homogeneous linear differential equations with constant exponentials" by Numan Yalcın (Gümüşhane University in Turkey) and Ercan Celik (Atatürk University in Turkey). [420] From the article: "Grossman  and  Katz,  introduced  the  non-Newtonian  calculus including the branches  of  geometric, anageometric, and biogeometric calculus, etc. ... In  this  study, the  solution  of  multiplicative  homogeneous differential  equations  with  constant [exponents] is researched. For this purpose, first of all, multiplicatively linearly independent, multiplicatively linearly dependent and  Wronskian  determinant of the functions are  expressed."

The geometric calculus was used in the article "The solution of multiplicative non-homogeneous linear differential equations" by Numan Yalcın (Gümüşhane University in Turkey) and Ercan Celik (Atatürk University in Turkey). [395]

The geometric calculus was used in the article "On the solution of multiplicative linear differential equations without right-hand side" by Ahmet Gokdogan (Gumushane University in Turkey), Numan Yalcin (Gumushane University in Turkey), and Ercan Celik (Atatürk University in Turkey). [396]

Several applications of discrete "multiplicative" [i.e., geometric] calculus have been made by Mohammad Jahanshahi (Azad Islamic University of Karadj in Iran) and his colleagues. [150, 151, 152, 202, 286]

"Non-Newtonian improper integrals" is the title of an article by Cenap Duyar and Murat Erdoğan (both from Ondokuz Mayıs University in Turkey). [387, 417] From [417]: "Non-Newtonian  calculus  provides  a  wide diversity  of mathematical  tools  for  use in  engineering,  mathematics  and  science.  The notion  of  non-Newtonian  calculus  was  firstly introduced and worked by Grossman and Katz. They published the book about fundamentals of  non-Newtonian  calculus  and  which  includes  some special  calculi such  as  geometric, harmonic, bigeometric, etc. ... As a result of these studies, it has arisen the need of examination the improper integrals on non-Newtonian calculus. Hence, in this study, we introduce the non-Newtonian improper integrals and show some convergence tests for them."

In his 1993 article “g-calculus”, Endre Pap (Novi Sad University in Yugoslavia) discussed a family of calculi similar to the subfamily of non-Newtonian calculi created by Grossman and Katz in 1967. Apparently Professor Pap was unaware of non-Newtonian calculus. His work has been used in nonlinear ordinary and partial differential equations, difference equations, generalized functions (distributions), idempotent analysis, measure theory, and Laplace transforms. [480]

The following Wikipedia articles have been included in Wikipedia’s “Category: Non-Newtonian calculus”: “List of derivatives and integrals in alternative calculi”, “Fractal derivative”, “Geometric standard deviation”, “Indefinite product”, “Log–log plot”, “Log-normal distribution”, “Logarithmic scale”, “Multiplicative calculus”, “Ordered exponential”, “Product integral”, and “Semi-log plot”. [467]

In Non-Newtonian Calculus [15], Grossman and Katz used nonclassical arithmetics (i.e., complete ordered fields distinct from the real number system) to define the concept of  “*-metric”, later called “non-Newtonian metric” by some authors. An important special case of *-metric, based on geometric arithmetic, is called “multiplicative metric” by some authors, and is discussed in The First Nonlinear System of Differential and Integral Calculus [11] (page 62), and in Bigeometric Calculus: A System with a Scale-Free Derivative [10] (page 73). Subsequently, beginning in 2012, the concepts non-Newtonian metric space and multiplicative metric space were both used by various researchers to devise new theorems in fixed-point theory, an important tool in mathematics, biology, chemistry, economics, engineering, game theory, and physics.  (For example: [117], [128], [182], [206], [213], [214], [234], [242], [243], [244], [249], [252], [254], [255], [257], [260], [261], [262], [263], [304], [305], [310], [311], [312], [313], [314], [317], [318], [320], [321], [325], [327], [335], [336], [337], [338], [339], [340], [341], [342], [343], [344], [346].  [353], [354], [355], [356], [357], [360], [362],[363], [364], [365], [366], [368], [371], [372], [375], [379], [386], [398], [402], [408], [409], [410], [411], [419], [424], [425], [426], [438], [439], [442], [448], [461], [462], [466], [473], [483], [484], and [486].) A comprehensive discussion of multiplicative metric spaces is included in the chapter "Survey on Metric Fixed Point Theory and Applications" by Yeol Je Cho (Gyeongsang National University in Korea), in the book Advances in Real and Complex Analysis with Applications [372]. From the publisher, Springer/Birkhäuser: "This book discusses a variety of topics in mathematics and engineering as well as their applications, clearly explaining the mathematical concepts in the simplest possible way and illustrating them with a number of solved examples. ...  It includes papers presented at the 24th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications (24ICFIDCAA), held at the Anand International College of Engineering, Jaipur, India, 22–26 August 2016. The book is a valuable resource for researchers in real and complex analysis." (Please see the page Nonclassical Arithmetics.)

Geometric arithmetic can be used to construct the geometric, anageometric, and bigeometric calculi. The well-known geometric average is the natural average in the geometric arithmetic. [15]

Harmonic arithmetic can be used to construct the harmonic, anaharmonic, and biharmonic calculi. The well-known harmonic average is the natural average in the harmonic arithmetic. [15] (Some concepts from harmonic arithmetic such as harmonic average, harmonic sum, and reciprocal are important in electrical circuit theory, combinatorial mathematics, the mathematics of real estate appraisal, and in financial arithmetic [489].)

Non-Newtonian calculus was used by Ahmet Faruk Cakmak (Yıldız Technical University in Turkey) and Feyzi Basar (Fatih University in Turkey) in their article "Some new results on sequence spaces with respect to non-Newtonian calculus", in which they discussed  the concept of 'non-Newtonian metric-space'. [122] From that article: " As alternative[s] to the classical calculus, Grossman and Katz (Non-Newtonian Calculus, Lee Press, Pigeon Cove, Massachusetts, 1972) introduced the non-Newtonian calculi, [including] the geometric, anageometric, and bigeometric calculi. ... Even you can create your own calculus by choosing different function[s] as generator[s]." 

Multiplicative metric-spaces and the geometric calculus are discussed in the article "Iterative algorithms and an overview of recent results for fixed-point problems" by Feng Gu (Hangzhou Normal University in China). [419] The article is Chapter 5 of Science: Hangzhou Normal University, a supplement to the journal Science of the American Association for the Advancement of Science (AAAS). From the article: "In 2008, Bashirov and colleagues ["Multiplicative calculus and its applications"] introduced the definition of multiplicative metric-spaces, studied multiplicative calculus, and derived a fundamental theorem of multiplicative calculus. In 2012, Florack and Van Assen ["Multiplicative calculus in biomedical image analysis"] provided an example of multiplicative calculus application in biomedical image analysis. In 2011, Bashirov and colleagues ["On modeling with multiplicative differential equations"] exploited the efficiency of multiplicative calculus as compared to that of Newtonian calculus. They showed that multiplicative differential equations are more suitable than ordinary differential equations to describe certain practical problems. Moreover, Bashirov and colleagues ["Multiplicative calculus in biomedical image analysis"] demonstrated some interesting applications for multiplicative calculus."

Geometric arithmetic and multiplicative metric-spaces are used to prove a result for dynamic programming (computer science) in the article "On the existence of fixed points of two rational type contractions in a dislocated quasi multiplicative metric-spaces and its application" by Ahmed H. Soliman (Al-Azhar University in Egypt), M. A. Ahmed (Assiut University in Egypt), and A. M. Zidan (Al-Azhar University). [364]

Geometric arithmetic and multiplicative metric-spaces are used in the article "Some critical remarks on the multiplicative metric-spaces and fixed point results" by Satish Shukla (Shri Vaishnav Institute of Technology & Science in India). [327]

Non-Newtonian metric-spaces and nonclassical arithmetics are used in the article "Some results of fixed point theorem in non-Newtonian metric spaces" by Anita Dahiya and Asha Rani (both from S. R. M. University Haryana, in India). [353]

Multiplicative metric-spaces and the geometric calculus are used in the article "Some fixed point theorems for mappings satisfying a general multiplicative contractive condition of integral type" (or briefly, "Fixed point theorems and multiplicative integral") by Badshah-e-Rome and Muhammad Sarwar (both from the University of Malakand in Pakistan). [311]

Geometric arithmetic and multiplicative metric-spaces are used in the article "Characterization of multiplicative metric-completeness" by Badshah-е-Rome and Muhammad Sarwar (both from the University of Malakand in Pakistan). [339]

Geometric arithmetic and multiplicative metric-spaces are used in the article "Extensions of the Banach contraction principle in multiplicative metric-spaces" by Badshah-е-Rome and Muhammad Sarwar (both from the University of Malakand in Pakistan). [346]

Geometric arithmetic and multiplicative metric-spaces are used in the article "Fixed point theorems in left and right dislocated quasi-multiplicative metric-spaces" by M.A. Ahmed (Assiut University in Egypt) and A.M. Zidan (Al-Azhar University in Egypt). [354]

Geometric arithmetic and multiplicative metric-spaces are used in the article "Fixed point theorems in multiplicative metric-spaces" by Avinash Chandra Upadhyaya (Deenbandhu Chhotu Ram University of Science and Technology in India). [356]

Multiplicative metric-spaces and the geometric calculus are used in the article "Some fixed point theorems in multiplicative cone b-metric spaces" by Akbar Zada, Rahim Shah, and Shahid Saifullah (all from the University of Peshawar in Pakistan). [338]

Multiplicative calculus, geometric arithmetic, and the geometric mean are the topics of the article "Necessary and sufficient conditions for geometric means of sequences in multiplicative calculus" by İbrahim Çanak (Ege University in Turkey), Umit Totur (Adnan Menderes University in Turkey), Sefa Anil Sezer (İstanbul Medeniyet University in Turkey). [326]

The geometric calculus, geometric arithmetic [15], and multiplicative metric-spaces are used in the 2017 doctoral thesis of Alia Shani Hassan Kurdi at University Politehnica of Bucharest in Romania. The thesis includes an existence theorem for the solution of a class of Fredholm multiplicative integral equations. [355] From the Abstract: "The content presented in this Thesis is mainly oriented on Iteration Theory (Chapter 1 and Chapter 2), Continuous Optimization (Chapter 3 and Chapter 4) and non-Newtonian calculus (Chapter 5). ... By using the ideas of Grossman and Katz [Non-Newtonian Calculus], Bashirov et al. ["Multiplicative calculus and its applications"] defined the notion of multiplicative metric." (The author thanked Yusuf Gurefe of Usak University in Turkey, who "kindly" agreed "to supervise my study in non-Newtonian calculus.")

The geometric calculus, geometric arithmetic [15], and multiplicative metric-spaces are used in the article "Fixed point theorems in b-multiplicative metric-spaces" by Muhammad Usman Ali (COMSATS Institute of Information Technology in Pakistan), Tayyab Kamran (National University of Sciences and Technolo in Pakistan), and Alia Kurdi (University Politehnica of Bucharest in Romania). The article includes an existence theorem for the solution of a class of Fredholm multiplicative integral equations. [368] From that article: "Grossman and Katz [Non-Newtonian Calculus] introduced a new kind of calculus called multiplicative (or non-Newtonian) calculus ... . By using the ideas of Grossman and Katz [Non-Newtonian Calculus], Bashirov et al. ["Multiplicative calculus and its applications"] defined the notion of multiplicative metric."

A new concept based on geometric arithmetic [15] and called "multiplicative normed linear space" was introduced in the article "Multiplicative normed linear space and its topological properties" by Renu Chugh, Ashish Nandal, and Naresh Kumar (all from Maharshi Dayanand University in India) . [342] From that article: "The aim of this article is to propose a new space called multiplicative normed linear space and discuss different topological properties of this space. We establish equivalence between multiplicative compactness and multiplicative sequentially compactness. ... Further we apply our results to prove fixed point existence theorems."

Geometric arithmetic [15] was used in two articles about "multiplicative parameters-applications in economics and finance" by Helena Jasiulewicz (Wrocław University of Environmental and Life Sciences in Poland) and Wojciech Kordecki (The Witelon State University of Applied Sciences in Poland). Among the concepts discussed: geometric mean, geometric-mean investment strategy, geometric standard deviation, lognormal distributions, and the Pareto distribution (a power-law probability distribution). From those articles: "When is it better to use arithmetic (additive) parameters and when geometric (multiplicative) ones?" [264, 265]

Application of geometric arithmetic [15] to wavelet analysis was made in the article "Geometric wavelet approximations and differencing" by Abdourrahmane Mahamane Atto, Emmanuel Trouve, Jean Marie Nicolas (the former two from Polytech Annecy-Chambery in France, and the latter from Télécom ParisTech in France). From that article: "This paper introduces the concept of geometric wavelets defined from multiplicative algebras. ... In particular, when the acquisition system yields a multiplicative interaction model involving a non-constant signal, then geometric representation frameworks such as that presented in this paper are expected to be more relevant than additive frameworks." [212]

Geometric arithmetic was used in an article about what could be called 'geometric Lebesque measure' by Cenap Duyar and Birsen Sagir (both from Ondokuz Mayıs University in Turkey). From the Abstract: "In this work, we investigate the corresponding results of Lebesgue measure in the sense of non-Newtonian calculus." [315]

Geometric arithmetic is used by Kaizhong Guan (University of South China, in China) in his article "Multiplicative convexity and its applications". [333] From the article: "It is known that the theory of multiplicatively convex functions is similar to that of classical convex functions. Some inequalities are easier to state using multiplicatively convex functions and some are easier to state using convex functions. However, in many cases the inequalities based on multiplicatively convexity are better than the direct application of the usual inequalities of convexity (or yield complementary information). Thus, there is strong interest in investigating the multiplicatively convex (concave) functions."

A study of infinite series in nonclassical arithmetics is discussed in the article "On non-Newtonian real number series" by Cenap Duyar and Murat Erdoğan (both from Ondokuz Mayıs University in Turkey). [389]

The First Systems of Weighted Differential and Integral Calculus [9] is discussed in the article “Accelerating the outlier detection methods for categorical data by using matrix of attribute value frequency” by Nur Rokhman, Subanar Seno Salem, and Edi Winarko (all from Gadjah Mada University in Indonesia). [468]

For convenience, scientists sometimes use "log-transforms" to facilitate their work. For example, here's a technique for approximating a positive discrete function f  by an exponential function: apply the classical method of least-squares to approximate log(f) by a linear function g, and then use exp(g) as the desired exponential function. This procedure is commonly used even though there is no apparent rationale for it. Interestingly, in Non-Newtonian Calculus [15] and in The First Nonlinear System of Differential and Integral Calculus [11], Grossman and Katz introduced a more intuitively appealing technique for obtaining the desired approximation. This technique could  aptly be called "the geometric method of least-squares" because it involves judicious use of geometric arithmetic [15] and geometric calculus [15].  It turns out that the geometric method of least-squares yields the same exponential function as the log-transform technique, thus providing a rationale for what otherwise seems to be a makeshift result. Furthermore, the geometric method of least-squares may well lead scientists to new, simple, and intuitively satisfying definitions and theorems formulated in terms of geometric arithmetic and geometric calculus. Two other examples (unrelated to least-squares methods) involving log-transforms are discussed in these two articles:  

• "Wavelet operators and multiplicative observation models—application" [299] by Abdourrahmane Mahamane Atto (University Savoie-Mont-Blanc in France), Emmanuel Trouve (University Savoie-Mont-Blanc), Jean Marie Nicolas (Telecom ParisTech in France), and Thu Trang Lê (University Savoie-Mont-Blanc).

• Gradient domain high dynamic range compression" [415] by Raanan Fattal, Dani Lischinski, and Michael Werman (all from The Hebrew University in Jerusalem).

In his article “Introduction to hypertype theory”, Ismael Ghalimi wrote: “Hypertype Theory is a new branch of type theory [used in mathematics, logic, and computer science] interested in the study of recursive groups and fields. Its main idea is that introducing a commutative expansion operator complementary to the traditional non-commutative exponentiation operator unveils simple and beautiful structures. ... The expansion operator can be used to simplify equations related to log-normal probability distributions. ... Taking a multiplication-centric approach akin to “Multiplicative calculus” [in Wikipedia] is expected to uncover many more applications. This intuition is supported by the fact that many observable processes are multiplicative rather than additive in nature.” [472]

The geometric derivative is included in Mathematica's "Stack Exchange" website, which has links to articles [2], [88], and [19]. [198]

Multiplicative calculus and non-Newtonian calculus were “Related Wiki Topics” for the (2018 - 2019) course “Introduction to Calculus” on Coursera, an online learning platform offering more than 3,000 academic courses from over 180 top universities and organizations in more than 125 countries. “Introduction to Calculus” was taught by Professor David Easdown of The University of Sydney in Australia. [465]

Multiplicative calculus is discussed in the article “What is Bitcoin? How to Buy Bitcoin Online?” at the All Tech Facts website. From the article: "Among the most popular ways to invest money in Dash and other similar currencies is through the use of what’s called an eToro platform. An eToro platform is essentially an automated trading system that makes it possible to trade Dash and other cryptosurfs like it without having to be involved in the trading process. An eToro platform uses what’s called a multiplicative algorithm to make profitable trades even when other factors, such as market conditions and current events, don’t favour or prevent profitable outcomes. While other algorithms may also do this, nothing compares to the experience of using a multiplicative calculus that’s been designed to be precise and effective for the long term." (EToro is an Israeli social trading and multi-asset brokerage company that focuses on providing financial and copy trading services.)