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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. (Since 24 June 2021, the website has not been updated regularly.)



"Your ideas [in Non-Newtonian Calculus] seem quite ingenious."

- Dirk J. Struik, Massachusetts Institute of Technology, USA; from his letter to Katz and Grossman, dated 20 April 1972.


"The monographs on non-Newtonian calculus by you and the Grossmans appear to be very useful and innovative."
-  Kenneth J. Arrow, Nobel-Laureate, Stanford University, USA; from his letter to Robert Katz, dated 26 November 1980.

 

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.”

- Mark Burgin (University of California at Los Angeles, USA) and Marek Czachor (Gdańsk University of Technology, Poland); from their 2020 book Non-Diophantine Arithmetics in Mathematics, Physics and Psychology [510].

     

"I am an evangelist for non-Newtonian calculus. ... Just know that I think you and your co-authors' legacy should match Newton/Leibnitz. ... Brilliant insights are often greeted with hostility.”

"You and Robert Katz opened up a whole new world.”

"It's an honor to know you. I'd like to offer a quick historical perspective you might like:

- Peter Carr, Chair of Finance and Risk Engineering at New York University’s Tandon School of Engineering, USA; from his e-mail to Michael Grossman in May of 2019.


"After finding your pioneering publications on non-Newtonian calculus, I decided that your groundbreaking results deserved acknowledgement and full recognition. ... Your Non-Newtonian Calculus is an eminent innovation.”

 - Mark Burgin, University of California at Los Angeles, USA; from his e-mail to Michael Grossman, dated 19 July 2018.

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"In 1972, Grossman and Katz [Non-Newtonian Calculus] proposed alternative calculi to the calculus of Newton and Leibnitz. ... This pioneering work initiated numerous studies."

- Agamirza E. Bashirov and Sajedeh Norozpour, both from Eastern Mediterranean University in North Cyprus; from their 2016 article [293].


"It is known that non-Newtonian  calculus models real life problems more accurately."

 - R. C. Mittal, Indian Institute of Technology, India; from the ResearchGate website on 12 November 2014. [218]


"There is enough here [in Non-Newtonian Calculus] to indicate that non-Newtonian calculi ... have considerable potential as alternative approaches to traditional problems. This very original piece of mathematics will surely expose a number of missed opportunities in the history of the subject."

 - Ivor Grattan-Guinness, Middlesex University, England; from his 1977 review of Non-Newtonian Calculus [101].


"The possibilities opened up by the [non-Newtonian] calculi seem to be immense."

-H. Gollmann; from his review of Non-Newtonian Calculus in Internationale Mathematische Nachrichten (Number 105, 1972), a publication of Österreichische Mathematische Gesellschaft in Vienna, Austria. [53]


"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.)

 - Max Cubillos, California Institute of Technology, USA; from his 2018 article "Modelling wave propagation without sampling restrictions using the multiplicative calculus I: Theoretical considerations" [391].


"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."

- Michael Valenzuela, University of Arizona, USA; from his 2016 doctoral dissertation "Machine learning, optimization, and anti-training with sacrificial data" [279]. (In computer science, machine learning is a branch of artificial intelligence.)


"Non-Newtonian Calculus, by Michael Grossman and Robert Katz, is a fascinating and (potentially) extremely important piece of mathematical theory. That a whole family of differential and integral calculi, parallel to but nonlinear with respect to ordinary Newtonian (or Leibnizian) calculus, should have remained undiscovered (or uninvented) for so long is astonishing -- but true. Every mathematician and worker with mathematics owes it to himself to look into the discoveries of Grossman and Katz. The theory has proved to be most valuable in several research studies in which I am engaged. I predict that non-Newtonian calculus will come to be recognized as the most important mathematical discovery of the Twentieth Century."

- James R. Meginniss, Claremont Graduate School and Harvey Mudd College, USA; from a 1980 memorandum to his colleagues.


"The purpose of this paper is to present a new theory of probability that is adapted to human behavior and decision making."

- James R. Meginniss, Claremont Graduate School and Harvey Mudd College, USA; from his 1980 article "Non-Newtonian calculus applied to probability, utility, and Bayesian analysis" [16].


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." (The expression "multiplicative calculus" refers here to the geometric calculus.)

- Ali Ozyapici (Cyprus International University in Cyprus), Yusuf Gurefe (Usak University in Turkey), and Emine Misirli (Ege University in Turkey); from their 2017 article "Generalization of Special Functions and its Applications to Multiplicative and Ordinary Fractional Derivatives" [334].


"There has been great research in geometric (multiplicative) and bigeometric calculus within recent years."

- Bulent Bilgehan (Girne American University in Cyprus), Bugce Eminaga (Girne American University in Cyprus), and Mustafa Riza (Eastern Mediterranean University in Cyprus); from their 2016 article "New solution method for electrical systems represented by ordinary differential equation" [250].


"This [Non-Newtonian Calculus] is an exciting little book ... The greatest value of these non-Newtonian calculi may prove to be their ability to yield simpler physical laws than the Newtonian calculus. Throughout, this book exhibits a clarity of vision characteristic of important mathematical creations. ... The authors have written this book for engineers and scientists, as well as for mathematicians. ... The writing is clear, concise, and very readable. No more than a working knowledge of [classical] calculus is assumed."

- David Pearce MacAdam, Cape Cod Community College, USA; from his 1973 review of Non-Newtonian Calculus [100].


“The power and efficiency of the non-Newtonian approach lies in its low-level starting point — the arithmetic.”

- Marek Czachor, from Gdańsk University of Technology in Poland); from his 2020 article [518].


  "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."

- Fernando Córdova-Lepe and Marco Mora,  both from Universidad Católica del Maule in Chile; from a translation (from Spanish) of the announcement (January of 2017) of the 17th International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE), 4-8 July 2017. [324]


"In [1967] Michael Grossman and Robert Katz gave definitions of a new kind of derivative and integral ...  and thus established a new calculus, called multiplicative calculus [the geometric calculus]. ... We think that [the geometric calculus] can especially be useful as a mathematical tool for economics and finance ... In the present paper our aim is to bring [the geometric] calculus to the attention of researchers ...  and [to] demonstrate its usefulness."

- Agamirza E. Bashirov (Eastern Mediterranean University in North Cyprus), Emine Misirli Kurpinar (Ege University in Turkey), and Ali Ozyapici (Ege University in Turkey); from their 2008 article "Multiplicative calculus and its applications" [2].


"After a long period of silence in the field of non-Newtonian calculus introduced by Grossman and Katz [Non-Newtonian Calculus] in 1972, the field experienced a revival with the mathematically comprehensive description of the geometric calculus by Bashirov et al. ["Multiplicative calculus and its applications", 2008] [2], which initiated a kick-start of numerous publications in this field."

- Mustafa Riza (Eastern Mediterranean University in North Cyprus) and Bugce Eminaga (Girne American University in Cyprus);  from their 2015 article "Bigeometric Calculus and Runge Kutta Method" [215].


"Grossman and Katz have shown that it is possible to create infinitely many calculi independently. They constructed a comprehensive family of calculi, including the Newtonian (or Leibnizian) calculus, the geometric calculus, the bigeometric calculus, and infinitely-many other calculi. ... The geometric and bigeometric calculi have become more and more popular in the past decade."

- Mustafa Riza (Eastern Mediterranean University in North Cyprus) and Bugce Eminaga (Girne American University in Cyprus);  from their 2015 article "Bigeometric Calculus and Runge Kutta Method" [215].


"We find applications of non-Newtonian calculus in the fields of probability, physics, image analysis, numerical analysis, non-linear dynamical systems, etc."

- Pranav Sharma and Sanjay Mishra, both from Lovely Professional University in India; from their 2016 article [325].


"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 ..."

- Wojbor Woyczynski, Case Western Reserve University, USA; from an abstract to his 2013 seminar "Fractional calculus for random fractals". [146]


"Non-Newtonian calculus for the dynamics of random fractal structures"

- Wojbor Woyczynski, Case Western Reserve University, USA; from his seminars (in 2011 and 2012) "Non-Newtonian calculus for the dynamics of random fractal structures". [90, 104]


"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."

- Wojbor Woyczynski, Case Western Reserve University, USA; from an abstract to his seminars (in 2011 and 2012) "Non-Newtonian calculus for the dynamics of random fractal structures". [90, 104]


"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."

- Martin Ostoja-Starzewski, University of Illinois at Urbana-Champaign, USA; from the 2013 media-upload "The inner workings of fractal materials", University of Illinois at Urbana-Champaign. [163]


"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.)

- Dorota Aniszewska and Marek Rybaczuk, both from Wroclaw University of Technology in Poland; from their 2008  article "Lyapunov type stability and Lyapunov exponent for exemplary multiplicative dynamical systems". [131]


" ... evolution of fractal characteristics will be examined with the help of dynamical system theory or more precisely in terms of multiplicative calculus." (The expression "multiplicative calculus" refers here to the bigeometric calculus.)

- Dorota Aniszewska and Marek Rybaczuk, both from Wroclaw University of Technology in Poland; from their 2009 article "Fractal characteristics of defects evolution in parallel fibre reinforced composite in quasi-static process of fracture". [184]


"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." (The expression "multiplicative calculus" refers here to the geometric calculus.)

- Luc Florack and Hans van Assen, both of Eindhoven University of Technology in The Netherlands; from their 2012 article "Multiplicative calculus in biomedical image analysis" [88].


"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.)

- Luc Florack, Eindhoven University of Technology in The Netherlands; from his 2012 article "Regularization of positive definite matrix fields based on multiplicative calculus" [96].


"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."

- Luc Florack, Tom Dela Haije, and Andrea Fuster, all from Eindhoven University of Technology in the Netherlands; from their 2015 article "Direction-controlled DTI [Diffusion Tensor Imaging] interpolation". [231]


"Our work is an example of 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."

- Nico Persch, Christopher Schroers, Simon Setzer, and Joachim Weickert (Gottfried Wilhelm Leibniz Prize winner), all from Saarland University in Germany; from their 2015 article on imaging science called "Physically inspired depth-from-defocus". [316]


"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 ... ."

- Martin Schmidt (Saarland University in Germany); from his 2018 doctoral dissertation "Linear scale-spaces in image processing: drift-diffusion and connections to mathematical morphology" at Saarland University in Germany. [414]


"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."

- Xiaohong Gong, Yali Zhou, Hao Zhou, and Yinfei Zheng, all from Zhejiang University at Hangzhou in China; from their 2014 article "Ultrasound image edge detection based on a novel multiplicative gradient and Canny operator". [211]


"Reference [..] introduces a novel gradient operator better suited for edge detection in ultrasound or SAR imaging [SAR: synthetic aperture radar]. This new operator, the multiplicative gradient, ... is developed using non-Newtonian calculus."

- 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 Iles (TUCN, Romania); from their 2015 article "A multiplicative gradient-based anisotropic diffusion approach for speckle noise removal". [268]


"This work presents a new operator of non-Newtoniantype which [has] shown [to] be more efficient in contour detection [in images with multiplicative noise] than the traditional operators. ... In our view, the work proposed in (Grossman and Katz, 1972) stands as a foundation ... Innovative applications of non-Newtonian calculus can be found in the field of Bayesian Analysis (Meginniss, 1980)."  [15, 16]

- Marco Mora, Fernando Córdova-Lepe, and Rodrigo Del-Valle, all of Universidad Católica del Maule in Chile; from their 2012 article [99].


"In our proseminar we'll learn some of the most exciting non-linear calculations and consider the applications for which they may be of particular interest. The applications range from rates of return and other growth processes to highly active areas of digital image processing."

-  Joachim Weickert (Gottfried Wilhelm Leibniz Prize winner), Saarland University in Germany; from his description of his non-Newtonian calculus course "Analysis beyond Newton and Leibniz", Saarland University, 2012. [106]


"In many circumstances, multiplicative calculus is highly natural; for example, the decay of a radioactive material and the unconstrained growth of a bacterial colony [yield] constant multiplicative derivatives."  (The expression "multiplicative calculus" refers here to the geometric calculus.)

- Christopher Olah, Thiel Fellow; from Christopher Olah's Blog, 10 June 2011. [134, 135, 177]


"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.”

- Marek Czachor, from Gdańsk University of Technology in Poland); from his 2020 article [524].


"An exponential function is a multiplicative derivative." (Maybe he meant: "Any exponential function has a constant multiplicative derivative." The expression "multiplicative derivative" refers here to the geometric derivative.)

- Ray Kurzweil, from his keynote speech at the Singularity Summit, October 13–14 of 2012. [134, 135, 322]



"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."

- Diana Andrada Filip (Babes-Bolyai University of Cluj-Napoca in Romania) and Cyrille Piatecki(Orléans University in France); from their 2014 article [82].


"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."

- Diana Andrada Filip (Babes-Bolyai University of Cluj-Napoca in Romania) and Cyrille Piatecki(Orléans University in France); from their 2014 article [181].


"... [non-Newtonian calculus] could help to acquire new insight on classical subjects, or solve directly some problems which could only be reached by approximations."

- Diana Andrada Filip (Babes-Bolyai University of Cluj-Napoca in Romania) and Cyrille Piatecki(Orléans University in France); from their 2014 article [181].


"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." (The expression "multiplicative calculus" refers here to the geometric calculus.)

- Diana Andrada Filip (Babes-Bolyai University of Cluj-Napoca in Romania) and Cyrille Piatecki(Orléans University in France); from their 2014 article [149].


"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."

- Hasan Özyapıcı (Eastern Mediterranean University in Cyprus), İlhan Dalcı (Eastern Mediterranean University in Cyprus), and Ali Ozyapici (Cyprus International University); from their article "Integrating accounting and multiplicative calculus: an effective estimation of learning curve". [290] (The expression "multiplicative calculus" refers here to the geometric calculus.)


"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.)"

- Agamirza E. Bashirov (Eastern Mediterranean University in North Cyprus), Emine Misirli (Ege University in Turkey), Yucel Tandogdu (Eastern Mediterranean University in North Cyprus) Ali Ozyapici, Lefke European University in Turkey); from their 2011 article [94].


"In 2011, Bashirov et al. ["On modeling with multiplicative differential equations"] exploit the efficiency of  [the geometric] calculus over the Newtonian calculus. They demonstrated that the [geometric calculus] differential equations are more suitable than the ordinary differential equations in investigating some problems in various fields. Furthermore, Bashirov et al. [" Multiplicative calculus and its applications"] illustrated the usefulness of [the geometric] calculus with some interesting applications."            

- Feng Gu (Hangzhou Normal University in China) and Yeol-Je Cho (Gyeongsang National University in Korea); from their 2015 article [249].


"Bigeometric calculus - a modelling tool"

- Mustafa Riza (Eastern Mediterranean University in North Cyprus) and Bugce Eminaga (Girne American University in Cyprus); from their 2014 article "Bigeometric calculus - a modelling tool" [178], which includes a new mathematical model for studying tumor therapy with oncolytic virus.


"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."

- Mustafa Riza (Eastern Mediterranean University in North Cyprus) and Bugce Eminaga (Girne American University in Cyprus); from their 2015 article "Bigeometric Calculus and Runge Kutta Method"[215], which 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.


"While one problem can be easily expressed in one calculus, the same problem can not be expressed as easily [in another]."

 - Emine Misirli and Yusuf Gurefe, both of Ege University in Turkey; from their 2009 lecture [123].


"If non-Newtonian calculus is employed together with classical calculus in the formulations, then many of the complicated phenomena in physics or engineering may be analyzed more easily."

- Ahmet Faruk Çakmak (Yıldız Technical University in Turkey) and Feyzi Basar (Fatih University in Turkey); from their 2014 article "Certain spaces of functions over the field of non-Newtonian complex numbers". [161]


"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." (The expression "multiplicative calculus" refers here to the geometric calculus.)

- Bulent Bilgehan, from Girne American University in Cyprus/Turkey; from his 2015 article "Efficient approximation for linear and non-linear signal representation" [222].


"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 proposed representation."

- Ali Ozyapici (Cyprus International University in Cyprus/Turkey) and Bulent Bilgehan (Girne American University in Cyprus/Turkey); from their 2015 article "Finite product representation via multiplicative calculus and its applications to exponential signal processing" [225]


"The main goal of this paper is chaos examination in systems described with multiplicative [bigeometric] differential equations."

- Dorota Aniszewska and Marek Rybaczuk, both from Wroclaw University of Technology in Poland; from their 2010 article "Chaos in multiplicative systems". [126]


"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."

- Ali Ozyapici, Mustafa Riza, Bulent Bilgehan, Agamirza E. Bashirov (the first and third authors are from Girne American University in Cyprus/Turkey, the second and fourth from Eastern Mediterranean University in North Cyprus); from the Abstract to their 2013 article [176].


"It seems plausible that people who need to study functions from this point of view might well be able to formulate problems more clearly by using [bigeometric] calculus instead of [classical] calculus."     

- Ralph P. Boas, Jr., Northwestern University, USA; from his 1984 review of  Bigeometric Calculus: A System with a Scale-Free Derivative [47].


"Non-Newtonian calculus allowed scientists to look from a different point of view to the problems encountered in science and engineering."

- Numan Yalcın (Gumushane University in Turkey), Ercan Celik (Ataturk University in Turkey), and Ahmet Gokdogan (Gumushane University in Turkey); from their 2016 article [295].


"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." 

- Agamirza E. Bashirov and Sajedeh Norozpour (both from Eastern Mediterranean University in North Cyprus); from their 2016 article "Riemann surface of complex logarithm and multiplicative calculus" [309].


"The more innovative parts of the Stern Review – the non-Newtonian calculus in Chapter 13, for instance – have yet to be submitted to learned journals." 

- Richard Tol, University of Sussex, England; from his forward to "What is Wrong with Stern?", a critique (4 September 2012) of the 2006 report "Stern Review on the Economics of Climate Change". [116, 165]


"... 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]

- Roberto Sotero Diaz, Hotchkiss Brain Institute of the University of Calgary, Canada; from his correspondence, dated 31 October 2014, with Grossman, Grossman, and Katz.


“Non-Diophantine arithmetics and non-Newtonian calculus seem counterintuitive only at a first encounter. They are as natural as non-Euclidean geometry, non-Boolean logic, or non-Kolmogorovian probability.” -  Marek Czachor, from Gdańsk University of Technology in Poland); from his 2020 article [502].


"This counterintuitive possibility opened by the non-Newtonian calculus is especially useful in fractal applications. ... It would not be very surprising if our fractal [non-Newtonian type] calculus found applications also in other branches of physics, where finite physical results are buried in apparently infinite theoretical predictions.“

- Marek Czachor, from Gdańsk University of Technology in Poland); from his 2019 article [487].


I am pretty sure non-Newtonian calculus would improve option-pricing models [in finance]."

- Raymond Tang, financial advisor, Etechadvisors Inc., Laguna Beach, CA, USA; from his correspondence, dated 15 April 2014, with Grossman and Grossman. (Also, please see [37].)


"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." "Elements of multiplicative calculus aiming to demonstrate the non-absoluteness of Newtonian calculus." (The expression "multiplicative calculus" refers here to the geometric calculus.)

- Agamirza E. Bashirov, Eastern Mediterranean University, North Cyprus; from the Abstract to Chapter 11 of his 2014 textbook Mathematical Analysis: Fundamentals, and  from a list of items in the publisher's (Academic Press) Description.[179]


"Non-Newtonian Calculus or Multiplicative [Geometric] calculus can be used as a tool wherever a problem is of exponential (relational) in nature. ... The results show that for certain family of initial value problem the non-Newtonian (Multiplicative) Runge-Kutta method gives better results than the Ordinary Runge-Kutta method."

- Zakaria Adnan, from Kwame Nkrumah University of Science and Technology in Ghana; from his 2016 doctoral dissertation "An analysis of Runge-Kutta method in non-Newtonian calculus". [278]


"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." [276]

- Khirod Boruah and Bipan Hazarika, both from Rajiv Gandhi University in India; from their 2016 article "Application of geometric calculus in numerical analysis and difference sequence spaces". [276]


"Based on M. Grossman in Bigeometric Calculus: A System with a Scale-Free Derivative and Grossman and Katz in Non-Newtonian Calculus, in this paper we discuss applications of bigeometric calculus in different branches of mathematics and economics."

- Khirod Boruah and Bipan Hazarika, both from Rajiv Gandhi University in India; from their 2016 article "Bigeometric calculus and its applications". [298]


"As readers of this blog are probably aware, I’m a rather big fan of multiplicative calculus ... So, it should come as no surprise that when I was given an opportunity to speak at Singularity Summit last fall [October of 2012], a conference largely concerned with exponential trends in technology, I decided to try to persuade people of the utility of multiplicative calculus and the value of mathematical abstractions."  (The expression "multiplicative calculus" refers here to the geometric calculus.)

- Christopher Olah, Thiel Fellow; from Christopher Olah's Blog, 07 May 2013. [134, 135, 164]


"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."

- Bülent Bilgehan (Near East University in Cyprus) and Stephen Ojo (Girne American University in Cyprus); from their 2018 article [446].


“I wish more students were made aware of this fine topic. It is also very important for scholars in statistics to be aware of the averaging techniques from this text, and the generalizations available.”

- Edwardo Lindeloff; from his review of Non-Newtonian Calculus, dated 21 May 2017, at Amazon.com.


"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."

-  Demet Binbaşıoǧlu (Gaziosmanpaşa University in Turkey), Serkan Demiriz (Gaziosmanpaşa University in Turkey), and Duran Türkoǧlu (Gazi University in Turkey); from their 2016 article [254].