# Multiplicative calculus

In mathematics, a multiplicative calculus is a system with two multiplicative operators, called a "multiplicative derivative" and a "multiplicative integral", which are inversely related in a manner analogous to the inverse relationship between the derivative and integral in the classical calculus of Newton and Leibniz. The multiplicative calculi provide alternatives to the classical calculus, which has an additive derivative and an additive integral.

There are infinitely many non-Newtonian multiplicative calculi, including the geometric calculus and the bigeometric calculus discussed below.[1] These calculi all have a derivative and/or integral that is not a linear operator.

The geometric calculus is useful in biomedical image analysis.[2][3][4][5]

## Multiplicative derivatives

### Geometric calculus

The classical derivative is

$f'(x) = \lim_{h \to 0}{f(x+h) - f(x)\over{h}}$

The geometric derivative is

$f^{*}(x) = \lim_{h \to 0}{ \left({f(x+h)\over{f(x)}}\right)^{1\over{h}} }$

(For the geometric derivative, it is assumed that all values of f are positive numbers.)

This simplifies[6] to

$f^{*}(x)=e^{f'(x)\over f(x)}$

for functions where the statement is meaningful. Notice that the exponent in the preceding expression represents the well-known logarithmic derivative.

In the geometric calculus, the exponential functions are the functions having a constant derivative.[1] Furthermore, just as the arithmetic average (of functions) is the 'natural' average in the classical calculus, the well-known geometric average is the 'natural' average in the geometric calculus.[1]

### Bigeometric calculus

A similar definition to the geometric derivative is the bigeometric derivative

$f^{*}(x) = \lim_{h \to 0}{ \left({f((1+h)x)\over{f(x)}}\right)^{1\over{h}} } = \lim_{k \to 1}{ \left({f(kx)\over{f(x)}}\right)^{1\over{\ln(k)}} }$

(For the bigeometric derivative, it is assumed that all arguments and all values of f are positive numbers.)

This simplifies[7] to

$f^{*}(x)=e^{xf'(x)\over f(x)}.$

for functions where the statement is meaningful. Notice that the exponent in the preceding expression represents the well-known elasticity concept, which is widely used in economics.

In the bigeometric calculus, the power functions are the functions having a constant derivative.[1] Furthermore, the bigeometric derivative is scale invariant (or scale free), i.e., it is invariant under all changes of scale (or unit) in function arguments and values.

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## Multiplicative integrals

Each multiplicative derivative has an associated multiplicative integral. For example, the geometric derivative and the bigeometric derivative are inversely-related to the geometric integral and the bigeometric integral, respectively.

Of course, each multiplicative integral is a multiplicative operator, but some product integrals are not multiplicative operators. (See Product integral#Basic definitions.)

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## Discrete calculus

Just as differential equations have a discrete analog in difference equations with the forward difference operator replacing the derivative, so too there is the forward ratio operator f(x + 1)/f(x) and recurrence relations can be formulated using this operator.[8][9][10] See also Indefinite product.

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## Complex analysis

Multiplicative versions of derivatives and integrals from complex analysis behave quite differently from the usual operators.[11][12][13]

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## History

Between 1967 and 1988, Jane Grossman, Michael Grossman, and Robert Katz produced a number of publications on a subject created in 1967 by the latter two, called "non-Newtonian calculus." The geometric calculus[14] and the bigeometric calculus[15] are among the infinitely many non-Newtonian calculi that are multiplicative.[1] (Infinitely many non-Newtonian calculi are not multiplicative.)

In 1972, Michael Grossman and Robert Katz completed their book Non-Newtonian Calculus. It includes discussions of nine specific non-Newtonian calculi, the general theory of non-Newtonian calculus, and heuristic guides for application. Subsequently, with Jane Grossman, they wrote several other books/articles on non-Newtonian calculus, and on related matters such as "weighted calculus",[16] "meta-calculus",[17] and averages/means.[18][19]

On page 82 of Non-Newtonian Calculus, published in 1972, Michael Grossman and Robert Katz wrote:

"However, since we have nowhere seen a discussion of even one specific non-Newtonian calculus, and since we have not found a notion that encompasses the *-average, we are inclined to the view that the non-Newtonian calculi have not been known and recognized heretofore. But only the mathematical community can decide that."
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## General theory of non-Newtonian calculus

### Construction: an outline

(This section is based on six sources.[1][2][6][20][21][22])

The construction of an arbitrary non-Newtonian calculus involves the real number system and an ordered pair * of arbitrary complete ordered fields.

Let R denote the set of all real numbers, and let A and B denote the respective realms of the two arbitrary complete ordered fields.

Assume that both A and B are subsets of R. (However, we are not assuming that the two arbitrary complete ordered fields are subfields of the real number system.) Consider an arbitrary function f with arguments in A and values in B.

By using the natural operations, natural orderings, and natural topologies for A and B, one can define the following (and other) concepts of the *-calculus: the *-limit of f at an argument a, f is *-continuous at a, f is *-continuous on a closed interval, the *-derivative of f at a, the *-average of a *-continuous function f on a closed interval, and the *-integral of a *-continuous function f on a closed interval.

Many, if not most, *-calculi are markedly different from the classical calculus, but the structure of each *-calculus is similar to that of the classical calculus. For example, each *-calculus has two Fundamental Theorems showing that the *-derivative and the *-integral are inversely related; and for each *-calculus, there is a special class of functions having a constant *-derivative. Furthermore, the classical calculus is one of the infinitely many *-calculi.

A non-Newtonian calculus is defined to be any *-calculus other than the classical calculus.

### Relationships to classical calculus

(This section is based on six sources.[1][2][6][20][21][22])

The *-derivative, *-average, and *-integral can be expressed in terms of their classical counterparts (and vice-versa). (However, as indicated in the Reception-section below, there are situations in which a specific non-Newtonian calculus may be more suitable than the classical calculus.[2][6][21][22][23][24])

Again, consider an arbitrary function f with arguments in A and values in B. Let α and β be the ordered-field isomorphisms from R onto A and B, respectively. Let α−1 and β−1 be their respective inverses.

Let D denote the classical derivative, and let D* denote the *-derivative. Finally, for each number t such that α(t) is in the domain of f, let F(t) = β−1(f(α(t))).

Theorem 1. For each number a in A, [D*f](a) exists if and only if [DF](α−1(a)) exists, and if they do exist, then [D*f](a) = β([DF](α−1(a))).

Theorem 2. Assume f is *-continuous on a closed interval (contained in A) from r to s, where r and s are in A. Then F is classically continuous on the closed interval (contained in R) from α−1(r) to α−1(s), and M* = β(M), where M* is the *-average of f from r to s, and M is the classical (i.e., arithmetic) average of F from α−1(r) to α−1(s).

Theorem 3. Assume f is *-continuous on a closed interval (contained in A) from r to s, where r and s are in A. Then S* = β(S), where S* is the *-integral of f from r to s, and S is the classical integral of F from α−1(r) to α−1(s).

### Examples

(This section is based on six sources.[1][2][6][20][21][22])

Let I be the identity function on R. Let j be the function on R such that j(x) = 1/x for each nonzero number x, and j(0) = 0. And let k be the function on R such that k(x) = √x for each nonnegative number x, and k(x) = -√(-x) for each negative number x.

Example 1. If α = I = β, then the *-calculus is the classical calculus.

Example 2. If α = I and β = exp, then the *-calculus is the geometric calculus.

Example 3. If α = exp = β, then the *-calculus is the bigeometric calculus.

Example 4. If α = exp and β = I, then the *-calculus is the so-called anageometric calculus.

Example 5. If α = I and β = j, then the *-calculus is the so-called harmonic calculus.

Example 6. If α = j = β, then the *-calculus is the so-called biharmonic calculus.

Example 7. If α = j and β = I, then the *-calculus is the so-called anaharmonic calculus.

Example 8. If α = I and β = k, then the *-calculus is the so-called quadratic calculus.

Example 9. If α = k = β, then the *-calculus is the so-called biquadratic calculus.

Example 10. If α = k and β = I, then the *-calculus is the so-called anaquadratic calculus.

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## Reception

• The First Nonlinear System of Differential And Integral Calculus,[14] a book about the geometric calculus, was reviewed in Mathematical Reviews in 1980 by Ralph P. Boas, Jr. He included the following assertion: "It is not yet clear whether the new calculus [geometric calculus] provides enough additional insight to justify its use on a large scale".
• Bigeometric Calculus: A System with a Scale-Free Derivative[15] was reviewed in Mathematical Reviews in 1984 by Ralph P. Boas, Jr. He included the following assertion: "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".
• Non-Newtonian Calculus,[1] a book including detailed discussions about the geometric calculus and the bigeometric calculus (both of which are non-Newtonian calculi), was reviewed by David Pearce MacAdam in the Journal of the Optical Society of America.[25] He included the following assertion: "The greatest value of these non-Newtonian calculi may prove to be their ability to yield simpler physical laws than the Newtonian calculus."
• Non-Newtonian Calculus, a book including detailed discussions about the geometric calculus and the bigeometric calculus (both of which are non-Newtonian calculi), was reviewed by H. Gollmann (Graz, Austria) in the journal Internationale Mathematische Nachrichten.[26] He included the following assertion: "The possibilities opened up by the new [non-Newtonian] calculi seem to be immense." (German: "Die durch die neuen Kalkule erschlossenen Möglichkeiten scheinen unermesslich.")
• Non-Newtonian Calculus, a book including detailed discussions about the geometric calculus and the bigeometric calculus (both of which are non-Newtonian calculi), was reviewed by Ivor Grattan-Guinness in Middlesex Math Notes.[27] He included the following assertions: "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."
• Non-Newtonian calculus was used by James R. Meginniss (Claremont Graduate School and Harvey Mudd College) to create a theory of probability that is adapted to human behavior and decision making.[21]
• Seminars concerning non-Newtonian calculus and the dynamics of random fractal structures were conducted by Wojbor Woycznski (Case Western Reserve University) at The Ohio State University[23] on 22 April 2011, and at Cleveland State University[28] on 2 May 2012. In the abstracts for the seminars he 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."
• A seminar concerning fractional calculus, random fractals, and non-Newtonian calculus was conducted by Wojbor Woycznski (Case Western Reserve University) at Case Western Reserve University on 3 April 2013.[29] In the abstract for the seminar he 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."
• The geometric calculus was used by Agamirza E. Bashirov (Eastern Mediterranean University in Cyprus), together with Emine Misirli Kurpinar and Ali Ozyapici (both of Ege University in Turkey), in an article on differential equations and calculus of variations.[6] In that article, they state: "We think that multiplicative calculus can especially be useful as a mathematical tool for economics and finance ... In the present paper our aim is to bring multiplicative calculus to the attention of researchers ... and to demonstrate its usefulness." (The "multiplicative calculus" referred to here is the geometric calculus.)
• The geometric calculus was used by Agamirza E. Bashirov, Emine Misirli, Yucel Tandogdu, and Ali Ozyapici in an article on modelling with multiplicative differential equations.[24] In that article they state: "In this study it becomes evident that the multiplicative calculus methodology has some advantages over additive calculus in modeling some processes in areas such as actuarial science, finance, economics, biology, demographics, etc." (The "multiplicative calculus" referred to here is the geometric calculus.)
• The geometric calculus was used by Diana Andrada Filip (Babes-Bolyai University of Cluj-Napoca in Romania) and Cyrille Piatecki (Orléans University in France) to re-postulate and analyse the neoclassical exogenous growth model in economics.[22] In that article they state: "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)[1] ... 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."
• A discussion concerning the advantages of using the geometric calculus in economic analysis is presented in an article by Diana Andrada Filip (Babes-Bolyai University of Cluj-Napoca in Romania) and Cyrille Piatecki (Orléans University in France).[30] 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." (The "multiplicative calculus" referred to here is the geometric calculus.)
• The geometric calculus and the bigeometric calculus were among the topics covered in a course on non-Newtonian calculus conducted in the summer-term of 2012 by Joachim Weickert, Laurent Hoeltgen, and other faculty from the Mathematical Image Analysis Group of Saarland University in Germany. Among the other topics covered were applications to digital image processing, rates of return, and growth processes.[5]
• A multiplicative calculus was used in the 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). In that article they state: "This work presents a new operator of non-Newtonian type 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, for its clarity of purpose."[31]
• The geometric calculus was used by Emine Misirli and Yusuf Gurefe (both of Ege University in Turkey) in their lecture "The new numerical algorithms for solving multiplicative differential equations".[32] In that presentation they stated: "While one problem can be easily expressed using one calculus, the same problem can not be expressed as easily [using another]."
• A multiplicative type of calculus for complex-valued functions of a complex variable was developed by Ali Uzer (Fatih University in Turkey).[11]
• The geometric calculus was used by Emine Misirli and Yusuf Gurefe (both of Ege University in Turkey) in an article on the numerical solution of multiplicative differential equations.[36]
• The geometric calculus was used by James D. Englehardt (University of Miami) and Ruochen Li (Shenzhen, China) in an article on pathogen counts in treated water.[37]
• Weighted non-Newtonian calculus[16] is cited by P. Arun Raj Kumar and S. Selvakumar (both of the National Institute of Technology, Tiruchirappalli in India) in their article "Detection of distributed denial of service attacks using an ensemble of adaptive and hybrid neuro-fuzzy systems".[40]
• The bigeometric calculus was used in an article on chaos in multiplicative dynamical systems by Dorota Aniszewska and Marek Rybaczuk (both from the Wroclaw University of Technology in Poland).[42]
• The bigeometric calculus was used in an article on multiplicative Lorenz systems by Dorota Aniszewska and Marek Rybaczuk (both from Wroclaw University of Technology).[43]
• The bigeometric calculus was used in an article on multiplicative dynamical systems by Dorota Aniszewska and Marek Rybaczuk (both from Wroclaw University of Technology).[44]
• 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).[45]
• The bigeometric calculus was used in an article on fractal dimension and dimensional spaces by Marek Rybaczuka (Wroclaw University of Technology in Poland), Alicja Kedziab (Medical Academy of Wroclaw in Poland), and Witold Zielinskia (Wroclaw University of Technology).[46]
• The geometric calculus and the bigeometric calculus are useful in the study of dimensional spaces. 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.[7][33][43][44][47]
• The geometric calculus was applied to functional analysis by Cengiz Türkmen and Feyzi Başar (both from Fatih University in Turkey).[50]
• The bigeometric calculus was used by Ahmet Faruk Çakmak in his lecture at the 2011 International Conference on Applied Analysis and Algebra at Yıldız Technical University in Istanbul, Turkey.[52]
• The geometric calculus was used by Uğur Kadak (Gazi University in Turkey) and Yusef Gurefe (Bozok University in Turkey) in their presentation at the 2012 Analysis and Applied Mathematics Seminar Series of Fatih University in Istanbul, Turkey.[53]
• The geometric calculus is the subject of an article by Dick Stanley in the journal Primus.[56] The same issue of Primus contains a paper by Duff Campbell: "Multiplicative calculus and student projects".[57]
• The geometric calculus was the subject of a seminar by Michael Coco of Lynchburg College.[58]
• The geometric calculus is the subject of an article by Michael E. Spivey of the University of Puget Sound.[59]
• The geometric calculus is the subject of an article by Alex B. Twist and Michael E. Spivey of the University of Puget Sound.[60]
• In 2008, the article "Multiplicative calculus and its applications",[6] concerning applications of the geometric calculus, was published in the Journal of Mathematical Analysis and Applications. The article was submitted by Steven G. Krantz and written by Agamirza E. Bashirov, Emine Misirli Kurpinar, and Ali Ozyapici. The following is an excerpt from a review[61] of that article by Gerard Lebourg:
"What happens to the old calculus when you restrict its application to positive functions and replace the differential ratio $[f(x+h)-f(x)]\over h$ with the multiplicative one $\left[{f(x+h)\over f(x)}\right]^{1/h}\text{?}$ Answer: the usual derivative $f'(x)$ is replaced with $f^{\star}(x)=\left[ \exp(\ln[f(x)])\right]'$. So you are left with some avatar of the classical calculus to unfold. The authors of this original paper do play this game. Their stated purpose is to promote this new kind of multiplicative calculus." (Note that $f^{\star}(x)=\left[ \exp(\ln[f(x)])\right]'$ should read $f^{\star}(x)=\exp[(\ln\circ f)'(x)]$.[6])
• The article "Multiplicative calculus and its applications" (see preceding item) was reviewed by Stefan G. Samko (University of Algarve, Portugal) in Zentralblatt MATH:[62]
"In this expository article the authors develop the basics of the so called multiplicative calculus, under which the definition of derivatives and integrals is given in terms of the operations of multiplication and division in contrast to addition and subtraction in the usual definitions. Such an approach was suggested in a book of M. Grossman and R. Katz [“Non-Newtonian Calculus”. Pigeon Cove, Mass.: Lee Press (1972; Zbl 0228.26002)]. Transforming multiplication to addition by logarithms, it is easy to see that for instance a multiplicative derivative equals to exp[(lnf)′]. The authors give also some applications where they consider the usage of the language of multiplicative calculus as more useful than the usage of the usual calculus."
• Bigeometric Calculus: A System with a Scale-Free Derivative[15] was reviewed in Mathematics Magazine in 1984. The review was preceded by the following statement: "Articles and books are selected for this section to call attention to interesting mathematical exposition that occurs outside the mainstream of the mathematics literature." The review included the following assertion: "This book compares [the classical and bigeometric calculi], shows their relationship, and suggests applications for which the latter might be more appropriate."
• Non-Newtonian Calculus,[1] a book including detailed discussions about the geometric calculus and the bigeometric calculus (both of which are non-Newtonian calculi), is used in the 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. The report "Stern Review on the Economics of Climate Change" was commissioned by the UK government and was written by a team led by Nicholas Stern (former Chief Economist at the World Bank).[64][65]
• Non-Newtonian Calculus, a book including detailed discussions about the geometric calculus and the bigeometric calculus (both of which are non-Newtonian calculi), is cited by Ivor Grattan-Guinness in his book The Rainbow of Mathematics: A History of the Mathematical Sciences .[66]
• Non-Newtonian Calculus, a book including detailed discussions about the geometric calculus and the bigeometric calculus (both of which are non-Newtonian calculi), is used in an article on sequence spaces by Ahmet Faruk Cakmak (Yıldız Technical University in Turkey) and Feyzi Basar (Fatih University in Turkey).[67] The abstract of the article begins with the statement: "As alternatives to classical calculus, Grossman and Katz (Non-Newtonian Calculus, 1972) introduced the non-Newtonian calculi consisting of the branches of geometric, anageometric, and bigeometric calculus, etc."
• Application of non-Newtonian calculus 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.[74]
• Application of non-Newtonian calculus to "continuous and bounded functions over the field of non-Newtonian complex numbers" was made by Zafer Cakir (Gumushane University, Turkey) in his lecture at the 2012 conference The Algerian-Turkish International Days on Mathematics, at University of Badji Mokhtar at Annaba, in Algeria.[75]
• Non-Newtonian calculus is one of the topics of discussion at the 2013 conference Algerian-Turkish International Days on Mathematics at Fatih University in Istanbul, Turkey.[76]
• Non-Newtonian Calculus is cited in Gordon Mackay's book Comparative Metamathematics. (The eighteen previous editions of Comparative Metamathematics are entitled The True Nature of Mathematics.)[77]
• Non-Newtonian calculus is cited in a book on popular-culture by Paul Dickson.[78]
• Multiplicative calculus was the subject of Christopher Olah's lecture at the Singularity Summit on 13 October 2012.[80]Singularity University's Singularity Summit is a conference on robotics, artificial intelligence, brain-computer interfacing, and other emerging technologies including genomics and regenerative medicine. Christopher Olah is a Thiel Fellow.[81]
• The geometric calculus was the topic of a presentation by Ali Ozyapici and Emine Misirli Kurpinar (both of Ege University in Turkey) at the International ISAAC Congress in August of 2007.[82]
• Multiplicative calculus was the topic of a presentation by Ali Ozyapici and Emine Misirli Kurpinar (both of Ege University in Turkey) at the International Congress of the Jangjeon Mathematical Society in August of 2008.[83]
• Knowledge of the geometric calculus ("multiplicative calculus") is a requirement for the master's degree in computer-engineering at Inonu University (Malatya, Turkey).[84]
• Non-Newtonian calculus was 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.[85]
• The geometric calculus is cited by Daniel Karrasch in his article "Hyperbolicity and invariant manifolds for finite time processes".[86]
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## References

1. Michael Grossman and Robert Katz. Non-Newtonian Calculus, ISBN 0912938013, 1972.
2. Luc Florack and Hans van Assen."Multiplicative calculus in biomedical image analysis", Journal of Mathematical Imaging and Vision, DOI: 10.1007/s10851-011-0275-1, 2011.
3. ^ a b Luc Florack."Regularization of positive definite matrix fields based on multiplicative calculus", Reference 9, Scale Space and Variational Methods in Computer Vision, Lecture Notes in Computer Science, Volume 6667/2012, pages 786-796, DOI: 10.1007/978-3-642-24785-9_66, Springer, 2012.
4. ^ a b Luc Florack."Regularization of positive definite matrix fields based on multiplicative calculus", Third International Conference on Scale Space and Variational Methods In Computer Vision, Ein-Gedi Resort, Dead Sea, Israel, Lecture Notes in Computer Science: 6667, ISBN 978-3-642-24784-2, Springer, 2012.
5. ^ a b Joachim Weickert and Laurent Hoeltgen. University Course: "Analysis beyond Newton and Leibniz", Saarland University in Germany, Mathematical Image Analysis Group, Summer of 2012.
6. Agamirza E. Bashirov, Emine Misirli Kurpinar, and Ali Ozyapici. "Multiplicative calculus and its applications", Journal of Mathematical Analysis and Applications, 2008.
7. ^ a b Marek Rybaczuk, Alicja Kedzia and Witold Zielinski (2001) The concept of physical and fractal dimension II. The differential calculus in dimensional spaces, Chaos, Solitons, & Fractals Volume 12, Issue 13, October 2001, pages 2537–2552
8. ^ M. Jahanshahi, N. Aliev and H. R. Khatami (2004). "An analytic-numerical method for solving difference equations with variable coefficients by discrete multiplicative integration"., 5—10 July 2004, Antalya, Turkey – Dynamical Systems and Applications, Proceedings, pp. 425—435
9. ^ H. R. Khatami & M. Jahanshahi & N. Aliev (2004). "An analytical method for some nonlinear difference equations by discrete multiplicative differentiation"., 5—10 July 2004, Antalya, Turkey – Dynamical Systems and Applications, Proceedings, pp. 455—462
10. ^ N. Aliev, N. Azizi and M. Jahanshahi (2007) "Invariant functions for discrete derivatives and their applications to solve non-homogenous linear and non-linear difference equations"., International Mathematical Forum, 2, 2007, no. 11, 533–542
11. ^ a b Ali Uzer."Multiplicative type complex calculus as an alternative to the classical calculus", Computers & Mathematics with Applications, DOI:10.1016/j.camwa.2010.08.089, 2010.
12. ^ a b Agamirza E. Bashirov and Mustafa Riza."On complex multiplicative differentiation", TWMS Journal of Applied and Engineering Mathematics, Volume 1, Number 1, pages 75-85, 2011.
13. ^ a b Agamirza E. Bashirov and Mustafa Riza."Complex multiplicative calculus", arXiv.org, Cornell University Library, arXiv:1103.1462v1, 2011.
14. ^ a b Michael Grossman. The First Nonlinear System of Differential And Integral Calculus, ISBN 0977117006, 1979.
15. ^ a b c Michael Grossman. Bigeometric Calculus: A System with a Scale-Free Derivative, ISBN 0977117030, 1983.
16. Jane Grossman, Michael Grossman, Robert Katz. The First Systems of Weighted Differential and Integral Calculus, ISBN 0977117014, 1980.
17. ^ Jane Grossman. Meta-Calculus: Differential and Integral, ISBN 0977117022, 1981.
18. ^ a b c Jane Grossman, Michael Grossman, and Robert Katz. Averages: A New Approach, ISBN 0977117049, 1983.
19. ^ a b c Michael Grossman, and Robert Katz. "A new approach to means of two positive numbers", International Journal of Mathematical Education in Science and Technology, Volume 17, Number 2, pages 205-208, Taylor and Francis, 1986..
20. ^ a b c Michael Grossman."An introduction to non-Newtonian calculus", International Journal of Mathematical Education in Science and Technology, Volume 10, #4 (Oct.-Dec., 1979), 525-528.
21. James R. Meginniss. "Non-Newtonian calculus applied to probability, utility, and Bayesian analysis", American Statistical Association: Proceedings of the Business and Economic Statistics Section, 1980.
22. Diana Andrada Filip and Cyrille Piatecki. "A non-Newtonian examination of the theory of exogenous economic growth", CNCSIS – UEFISCSU(project number PNII IDEI 2366/2008) and LEO, 2010.
23. ^ a b Wojbor Woycznski."Non-Newtonian calculus for the dynamics of random fractal structures: linear and nonlinear", seminar at The Ohio State University on 22 April 2011.
24. ^ a b Agamirza E. Bashirov, Emine Misirli, Yucel Tandogdu, and Ali Ozyapici."On modelling with multiplicative differential equations", Applied Mathematics - A Journal of Chinese Universities, Volume 26, Number 4, pages 425-428, DOI: 10.1007/s11766-011-2767-6, Springer, 2011.
25. ^ David Pearce MacAdam.Journal of the Optical Society of America, The Optical Society, Volume 63, January of 1973.
26. ^ H. Gollmann.Internationale Mathematische Nachrichten, Volumes 27 - 29, page 44, 1973.
27. ^ Ivor Grattan-Guinness.Middlesex Math Notes, Middlesex University, London, England, Volume 3, pages 47 - 50, 1977.
28. ^ Wojbor Woycznski."Non-Newtonian calculus for the dynamics of random fractal structures: linear and nonlinear", seminar at Cleveland State University on 2 May 2012.
29. ^ Wojbor Woycznski."Fractional calculus for random fractals", seminar at Case Western Reserve University on 3 April 2013.
30. ^ Diana Andrada Filip and Cyrille Piatecki. "In defense of a non-Newtonian economic analysis", http://www.univ-orleans.fr/leo/infer/PIATECKI.pdf, CNCSIS – UEFISCSU (Babes-Bolyai University of Cluj-Napoca, Romania) and LEO (Orléans University, France), 2013.
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