Talk:PlanetPhysics/Non Newtonian Calculi 2

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INTRODUCTION

The non-Newtonian calculi provide a wide variety of mathematical tools for use in science, engineering, and mathematics. They appear to have considerable potential for use as alternatives to the classical calculus of Newton and Leibniz. [2, 6, 12, 15, 16]

BRIEF DESCRIPTION

There are infinitely many non-Newtonian calculi. Like the classical calculus, each of them possesses (among other things): a derivative, an integral, a natural average, a special class of \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} having a constant derivative, and two Fundamental \htmladdnormallink{theorems}{http://planetphysics.us/encyclopedia/Formula.html} which reveal that the derivative and integral are 'inversely' related. Nevertheless, the non-Newtonian calculi are different from the classical calculus.

For example, in the classical calculus, the derivative and integral are \htmladdnormallink{linear operators}{http://planetphysics.us/encyclopedia/Commutator.html}, i.e., they are additive and homogeneous. This contrasts sharply with the many non-Newtonian calculi having a nonlinear derivative or integral. Indeed, the derivative and integral in each of the following non-Newtonian calculi are nonlinear \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html}: the "geometric calculus", the "bigeometric calculus", the "harmonic calculus", the "biharmonic calculus", the "quadratic calculus", and the "biquadratic calculus". In fact, in each of the former two calculi, the derivative and integral are multiplicative.

Of course in the classical calculus, the linear functions are the functions having a constant derivative. However, in the geometric calculus, the exponential functions are the functions having a constant derivative. And in the bigeometric calculus, the \htmladdnormallink{power functions}{http://planetphysics.us/encyclopedia/PowerFunction.html} are the functions having a constant derivative. (The geometric derivative and the bigeometric derivative are closely related to the well-known logarithmic derivative and elasticity, respectively.)

The well-known arithmetic average (of functions) is the natural average in the classical calculus, but the well-known geometric average is the natural average in the geometric calculus. And the well-known harmonic average and quadratic average (or root mean \htmladdnormallink{square}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html}) are closely related to the natural averages in the harmonic and quadratic calculi, respectively.

Furthermore, unlike the classical derivative, 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.

HISTORY

The non-Newtonian calculi were created in the period from 1967 to 1970 by Michael Grossman and Robert Katz. In August of 1970, they constructed a comprehensive family of calculi consisting of the infinitely many calculi they created in July of 1967 and infinitely many others. Included in the family are the classical calculus, the geometric calculus (July of 1967), and the bigeometric calculus (August of 1970). All of the calculi can be described simultaneously within the framework of a \htmladdnormallink{general theory}{http://planetphysics.us/encyclopedia/GeneralTheory.html}. They decided to use the adjective "\htmladdnormallink{non-Newtonian}{http://planetphysics.us/encyclopedia/NonNewtonian2.html}" to indicate any of the calculi other than the classical calculus.

In 1972, Grossman and Katz completed their book "Non-Newtonian Calculus"[15]. It contains discussions of nine specific non-Newtonian calculi (including the geometric calculus and the bigeometric calculus), 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", "meta-calculus", averages, and means. [7 - 15, 34, 35]

Michael Grossman and Robert Katz knew nothing about non-Newtonian calculus prior to 14 July 1967, when they began their development of that subject. Indeed, in their book "Non-Newtonian Calculus" (1972), they included the following paragraph (page 82): "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."

Note. The six books by Grossman, Grossman, and Katz on non-Newtonian calculus and related matters are available at some academic libraries, public libraries, and book stores such as Amazon.com. On the Internet, each of the books can be read (free of \htmladdnormallink{charge}{http://planetphysics.us/encyclopedia/Charge.html}) at Google Book Search, and each of them can be read and/or downloaded (free of charge) at HathiTrust.

APPLICATIONS AND CITATIONS

Various applications and citations are worth noting, including the following.

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. [16]

Several applications of non-Newtonian calculus were discovered by Agamirza E. Bashirov and Mustafa Riza (both of Eastern Mediterranean University in Cyprus), and Emine Misirli Kurpinar, Ali Ozyapici, and Yusuf Gurefe (all of Ege University in Turkey). Their \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} includes applications to \htmladdnormallink{differential equations}{http://planetphysics.us/encyclopedia/DifferentialEquations.html}, calculus of variations, finite-difference methods, and complex analysis. [2, 24, 27, 33, 84, 87] (The article [2] was "submitted by Steven G. Krantz" and published in 2008 by the Journal of Mathematical Analysis and Applications.)

An application of non-Newtonian calculus to the study of biomedical image analysis was made by Luc Florack and Hans van Assen (both of the Eindhoven University of Technology in the Netherlands). [88]

Non-Newtonian calculus was used by Ali Uzer (Fatih University in Turkey) to develop a multiplicative \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} of calculus for complex-valued functions of a complex variable. [78]

Non-Newtonian calculus was used by Diana Andrada Filip (Babes-Bolyai University of Cluj-Napoca, Romania) and Cyrille Piatecki (LEO, Orleans University, France) to re-postulate and analyse the neoclassical exogenous growth model in economics. [82]

The non-Newtonian natural averages 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 \htmladdnormallink{power}{http://planetphysics.us/encyclopedia/Power.html} 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 four articles [29-32].

A seminar concerning non-Newtonian calculus and the study of the \htmladdnormallink{dynamics}{http://planetphysics.us/encyclopedia/ClassicalMechanics2.html} of random fractal structures was conducted by Wojbor Woycznski (Case Western Reserve University) at The Ohio State University on 22 April 2011. [90]

An application of non-Newtonian calculus to information technology was made in 2008 by S. L. Blyumin of the Lipetsk State Technical University in Russia. [23]

An application of non-Newtonian calculus to the study of pathogen counts in treated water was made by James D. Englehardt (University of Miami) and Ruochen Li (Shenzhen, China). [85]

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

An application of the bigeometric derivative to the theory of elasticity in economics was made by Fernando Cordova-Lepe (Universidad Catolica del Maule in Chile). (He referred to the bigeometric derivative as the "multiplicative derivative.") [3,4] Elasticity and its relationship to the bigeometric derivative is also discussed in Non-Newtonian Calculus [15] and Bigeometric Calculus: A \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} with a Scale-Free Derivative [10].

Non-Newtonian calculus may have application in studies of growth, and in situations involving discontinuous phenomena. [34, 35]

The geometric calculus and/or the bigeometric calculus may have application to \htmladdnormallink{dynamical systems}{http://planetphysics.us/encyclopedia/ContinuousGroupoidHomomorphism.html}, chaos theory, dimensional spaces, and fractal theory. [1, 5, 18]

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

The geometric calculus is cited in a book on the phenomena of growth and structure-building by Manfred Peschel and Werner Mende. [25]

Non-Newtonian calculus is cited in an article on atmospheric \htmladdnormallink{temperature}{http://planetphysics.us/encyclopedia/BoltzmannConstant.html} by Robert G. Hohlfeld, Thomas W. Drueding, and John F. Ebersole. [89]

Non-Newtonian calculus is cited in a book on the \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} crisis by R. Gagliardi and Jerry Pournelle. [26]

"Non-Newtonian Calculus" is cited in a doctoral thesis on nonlinear dynamical systems by David Malkin at University College London. [36]

"Non-Newtonian Calculus" is cited in an article on petroleum engineering by Raymond W. K. Tang and William E. Brigham (both of Stanford University). [37]

Non-Newtonian calculus is mentioned in a book on popular-culture by Paul Dickson. [28]

Non-Newtonian calculus is mentioned in the journal Science Education International. [38]

Non-Newtonian calculus is mentioned in the journal Ciencia e cultura. [39]

Non-Newtonian calculus is mentioned in the journal American Statistical Association: 1997 Proceedings of the \htmladdnormallink{section}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} on Bayesian Statistical Science. [40]

"Non-Newtonian Calculus" is mentioned in the Australian Journal of Statistics. [73]

"Non-Newtonian Calculus" is mentioned in the journal Physique au Canada. [83]

"Non-Newtonian Calculus" is mentioned in the journal Synthese. [74]

"Non-Newtonian Calculus" is mentioned in the journal Mathematical Education. [75]

"Non-Newtonian Calculus" is mentioned in the the journal Institute of Mathematical Statistics Bulletin. [76]

"Non-Newtonian Calculus" was reviewed by Otakar Zich in the journal Kybernetika. [45]

"Non-Newtonian Calculus" was reviewed in the magazine Choice. [41]

"Non-Newtonian Calculus" was reviewed in the journal Search. [77]

"Non-Newtonian Calculus" was reviewed in the journal Wissenschaftliche Zeitschrift: Mathematisch-Naturwissenschaftliche Reihe. [51]

"Non-Newtonian Calculus" was reviewed by M. Dutta in the Indian Journal of History of Science. [42]

"Non-Newtonian Calculus" was reviewed by Karel Berka in the journal Theory and Decision. [44]

"Non-Newtonian Calculus" was reviewed by David Preiss in the journal Aplikace Matematiky. [46]

"Non-Newtonian Calculus" was reviewed in the journal Physikalische Blatter. [62]

"Non-Newtonian Calculus" was reviewed in the journal "Scientia"; Rivista di Scienza. [63]

"Non-Newtonian Calculus" was reviewed in the journal Science Weekly. [64]

"Non-Newtonian Calculus" was reviewed in the journal Philosophia mathematica. [65]

"Non-Newtonian Calculus" was reviewed in the journal Annals of Science. [66]

"Non-Newtonian Calculus" was reviewed in the journal Science Progress. [67]

"Non-Newtonian Calculus" was reviewed in the journal Revue du CETHEDEC. [68]

"Non-Newtonian Calculus" was reviewed in the journal Allgemeines Statistisches Archiv. [69]

"Non-Newtonian Calculus" was reviewed in the journal Il Nuovo Cimento della Societa Italiana di Fisica: A. [70]

"Non-Newtonian Calculus" was reviewed in the journal Bollettino della Unione Matematica Italiana. [71]

"Non-Newtonian Calculus" was reviewed in the journal Cahiers du Centre d'Etudes de Recherche Operationnelle. [72]

"Non-Newtonian Calculus" was reviewed in the journal American Mathematical Monthly. [48]

"The First Nonlinear System of Differential And Integral Calculus" [11], a book about the geometric calculus, was reviewed in the journal American Mathematical Monthly. [52]

"Bigeometric Calculus: A System with a Scale-Free Derivative" [10] was reviewed in Mathematics Magazine. [49]

"Bigeometric Calculus: A System with a Scale-Free Derivative" was reviewed in the journal The Mathematics Student. [58]

"The First Systems of Weighted Differential and Integral Calculus" [9] was reviewed in the journal Praxis der Mathematik. [79]

"Meta-Calculus: Differential and Integral" [7] was reviewed in the journal Indian Journal of \htmladdnormallink{theoretical physics}{http://planetphysics.us/encyclopedia/NonNewtonian2.html}. [80]

The article "An introduction to non-Newtonian calculus" [12] was reviewed by K. Strubecker in the journal Zentralblatt Math (Zbl 0418.26008) [43].

The article "A new approach to means of two positive numbers" [14] was reviewed in Zentralblatt Math (Zbl 0586.26014) [43].

Each of the following three books was reviewed by K. Strubecker in Zentralblatt MATH [43]. 1) "Non-Newtonian Calculus" [15]: Zbl 0228.26002. 2) "The First Systems of Weighted Differential and Integral Calculus" [9]: Zbl 0443.26005. 3) "Meta-Calculus: Differential and Integral" [7]: Zbl 0493.26001.

The article "A new approach to means of two positive numbers" [14] was reviewed in the journal ZDM (1986c.10787) [50].

Each of the following five books was reviewed in ZDM [50]. 1) "Non-Newtonian Calculus"[15]: 1982a.00259. 2) "The First Nonlinear System of Differential and Integral Calculus" [11]: 1982a.00243. 3) "The First Systems of Weighted Differential and Integral Calculus" [9]: 1982a.00248. 4) "Bigeometric Calculus: A System with a Scale-Free Derivative" [10]: 19861.06868. 5) "Averages: A New Approach" [8]: 19861.06873.

Each of the following six books was reviewed in the journal Internationale Mathematische Nachrichten [53]. 1) "Non-Newtonian Calculus": Number 105, 1972. 2) "The First Nonlinear System of Differential and Integral Calculus": \htmladdnormallink{volumes}{http://planetphysics.us/encyclopedia/Volume.html} 35-36, page 42, 1981. 3) "The First Systems of Weighted Differential and Integral Calculus": Volumes 35-36, page 40, 1981. 4) "Meta-Calculus: Differential and Integral": Volumes 35-36, page 140, 1981. 5) "Bigeometric Calculus: A System with a Scale-Free Derivative": Volumes 37-38, page 266, 1983. 6) "Averages: A New Approach": Volumes 37-38, page 266, 1983.

Each of the following six books was reviewed in the journal Scientific Annals of Alexandru Ioan Cuza University of Iasi: Mathematics Section. [55] 1) "Non-Newtonian Calculus": Volumes 17-18, 1972. 2) "The First Nonlinear System of Differential and Integral Calculus": Volumes 26-27, 1980. 3) "The First Systems of Weighted Differential and Integral Calculus": Volumes 27-28, 1981. 4) "Meta-Calculus: Differential and Integral": Volumes 28-29, 1982. 5) "Bigeometric Calculus: A System with a Scale-Free Derivative": Volumes 29-30, 1983. 6) "Averages: A New Approach": Volumes 29-30, 1983.

Each of the following two books was reviewed in the journal Publicationes Mathematicae. [56] 1) "Non-Newtonian Calculus": Volume 19, page 351, 1972. 2) "Bigeometric Calculus: A System with a Scale-Free Derivative": Volume 32, page 282, 1985.

Each of the following three books was reviewed in the journal Nieuw Tijdschrift Voor Wiskunde. [57] 1) "The First Nonlinear System of Differential And Integral Calculus": Volume 68, page 104, 1981. 2) "The First Systems of Weighted Differential and Integral Calculus": Volumes 69-70, page 235, 1982. 3) "Meta-Calculus: Differential and Integral": Volumes 69-70, page 236, 1982.

Each of the following two books was reviewed by Leo Barsotti in the journal Boletim da Sociedade Paranaense de Matematica. [54] 1) "The First Nonlinear System of Differential and Integral Calculus": Volume 2, page 32, 1981. 2) "The First Systems of Weighted Differential and Integral Calculus": Volume 2, pages 32-33, 1981.

Each of the following three books was reviewed in the journal L'Enseignement Mathematique. [59] 1) "The First Nonlinear System of Differential and Integral Calculus": page 52, 1980. 2) "Bigeometric Calculus: A System with a Scale-Free Derivative": page 83, 1982. 3) "Averages: A New Approach": page 83, 1982.

Each of the following two books was reviewed in the journal Acta Scientiarum Mathematicarum. [60] 1) "Non-Newtonian Calculus": Volume 33, page 361, 1972. 2) "The First Nonlinear System of Differential and Integral Calculus": Volumes 42-43, page 225, 1980.

Each of the following six books was reviewed in the journal Industrial Mathematics. [61] 1) "Non-Newtonian Calculus": Volumes 43-45, page 91, 1994 . 2) "The First Nonlinear System of Differential and Integral Calculus": Volumes 28-30, page 143, 1978. 3) "The First Systems of Weighted Differential and Integral Calculus": Volumes 31-33, page 66, 1981. 4) "Meta-Calculus: Differential and Integral": Volumes 31-33, page 83, 1981. 5) "Bigeometric Calculus: A System with a Scale-Free Derivative": Volumes 33-34, page 91, 1983. 6) "Averages: A New Approach": Volumes 33-34, page 91, 1983.

Each of the following two books was reviewed in the journal Economic Books: Current Selections. [81] 1) "The First Systems of Weighted Differential and Integral Calculus": Volume 9, page 29, 1982. 2) "Meta-Calculus: Differential and Integral": Volume 9, page 29, 1982.

"Non-Newtonian Calculus" was reviewed in the journal Mathematical Reviews in 1978. [47]

Each of the following five books was reviewed by Ralph P. Boas, Jr. in Mathematical Reviews [47]. 1) "The First Nonlinear System of Differential and Integral Calculus" [11]: Mathematical Reviews, 1980. 2) "The First Systems of Weighted Differential and Integral Calculus" [9]: Mathematical Reviews, 1981. 3) "Meta-Calculus: Differential and Integral" [7]: Mathematical Reviews, 1982. 4) "Bigeometric Calculus: A System with a Scale-Free Derivative" [10]: Mathematical Reviews, 1984. 5) "Averages: A New Approach" [8]: Mathematical Reviews, 1984.

Note. Other reviews are indicated in the COMMENTS section below.

Note. It is natural to speculate about future applications of non-Newtonian calculus and related matters such as "weighted calculus" and "meta-calculus". Perhaps scientists, engineers, and mathematicians will use them to define new \htmladdnormallink{concepts}{http://planetphysics.us/encyclopedia/PreciseIdea.html}, to yield new or simpler laws, or to formulate or solve problems.

REFERENCES

[1] Dorota Aniszewska. "Multiplicative Runge-Kutta methods.", Nonlinear Dynamics, Volume 50, Numbers 1-2, 2007.

[2] Agamirza E. Bashirov, Emine Misirli Kurpinar, and Ali Ozyapici. "Multiplicative calculus and its applications", Journal of Mathematical Analysis and Applications, Volume 337, Issue 1, pages 36 - 48, January 2008.

[3] Fernando Cordova-Lepe. "From quotient \htmladdnormallink{operation}{http://planetphysics.us/encyclopedia/Cod.html} toward a proportional calculus", Journal of Mathematics, Game Theory and Algebra, 2004.

[4] Fernando Cordova-Lepe. "The multiplicative derivative as a measure of elasticity in economics", TMAT Revista Latinoamericana de Ciencias e Ingenieria, Volume 2, Number 3, 2006.

[5] Felix R. Gantmacher. "The Theory of \htmladdnormallink{matrices",}{http://planetphysics.us/encyclopedia/Matrix.html} Volumes 1 and 2, Chelsea Publishing Company, 1959.

[6] Ivor Grattan-Guinnness. "The Rainbow of Mathematics: A History of the Mathematical Sciences", pages 332 and 774, ISBN 0393320308, 2000.

[7] Jane Grossman. "Meta-Calculus: Differential and Integral", ISBN 0977117022, 1981.

[8] Jane Grossman, Michael Grossman, and Robert Katz. "Averages: A New Approach", ISBN 0977117049, 1983.

[9] Jane Grossman, Michael Grossman, Robert Katz. "The First Systems of Weighted Differential and Integral Calculus", ISBN 0977117014, 1980.

[10] Michael Grossman. "Bigeometric Calculus: A System with a Scale-Free Derivative", ISBN 0977117030, 1983.

[11] Michael Grossman. "The First Nonlinear System of Differential and Integral Calculus", ISBN 0977117006, 1979.

[12] Michael Grossman. "An introduction to non-Newtonian calculus", International Journal of Mathematical Education in Science and Technology, Volume 10, Number 4, pages 525-528, 1979.

[13] Michael Grossman and Robert Katz, "Isomorphic calculi", International Journal of Mathematical Education in Science and Technology, Volume 15, Number 2, pages 253 - 263, 1984.

[14] 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, 1986.

[15] Michael Grossman and Robert Katz. "Non-Newtonian Calculus", ISBN 0912938013, Lee Press, 1972.

[16] James R. Meginniss. "Non-Newtonian calculus applied to probability, utility, and Bayesian analysis", American Statistical Association: Proceedings of the Business and Economic Statistics Section, pages 405 - 410, 1980.

[17] "Vito Volterra". Wikipedia article (Internet).

[18] M. Rybaczuk and P. Stoppel. "The fractal growth of fatigue defects in materials", International Journal of Fracture, Volume 103, Number 1, 2000.

[19] Dick Stanley. "A multiplicative calculus", Primus, Volume 9, Issue 4, 1999.

[20] Jane Tang. "On the construction and interpretation of means", International Journal of Mathematical Education in Science and Technology, Volume 14, Number 1, pages 55 - 57, 1983.

[21] "Multiplicative calculus". Wikipedia article (Internet).

[22] "Product integral". Wikipedia article (Internet).

[23] S. L. Blyumin. "Discrete vs. continuous, in information technology: quantum calculus and its alternatives", Reference 13, Lipetsk State Technical University, 2008. (For a link, see http://sites.google.com/site/nonnewtoniancalculus/Home)

[24] Mustafa Riza, Ali Ozyapici, and Emine Misirli Kurpinar. "Multiplicative finite difference methods", Quarterly of Applied Mathematics, Volume 67, pages 745 - 754, May 2009.

[25] Manfred Peschel and Werner Mende. "The Predator-Prey Model: Do We Live in a Volterra World?", ISBN 0387818480, Springer, 1986.

[26] R. Gagliardi and Jerry Pournelle. "The Mathematics of the Energy Crisis", page 76, Intergalactic Pub. Co., 1978.

[27] Ali Ozyapici and Emine Misirli Kurpinar. "Notes on Multiplicative Calculus", 20th International Congress of the Jangjeon Mathematical Society, No. 67, August 2008.

[28] Paul Dickson. "The New Official Rules", page 113, ISBN 0201172763, Addison-Wesley Publishing Company, 1989.

[29] Horst Alzer. "Bestmogliche abschatzungen fur spezielle mittelwerte", Reference 19; Univ. u Novom Sadu, Zb. Rad. Prirod.-Mat. Fak., Ser. Mat. 23/1; 1993.

[30] V. S. Kalnitsky. "Means generating the conic sections and the third degree polynomials", Reference 7, Saint Petersburg Mathematical Society Preprint 2004-04, 2004.

[31] Methanias Colaco Junior, Manoel Mendonca, and Francisco Rodrigues. "Mining software change history in an industrial environment", Reference 20, XXIII Brazilian Symposium on Software Engineering, 2009.

[32] Nicolas Carels and Diego Frias. "Classifying coding \htmladdnormallink{DNA}{http://planetphysics.us/encyclopedia/FCS3.html} with nucleotide statistics", Reference 36, Bioinformatics and Biology Insights 2009:3, Libertas Academica, 2009.

[33] Ali Ozyapici and Emine Misirli Kurpinar. "Exponential Approximation on Multiplicative Calculus", 6th ISAAC Congress, page 471, 2007.

[34] Jane Grossman, Michael Grossman, and Robert Katz. "Which growth rate?", International Journal of Mathematical Education in Science and Technology, Volume 18, Number 1, pages 151 - 154, 1987.

[35] Michael Grossman. "Calculus and discontinuous phenomena", International Journal of Mathematical Education in Science and Technology, Volume 19, Number 5, pages 777 - 779, 1988.

[36] David Malkin. "The Evolutionary Impact of Gradual Complexification on Complex Systems", doctoral thesis at University College London's \htmladdnormallink{computer}{http://planetphysics.us/encyclopedia/Program3.html} Science Department, 2009.

[37] Raymond W. K. Tang and William E. Brigham. "Transient pressure analysis in composite reservoirs", Reference 18, Stanford University: Petroleum Research Institute (with United States Department of Energy), 1982.

[38] Science Education International, International Council of Associations for Science Education, Volumes 2-3, page 24, 1991.

[39] Ciencia e Cultura, Sociedade Brasileira para o Progresso da Ciencia, Volume 32, Issues 5-8, page 829, 1980.

[40] American Statistical Association: 1997 Proceedings of the Section on Bayesian Statistical Science, page 176, 1997.

[41] Choice, Association of College and Research Libraries, American Library Association, Volume 9, Issues 8 - 12, 1972.

[42] Indian Journal of History of Science, Indian National Science Academy, Volumes 6 - 8, page 154, 1971 - 1973.

[43] Zentralblatt MATH, FIZ Karlsruhe.

[44] Theory and Decision, Springer, Volume 6, page 237, 1975.

[45] Kybernetika, Ceskoslovenska Kyberneticka Spolecnost, Volume 9, page 155, 1973.

[46] Aplikace Matematiky, Ceskoslovenska Akademie Ved. Matematicky Ustav, Volume 18, page 208, 1973.

[47] Mathematical Reviews, American Mathematical Society.

[48] American Mathematical Monthly, Mathematical Association of America, May of 1973.

[49] Mathematics Magazine, Mathematical Association of America, Volume 57, Number 2, page 119, 1984.

[50] ZDM, Springer.

[51] Wissenschaftliche Zeitschrift: Mathematisch-Naturwissenschaftliche Reihe, University of Leipzig, Volume 22, page 97, 1973.

[52] American Mathematical Monthly, Mathematical Association of America, June/July of 1980.

[53] Internationale Mathematische Nachrichten, Osterreichische Mathematische Gesellschaft.

[54] Boletim da Sociedade Paranaense de Matematica, Sociedade Paranaense de Matematica.

[55] Analele Stiintifice ale Universitatii "Al. I. Cuza" din Iasi: Mathematics Section, Universitatea "Al. I Cuza".

[56] Publicationes Mathematicae, Kossuth Lajos Tudomanyegyetem: Matematikai Intezet.

[57] Nieuw Tijdschrift Voor Wiskunde, P. Noordhoff.

[58] The Mathematics Student, Indian Mathematical Society, Volumes 53-54, page 57, 1987.

[59] L'Enseignement Mathematique, International Commission on the Teaching of Mathematics.

[60] Acta Scientiarum Mathematicarum, Institutum Bolyaianum Universitatis Szegediensis.

[61] Industrial Mathematics, Industrial Mathematics Society.

[62] Physikalische Blatter, Physik Verlag, Volume 29, page 48, 1973.

[63] "Scientia"; Rivista di Scienza, ResearchGATE, Volume 107, page 919, 1972.

[64] Science Weekly, American Association for the Advancement of Science,Volume 176, page 954, 1972.

[65] Philosophia mathematica, Canadian Society for the History and Philosophy of Mathematics, Volumes 9-14, page 96, 1972.

[66] Annals of Science, British Society for the History of Science, Volumes 29-30, page 424, 1972.

[67] Science Progress, Science Progress, Volume 60, page 428, 1972.

[68] Revue du CETHEDEC, Centre d'Etudes Theoriques de la \htmladdnormallink{detection}{http://planetphysics.us/encyclopedia/MolecularOrbitals.html} et des Communications, Volume 9, page 110, 1972.

[69] Allgemeines Statistisches Archiv, Deutsche Statistische Gesellschaft, Volumes 56-57, page 403, 1972.

[70] Il Nuovo Cimento della Societa Italiana di Fisica: A, Societa Italiana di Fisica, page 851, 1972.

[71] Bollettino della Unione Matematica Italiana, Unione Matematica Italiana, page 289, 1972.

[72] Cahiers du Centre d'Etudes de Recherche Operationnelle, Centre d'Etudes de Recherche Operationnelle, Volumes 14-15, page 85, 1972.

[73] Australian Journal of Statistics. Statistical Society of Australia, Volumes 14-15, 1972.

[74] Synthese, D. Reidel Publishing Company, Volume 26, page 181, 1973.

[75] Mathematical Education, India University Grants Commission, Volume 2, 1985.

[76] Institute of Mathematical Statistics Bulletin, Institute of Mathematical Statistics, Volumes 1-2, 1972.

[77] Search, ANZAAS, Volume 3, page 457, 1972.

[78] Ali Uzer. "Multiplicative type complex calculus as an alternative to the classical calculus", Computers and Mathematics with Applications, 2010.

[79] Praxis der Mathematik, Aulis Verlag Deubner, Volume 23, page 94, 1981.

[80] Indian Journal of Theoretical Physics, Institute of Theoretical Physics (India), Volume 31, page 176, 1983.

[81] Economic Books: Current Selections, University of Pittsburgh, Department of Economics, Volume 9, page 29, 1982.

[82] 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 Laboratoire d'Economie d'Orleans (LEO), 2010.

[83] Physique au Canada, Canadian Association of Physicists, Volumes 27-28, page 88, 1971.

[84] Emine Misirli and Yusuf Gurefe. "Multiplicative Adams Bashforth Moulton methods", Numerical \htmladdnormallink{algorithms}{http://planetphysics.us/encyclopedia/RecursiveFunction.html}, doi: 10.1007/s11075-010-9437-2, Volume 55, 2010.

[85] James D. Englehardt and Ruochen Li. "The discrete Weibull distribution: an alternative for correlated counts with confirmation for microbial counts in water", Risk Analysis, doi: 10.1111/j.1539-6924.2010.01520.x, 2010.

[86] David Baqaee. "Intertemporal choice: a Nash bargaining approach", Reserve Bank of New Zealand, Research: Discussion Paper Series, ISSN 1177-7567, DP2010/08, September 2010.

[87] Agamirza E. Bashirov and Mustafa Riza. "Complex multiplicative calculus", arXiv.org, Cornell University Library, arXiv:1103.1462v1, 2011.

[88] 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.

[89] Robert G. Hohlfeld, Thomas W. Drueding, and John F. Ebersole. "Application of optical measure theory to atmospheric temperature sounding from TOVS radiances", U.S. Air \htmladdnormallink{force}{http://planetphysics.us/encyclopedia/Thrust.html} Geophysics Laboratory, Atmospheric Sciences Division, GL-TR-89-0120, 1989.

[90] 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.

ADDITIONAL READING

Google Book Search: "Non-Newtonian Calculus"

HathiTrust: http://babel.hathitrust.org/cgi/mb?a=listis;c=216746186

Non-Newtonian Calculus website: http://sites.google.com/site/nonnewtoniancalculus/

COMMENTS

"Your ideas [in "Non-Newtonian Calculus"] seem quite ingenious." - Professor Dirk J. Struik, Massachusetts Institute of Technology, USA.

"[Your books] on non-Newtonian calculus ... appear to be very useful and innovative." - Professor Kenneth J. Arrow, Nobel-Laureate, Stanford University, USA.

"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." - Professor James R. Meginniss, Claremont Graduate School and Harvey Mudd College, USA.

"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." - Professor Ivor Grattan-Guinness, Middlesex University, England.

"The possibilities opened up by the [non-Newtonian] calculi seem to be immense." - Professor H. Gollmann, Graz, Austria.

"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 \htmladdnormallink{physical laws}{http://planetphysics.us/encyclopedia/PrincipleOfCorrespondingStates.html} 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." - Professor David Pearce MacAdam, Cape \htmladdnormallink{cod}{http://planetphysics.us/encyclopedia/Cod.html} Community College, USA.

"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." - Professor Ralph P. Boas, Jr., Northwestern University, USA.

"We think that [the geometric calculus] can especially be useful as a mathematical tool for economics and finance ... ." - Professor Agamirza E. Bashirov, Eastern Mediterranean University, Cyprus; Professor Emine Misirli Kurpinar, Ege University, Turkey; Professor Ali Ozyapici, Ege University, Turkey.

"In this paper, we have tried to present how a non-Newtonian calculus could be applied to repostulate and analyse the neoclassical exogenous growth model [in economics]. ... In fact, one must acknowledge that it's only under the effort of Grossman and 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 \htmladdnormallink{fields}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} of economic analysis which can be profoundly re-thought in the light of this discovery." - Professor Diana Andrada Filip, Babes-Bolyai University of Cluj-Napoca, Romania; Professor Cyrille Piatecki, Orleans University, France.

"We advocate the use of an alternative calculus [the geometric calculus] in biomedical image analysis ... . The use of [that] calculus has been advocated in other contexts, such as in the theory of survival analysis and Markov processes ... ." - Professors Luc Florack and Hans van Assen, Eindhoven University of Technology, The Netherlands.

SOURCES. The comments by Professors Struik, Arrow, and Meginniss are excerpts from their correspondence with Grossman, Grossman, and Katz. The comments by Professors Grattan-Guinness, Gollmann, and MacAdam are excerpts from their reviews of the book "Non-Newtonian Calculus" in Middlesex Math Notes (1977), Internationale Mathematische Nachrichten (1972), and Journal of the Optical Society of America (1973), respectively. The comment by Professor Boas is an excerpt from his review of the book "Bigeometric Calculus: A System with a Scale-Free Derivative" in Mathematical Reviews (1984). The comment by Professors Bashirov, Misirli Kurpinar, and Ozyapici is an excerpt from their article "Multiplicative calculus and its applications" in the Journal of Mathematical Analysis and Applications (2008). The comments by Professors Andrada Filip and Piatecki are excerpts from their article "A non-Newtonian examination of the theory of exogenous economic growth" in CNCSIS - UEFISCSU (project number PNII IDEI 2366/2008) and Laboratoire d'Economie d'Orl\'eans (LEO) (2010). The comments by Professors Florack and van Assen are excerpts from their article "Multiplicative calculus in biomedical image analysis" in Journal of Mathematical Imaging and Vision, published with open access at Springerlink.com (2011).

ACKNOWLEDGEMENT Thanks to David Lukas and Kenneth Lukas for constructing previous versions of this website, and for their expert advice on website construction.

CONTACT

Name: Michael Grossman

E-mail: smithpith@yahoo.com

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