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 functions having a constant derivative, and two Fundamental Theorems which reveal that the derivative and integral are 'inversely' related. Nevertheless, many non-Newtonian calculi are markedly different from the classical calculus.
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 power functions 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 square) 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
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 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. In 2008, Michael Grossman encountered discussions that led him to wonder if a multiplicative (perhaps non-Newtonian) integral or derivative had been developed by Vito Volterra, who lived from 1860 to 1940. [1, 5, 17, 18, 22]
Note. The six books by Grossman, Grossman, and Katz on non-Newtonian calculus and related matters are listed below, and 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 charge) at Google Books, and each of them can be read and/or downloaded (free of charge) at HathiTrust.
(1) Michael Grossman and Robert Katz: Non-Newtonian Calculus, ISBN 0912938013, 1972. [15]
(2) Michael Grossman: The First Nonlinear System of Differential and Integral Calculus, ISBN 0977117006, 1979. [11]
(3) Jane Grossman, Michael Grossman, Robert Katz: The First Systems of Weighted Differential and Integral Calculus, ISBN 0977117014, 1980. [9]
(4) Jane Grossman: Meta-Calculus: Differential and Integral, ISBN 0977117022, 1981. [7]
(5) Michael Grossman: Bigeometric Calculus: A System with a Scale-Free Derivative, ISBN 0977117030, 1983. [10]
(6) Jane Grossman, Michael Grossman, and Robert Katz: Averages: A New Approach, ISBN 0977117049, 1983. [8]
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, Mustafa Riza, and Yucel Tandogdu (all of Eastern Mediterranean University in Turkey); Emine Misirli Kurpinar and Yusuf Gurefe (both of Ege University in Turkey); and Ali Ozyapici (Lefke European University in Turkey). Their work has application to differential equations, calculus of variations, finite-difference methods, complex analysis, actuarial science, finance, economics, biology, and demographics. [2, 24, 27, 33, 84, 87, 94, 95] (The article [2] was "submitted by Steven G. Krantz" and published in 2008 by the Journal of Mathematical Analysis and Applications.)
Non-Newtonian calculus was used by Diana Andrada Filip (Babes-Bolyai University of Cluj-Napoca, Romania) and Cyrille Piatecki (LEO, Orléans University, France) to re-postulate and analyse the neoclassical exogenous growth model in economics. [82]
Non-Newtonian calculus was used in the study of biomedical image analysis by Luc Florack and Hans van Assen (both of the Eindhoven University of Technology in the Netherlands). Their work has application to "complex imaging frameworks, such as diffusion tensor imaging". [88, 96]
Non-Newtonian calculus was used by Ali Uzer (Fatih University in Turkey) to develop a multiplicative type of calculus for complex-valued functions of a complex variable. [78]
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 power means, the logarithmic mean, the identric mean, and the Stolarsky mean. The family of means was used to yield simple proofs of some familiar inequalities. [14] Publications [8, 14] about that family are cited in four articles [29-32].
A seminar concerning non-Newtonian calculus and the study of the dynamics of random fractal structures was conducted by Wojbor Woycznski (Case Western Reserve University) at The Ohio State University on 22 April 2011. [90]
Application of non-Newtonian calculus to information technology was made in 2011 by S. L. Blyumin of the Lipetsk State Technical University in Russia. [23]
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]
Application of the bigeometric derivative to the theory of elasticity in economics was made by Fernando Córdova-Lepe (Universidad Católica del Maule in Chile) . (He referred to the bigeometric derivative as the "multiplicative derivative.") [3, 4] Elasticity and its relationship to the bigeometric derivative is also discussed in Non-Newtonian Calculus [15] and Bigeometric Calculus: A System with a Scale-Free Derivative [10].
The geometric integral is useful in stochastics. [22]
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 dynamical systems, chaos theory, dimensional spaces, and fractal theory. [1, 5, 18]
Non-Newtonian calculus was used by Stanley Paul Palasek (Sonoran Science Academy in Tucson, Arizona) in a biology project on opioid peptide delivery at an Intel® International Science and Engineering Fair. [97]
Non-Newtonian Calculus [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 First Systems of Weighted Differential and Integral Calculus [9] is cited by the authors indicated below in their article on the global burden of cholera: Mohammad Ali, Anna Lena Lopez, Young Ae You, Young Eun Kim, Binod Sah, Brian Maskery, and John Clemens (all of the United Nations' International Vaccine Institute, Snu Research Park, San 4-8 Nakseongdae-dong Gwanak-gu, Seoul, Korea, 151 - 919). [98]
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 temperature by Robert G. Hohlfeld, Thomas W. Drueding, and John F. Ebersole. [89]
Non-Newtonian calculus is cited in a book on the energy 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 the eBook Economic Statistics. [91]
Non-Newtonian Calculus was cited by Karol Kosar and Ivan Kupka in a lecture at a student conference at Comenius University in Slovakia. [93]
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 Ciência e cultura. [39]
Non-Newtonian calculus is mentioned in the journal American Statistical Association: 1997 Proceedings of the Section 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 is cited 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 Blätter. [62]
Non-Newtonian Calculus was reviewed in the journal "Scientia"; Rivista di Scienza. [63]
Non-Newtonian Calculus is cited in the journal Science Weekly. [64]
Non-Newtonian Calculus was reviewed in the journal Philosophia mathematica. [65]
Non-Newtonian Calculus is cited in the journal Annals of Science. [66]
Non-Newtonian Calculus is cited in the journal Science Progress. [67]
Non-Newtonian Calculus was reviewed in the journal Revue du CETHEDEC. [68]
Non-Newtonian Calculus is cited in the journal Allgemeines Statistisches Archiv. [69]
Non-Newtonian Calculus is cited 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 Opérationnelle. [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] is cited in the journal Praxis der Mathematik. [79]
Meta-Calculus: Differential and Integral [7] is cited in the journal Indian Journal of Theoretical Physics. [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.
1) Non-Newtonian Calculus: Number 105, 1972.
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 Iaşi: Mathematics Section. [55]
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 is cited 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 Matemática. [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 Mathématique. [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 is cited 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 is cited 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 is cited 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.
Each of the following two books was reviewed by P. Wilker in the journal Revue de mathématique élémentaires. [92]
1) The First Nonlinear System of Differential and Integral Calculus, Volumes 37-40.
2) The First Systems of Weighted Differential and Integral Calculus: Volumes 37-40.
Non-Newtonian Calculus [15] 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’s 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 concepts, to yield new or simpler laws, or to formulate or solve problems.
REFERENCES
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[3] Fernando Córdova-Lepe. "From quotient operation toward a proportional calculus", Journal of Mathematics, Game Theory and Algebra, 2004.
[4] Fernando Córdova-Lepe. "The multiplicative derivative as a measure of elasticity in economics", TMAT Revista Latinoamericana de Ciencias e Ingeniería, Volume 2, Number 3, 2006.
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[6] Ivor Grattan-Guinnness. The Rainbow of Mathematics: A History of the Mathematical Sciences, pages 332 and 774, ISBN 0393320308, W. W. Norton & Company, 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.
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[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?, page 246, ISBN 0387818480, Springer, 1986.
[27] Ali Ozyapici and Emine Misirli Kurpinar. "Notes on Multiplicative Calculus", 20th International Congress of the Jangjeon Mathematical Society, #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.
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[31] Methanias Colaço Júnior, Manoel Mendonça, 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 DNA with nucleotide statistics", Reference 36, Bioinformatics and Biology Insights 2009:3, Libertas Academica, pages 141 - 154, 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 Computer 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.
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[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, Österreichische Mathematische Gesellschaft.
[54] Boletim da Sociedade Paranaense de Matemática, Sociedade Paranaense de Matemática.
[55] Analele științifice ale Universității "Al. I. Cuza" din Iași: Secțiunea Matematică, Universitatea "Al. I Cuza".
[56] Publicationes Mathematicae, Kossuth Lajos Tudományegyetem: Matematikai Intézet.
[57] Nieuw Tijdschrift Voor Wiskunde, P. Noordhoff.
[58] The Mathematics Student, Indian Mathematical Society, Volumes 53-54, page 57, 1987.
[59] L'Enseignement Mathématique, International Commission on the Teaching of Mathematics.
[60] Acta Scientiarum Mathematicarum, Institutum Bolyaianum Universitatis Szegediensis.
[61] Industrial Mathematics, Industrial Mathematics Society.
[62] Physikalische Blätter, 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 Théoriques de la Détection et des Communications, Volume 9, page 110, 1972.
[69] Allgemeines Statistisches Archiv, Deutsche Statistische Gesellschaft, Volumes 56-57, page 403, 1972.
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[71] Bollettino della Unione Matematica Italiana, Unione Matematica Italiana, page 289, 1972.
[72] Cahiers du Centre d'Etudes de Recherche Opérationnelle, Centre d'Etudes de Recherche Opérationnelle, Volumes 14-15, page 85, 1972.
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[77] Search, Australian & New Zealand Association for the Advancement of Science (ANZAAS), Volume 3, page 457, 1972.
[78] 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.
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[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’Orléans (LEO), 2010.
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[84] Emine Misirli and Yusuf Gurefe. "Multiplicative Adams Bashforth–Moulton methods", Numerical Algorithms, 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 Force 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.
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[94] Agamirza E. Bashirov, Emine Misirli, Yucel Tandogdu, and Ali Ozyapici. "On modeling with multiplicative differential equations", Applied Mathematics-A Journal of Chinese Universities, Volume 26, Number 4, pages 425-428, Springer, 2011.
[95] 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.
[96] 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, Springer, Volume 6667/2012, pages 786-796, DOI: 10.1007/978-3-642-24785-9_66, 2012.
[97] Stanley Paul Palasek. "Nonlinear modeling and optimization of peptide delivery", Intel® International Science and Engineering Fair, PH029, Society for Science and the Public, 2011.
[98] Mohammad Ali, Anna Lena Lopez, Young Ae You, Young Eun Kim, Binod Sah, Brian Maskery, and John Clemens. "The global burden of cholera", Bulletin of the World Health Organization, Research Article ID: BLT.11.093427, United Nations, 2012.
ADDITIONAL READING
S. L. Blyumin. "БИНАРНЫЕ АРИФМЕТИЧЕСКИЕ ОПЕРАЦИИ, ФОРМУЛЫКОНЕЧНЫХ РАЗЛИЧЕНИЙ, ФУНКЦИОНАЛЬНЫЕ УРАВНЕНИЯ ГОМОМОРФИЗМОВ".
Gauss Quote re "new calculi" (in "Memorabilia Mathematica" by Moritz, #1215).
Robert Katz. Axiomatic Analysis, D. C. Heath and Company, 1964.
Robert Edouard Moritz. "Quotientiation, an extension of the differentiation process", Proceedings of the Nebraska Academy of Sciences, Volume VII, pages 112 - 117, 1897 - 1890.
Non-Newtonian Calculus website: http://sites.google.com/site/nonnewtoniancalcus/
Non-Newtonian Calculus at WordPress.com.
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.
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.
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.
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
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."
Professor David Pearce MacAdam
Cape Cod 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 study it becomes evident that the [geometric] calculus methodology has some
advantages over [classical] calculus in modeling some processes in areas such as actuarial science,
finance, economics, biology, demographics, etc."
Professor Agamirza E. Bashirov
Eastern Mediterranean University, Turkey/
Professor Emine Misirli
Ege University, Turkey/
Professor Yucel Tandogdu
Eastern Mediterranean University, Turkey/
Professor Ali Ozyapici
Lefke European 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 & 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."
Professor Diana Andrada Filip
Babes-Bolyai University of Cluj-Napoca, Romania/
Professor Cyrille Piatecki
Orléans University, France
"We
advocate the use of [the geometric calculus] in
biomedical image analysis ... . "
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 comment by
Professors Bashirov, Misirli, Tandogdu, and Ozyapici is an excerpt from
their article "On modeling with multiplicative differential equations"
in the journal Applied Mathematics-A Journal of Chinese Universities (2011). 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’Orléans (LEO) (2010). The
comment by Professors Florack and van Assen is an excerpt from their
article "Multiplicative calculus in biomedical image analysis" in
Journal of Mathematical Imaging and Vision, published with open access
at Springerlink.com (2011).
QUOTATIONS
"For
each successive class of phenomena, a new calculus or a new geometry,
as the case might be, which might prove not wholly inadequate to the
subtlety of nature." "In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, indeed to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius."
"We fashion our geometry on the properties of a straight line because that seems to us to be the simplest of all. But really all lines that are continuous and of a uniform nature are just as simple as one another. Another kind of mind which might form an equally clear mental perception of some property of any one of these curves, as we do of the congruence of a straight line, might believe these curves to be the simplest of all, and from that property of these curves build up the elements of a very different geometry, referring all other curves to that one, just as we compare them to a straight line. Indeed, these minds, if they noticed and formed an extremely clear perception of some property of, say, the parabola, would not seek, as our geometers do, to rectify the parabola, they would endeavor, if one may coin the expression, to parabolify the straight line."
CONTACT
E-mail: smithpith@yahoo.com
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Last edit: 2/6/2012.
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