NonNewtonianCalculus 2009-04-29 20:48:19 2010-07-21 16:57:09 Topic calculus non-Newtonian nonlinear multiplicative derivative integral average means exponential function power function geometric anageometric bigeometric root mean square quadratic anaquadratic biquadratic harmonic anaharmonic biharmonic % this is the default PlanetPhysics preamble. as your \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % define commands here \subsection{INTRODUCTION} Non-Newtonian calculus is a mathematical theory that provides scientists, engineers, and mathematicians with alternatives to the classical calculus of Isaac Newton and Gottfried Leibniz. [2,6,12,15,16] The non-Newtonian calculi were created in the period from 1967 to 1970 by Michael Grossman and Robert Katz. 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. However, the non-Newtonian calculi are markedly different from the classical calculus. For example, in the classical calculus, the derivative and integral are linear operators, that is, 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 operators: 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 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. \subsection{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 (Eastern Mediterranean University in Cyprus), together with Emine Misirli Kurpinar and Ali Ozyapici (Ege University in Turkey). Their work includes applications to differential equations, calculus of variations, and finite-difference methods. [2,24,27,33] The family of all non-Newtonian natural averages was used to construct a family of means of two positive numbers. 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. [8,14] 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]. 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] The geometric calculus and/or the bigeometric calculus may have application to dynamical systems, chaos theory, dimensional spaces, and fractal theory. [1,5,18,21] 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 System with a Scale-Free Derivative [10]. 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] Non-Newtonian 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 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 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: 1998 Proceedings of the Section on Bayesian Statistical Science. [40] Non-Newtonian Calculus was reviewed in the magazine Choice. [41] Non-Newtonian Calculus was reviewed by M. Dutta in the Indian Journal of History of Science. [42] The article "An introduction to non-Newtonian calculus" [12] was reviewed in the journal Zentralblatt fur Mathematik und Ihre Grenzgebiete. [43] Non-Newtonian Calculus was reviewed by Karel Berka in the journal Theory and Decision. [44] Non-Newtonian Calculus was reviewed by Otakar Zich in the journal Kybernetika. [45] Non-Newtonian Calculus was reviewed by David Preiss in the journal Aplikace Matematiky. [46] Non-Newtonian Calculus was reviewed in the journal Mathematical Reviews. [47] Non-Newtonian Calculus was reviewed in the journal The American Mathematical Monthly. [48] Bigeometric Calculus: A System with a Scale-Free Derivative [10] was reviewed in Mathematics Magazine. [49] Note. Several other reviews are indicated in the COMMENTS section below. Note. It’s natural to speculate about future applications of non-Newtonian calculus, 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. Note. Non-Newtonian calculus may have application in studies of growth, and in situations involving discontinuous phenomena. [34, 35] \subsection{CONSTRUCTION OF A NON-NEWTONIAN CALCULUS: AN OUTLINE} The construction of an arbitrary non-Newtonian calculus involves the real number system and an ordered pair $*$ of arbitrary complete ordered fields. [12,15] 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, $*$-continuity of $f$ at a, *-continuity of f on a closed interval, the *-derivative of f at a, the *-integral of a *-continuous function f on a closed interval, and the *-average of a *-continuous function f on a closed interval. (The *-average is the natural average of the $*$-calculus.) It turns out that the structure of the $*$-calculus is similar to that of the classical calculus. For example, there are two Fundamental Theorems of *-calculus, which show that the *-derivative and the *-integral are inversely related. And there is a special class of functions having a constant *-derivative. There are infinitely many *-calculi, and the classical calculus is one of them. Each of the others is called a "non-Newtonian calculus". \subsection{RELATIONSHIPS TO CLASSICAL CALCULUS} The $*$-derivative, $*$-average, and $*$-integral can be expressed in terms of their classical counterparts (and vice-versa). [15] Again, consider an arbitrary function f with arguments in A and values in B Let G and H be the ordered-field isomorphisms from R onto A and B, respectively. Let g and h be their respective inverses. Let D denote the classical derivative, and let D* denote the *-derivative. Finally, for each number t such that G(t) is in the domain of f, let F(t) = h(f(G(t))). Theorem 1. For each number a in A, [D*f](a) exists if and only if [DF](g(a)) exists, and if they do exist, then [D*f](a) = H([DF](g(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 g(r) to g(s), and M* = H(M), where M* is the *-average of f from r to s, and M is the classical (i.e., arithmetic) average of F from g(r) to g(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* = H(S), where S* is the *-integral of f from r to s, and S is the classical integral of F from g(r) to g(s). \subsection{EXAMPLES} Let I be the identity function on R. Let j be the function on R such that j(x) equals the square root of x for each nonnegative number x, and j(x) equals the negative of the square root of $-x$ for each negative number x. And let k be the function on R such that $k(0) = 0$ and $k(x) = 1/x$ for each nonzero number x. Example 1. In the case where G = I = H, the *-calculus is the classical calculus. Example 2. In the case where G = I and H = exp, the *-calculus is the geometric calculus. Example 3. In the case where G = exp = H, the *-calculus is the bigeometric calculus. Example 4. In the case where G =exp and H = I, the *-calculus is the so-called anageometric calculus. Example 5. In the case where G = I and H = j, the *-calculus is the quadratic calculus. Example 6. In the case where G = j = H, the *-calculus is the biquadratic calculus. Example 7. In the case where G = j and H = I, the *-calculus is the so-called anaquadratic calculus. Example 8. In the case where G = I and H = k, the *-calculus is the harmonic calculus. Example 9. In the case where G = k = H, the *-calculus is the biharmonic calculus. Example 10. In the case where G = k and H = I, the *-calculus is the so-called anaharmonic calculus. \subsection{HISTORY} The non-Newtonian calculi were created by Michael Grossman and Robert Katz. In August of 1970, they constructed a comprehensive family of calculi, which includes the classical calculus, the geometric calculus, the bigeometric calculus, and infinitely-many other calculi that they constructed in July of 1967. All of these calculi can be described simultaneously within the framework of a general theory. They decided to use the adjective "non-Newtonian" to indicate any of the calculi other than the classical calculus. In 1972, Grossman and Katz completed their book "Non-Newtonian Calculus" [15]. 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", "meta-calculus", averages, and means. [7 - 15, 34, 35] Note. Grossman and 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. In 2008, Michael Grossman found information suggesting that a multiplicative (perhaps non-Newtonian) integral or derivative might have been developed by Vito Volterra, who lived from 1860 to 1940. [1,5,17,18,21,22] 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 charge) at Google Book Search, and each of them can be read and/or downloaded (free of charge) at HathiTrust. \subsection{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 operation 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 Ingeniería, Volume 2, Number 3, 2006. [5] Felix R. Gantmacher. "The Theory of Matrices", 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, vol 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", 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 DNA 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", ISAAC Congress, 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. [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 Ciência, Volume 32, Issues 5-8, page 829, 1980. [40] American Statistical Association: 1998 Proceedings of the Section on Bayesian Statistical Science, page 176, 1998. [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 für Mathematik und Ihre Grenzgebiete, Akademie der Wissenschaften der DDR, Volumes 417-418, page 136, 1980. [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, Volume 55, #3180, February 1978. [48] The American Mathematical Monthly, Mathematical Association of America, May 1973. [49] Mathematics Magazine, Mathematical Association of America, Volume 57, Number 2, page 119, 1984. \subsection{ADDITIONAL READING} Robert Katz. "Axiomatic Analysis", D. C. Heath and Company, 1964. \subsection{LINK} Non-Newtonian Calculus website: http://sites.google.com/site/nonnewtoniancalculus/Home \subsection{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 ["Non-Newtonian Calculus" is] original and thought provoking. Professor Otakar Zich Charles University in Prague, Czech Republic 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 Note. The comments by Professors Struik, Arrow, and Meginniss are excerpts from their correspondence with Grossman, Grossman, and Katz. The comments by Professors Grattan-Guinness, Zich, Gollmann, and MacAdam are excerpts from their reviews of the book "Non-Newtonian Calculus" in Middlesex Math Notes (1977), Kybernetika (1973), 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). \subsection{CONTACT} Name: Michael Grossman E-mail: smithpith@yahoo.com