Multiplicative calculus

Multiplicative calculus

In mathematics, multiplicative calculus refers to a number of calculi whose derivative and integral are multiplicative as compared to the classical (or conventional) calculus which is additive and linear. Different examples are given below. (Accordingly, the expression "the" multiplicative calculus" should be avoided.)

Multiplicative derivatives

Geometric calculus

The classical derivative is

:f'(x) = lim_{h o 0}{f(x+h) - f(x)over{h

The geometric derivative is

:f^{*}(x) = lim_{h o 0}{ left({f(x+h)over{f(x) ight)^{1over{h }

This simplifies [http://coco-m.web.lynchburg.edu/multiplicative%20calculus.pdf Multiplicative Calculus] ] Verify credibility|date=October 2008 to

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

for functions where the statement is meaningful.

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

Agamirza E. Bashirov (Eastern Mediterranean University in Cyprus), together with Emine Misirli Kurpinar and Ali Ozyapici (Ege University in Turkey), applied the geometric calculus to differential equations and calculus of variations.Bashirov, Misirli Kurpinar, and Ozyapici. [http://linkinghub.elsevier.com/retrieve/pii/S0022247X07003824 "Multiplicative calculus and its applications"] , Journal of Mathematical Analysis and Applications, 2008.]

Bigeometric calculus

A similar definition to the geometric derivative is the bigeometric derivative

:f^{*}(x) = lim_{h o 0}{ left({f((1+h)x)over{f(x) ight)^{1over{h } = lim_{k o 1}{ left({f(kx)over{f(x) ight)^{1over{ln(k) }

which simplifies to

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

for functions where the statement is meaningful.

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

Fernando Córdova-Lepe (Universidad Católica del Maule in Chile) applied the bigeometric derivative to the theory of elasticity in economics. [Fernando Córdova-Lepe. "From quotient operation toward a proportional calculus", Journal of Mathematics, Game Theory and Algebra, 2004.] Fernando Córdova-Lepe. [http://www.tmat.cl/articulocordova.html "The Multiplicative Derivative as a Measure of Elasticity in Economics"] .] Verify credibility|date=October 2008. (He refers to the bigeometric derivative as "the multiplicative derivative.")

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.

Fractals and Chaos

This paragraph describes the findings of Marek Rybaczuk, Alicja Kedzia and Witold Zielinski. They say that in dimensional spaces (in a similar way to physical quantities) you can multiply and divide quantities which have different dimensions but you cannot add and subtract quantities with different dimensions. This means that the classical additive derivative is undefined because the difference f(x+deltax)-f(x) has no value. However in dimensional spaces the multiplicative derivatives mentioned below remain well-defined. Multiplicative dynamical systems can become chaotic even when the corresponding classical additive system does not because the additive and multiplicative derivatives become inequivalent if the variables involved also have a varying fractal dimension. Marek Rybaczuk, Alicja Kedzia and Witold Zielinski (2001) [http://linkinghub.elsevier.com/retrieve/pii/S0960077900002319 The concept of physical and fractal dimension II. The differential calculus in dimensional spaces] , "Chaos, Solitons, & Fractals"Volume 12, Issue 13, October 2001, Pages 2537-2552 ] M. Rybaczuk and P. Stoppel (2000) [http://www.springerlink.com/index/R06853J7R1507T55.pdf "The fractal growth of fatigue defects in materials"] , International Journal of Fracture, Volume 103, Number 1 / May, 2000 ] Dorota Aniszewska (2007) [http://www.springerlink.com/index/TJ7T5G1623480442.pdf "Multiplicative Runge–Kutta methods"] , Nonlinear Dynamics, Volume 50, Numbers 1-2 / October, 2007] Dorota Aniszewska and Marek Rybaczuk (2005) [http://linkinghub.elsevier.com/retrieve/pii/S0960077904006319 Analysis of the multiplicative Lorenz system ] , "Chaos, Solitons & Fractals"Volume 25, Issue 1, July 2005, Pages 79-90] Dorota Aniszewska and Marek Rybaczuk (2008) [http://www.springerlink.com/content/w712343n31305787/ Lyapunov type stability and Lyapunov exponent for exemplary multiplicative dynamical systems ] , "Nonlinear Dynamics", Published online: 25 January 2008 ]

Only a few authors have used these multiplicative derivatives, whereas the similar logarithmic derivative is widely used.

Multiplicative Integrals

Just like the usual calculus each multiplicative derivative has an associated integral.

Multiplicative integrals (usually known as product integrals) are applied to matrix-valued functions and various other noncommutative spaces. The noncommutativity of the multiplication operation means that trying to express the multiplicative integral in terms of the usual additive integral fails.Carl M. Pearcy (1974) "Topics in Operator Theory", AMS Bookstore, ISBN 9780821815137]

History

(This history does not at present describe the large number of publications about product integrals.)

In 1887, Vito Volterra proposed a "multiplicative calculus" starting with a multiplicative integral which could also be defined in the complex plane.F.R. Gantmacher (1959) "The Theory of Matrices", volumes 1 and 2.] This was later reformulated in a simpler manner with a multiplicative derivative as a starting point..

In 1901, Robert Edouard Moritz published "Quotientiation, an extension of the differentiation process," Robert Edouard Moritz, [http://www.emis.de/cgi-bin/JFM-item?33.0303.01 Quotientiation, an extension of the differentiation process] , Proceedings of the Nebraska Academy of Sciences (1901), 112-117, [JFM 33.0303.01] . ] which defines the operator qy/qx =lim k->1 { logk (f(kx)/f(x))}.

Between 1972 and 1983, Michael Grossman, Robert Katz, and Jane Grossman produced a number of publications on Non-Newtonian calculus and related matters that describe, among other things, an infinite family of calculi discovered by the former two between 1967 and 1970. The family includes the classical calculus, the geometric calculus and the bigeometric calculus. These authors also give a guide to when it may be preferable to use one particular type of calculus. In 1999, the geometric calculus is rediscovered by Dick Stanley who published the paper "A multiplicative calculus"Dick Stanley (1999) [http://www.informaworld.com/smpp/title~content=g781286889~db=all "A multiplicative calculus"] , Primus vol 9, issue 4.] in the journal Primus. The same issue also contains a paper by Duff Campbell: "Multiplicative calculus and student projects".Duff Campbell (1999). [http://www.informaworld.com/smpp/title~content=g781286889~db=all "Multiplicative calculus and student projects"] , Primus vol 9, issue 4. ]

In 2004, a multiplicative derivative (the bigeometric derivative) is defined by F.Córdova-Lepe who calls it "proportional calculus".Verify credibility|date=October 2008

Discrete calculus

Just as differential equations have a discrete analog in difference equations with the forward difference operator replacing the derivative, so too there is the discrete multiplicative derivative f(x+1)/f(x) and recurrence relations can be formulated using this operator. [M. Jahanshahi, N. Aliev and H. R. Khatami (2004). [http://faculty.uaeu.ac.ae/hakca/papers/jahanshahi.pdf "An analytic-numerical method for solving difference equations with variable coefficients by discrete multiplicative integration".] ] Verify credibility|date=October 2008 [H. R. Khatami & M. Jahanshahi & N. Aliev (2004). [http://faculty.uaeu.ac.ae/hakca/papers/khatami.pdf "An analytical method for some nonlinear difference equations by discrete multiplicative differentiation".] ] Verify credibility|date=October 2008 [N. Aliev, N. Azizi and M. Jahanshahi (2007) [http://www.m-hikari.com/imf-password2007/9-12-2007/jahanshahiIMF9-12-2007-1.pdf "Invariant functions for discrete derivatives and their applications to solve non-homogenous linear and non-linear difference equations".] ] Verify credibility|date=October 2008

References


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