non newtonian calculus
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Non-Newtonian calculusFrom Wikipedia, the free encyclopedia
Contents
1 Indefinite product 11.1 Period rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Connection to indefinite sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Alternative usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 List of indefinite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 List of derivatives and integrals in alternative calculi 42.1 Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Multiplicative calculus 63.1 Multiplicative derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.1.1 Geometric calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1.2 Bigeometric calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Multiplicative integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Discrete calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.5 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.6 General theory of non-Newtonian calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.6.1 Construction: an outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.6.2 Relationships to classical calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.6.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.7 Reception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Product integral 20
i
ii CONTENTS
4.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.1.1 Type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.1.2 Type II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.6 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Chapter 1
Indefinite product
In mathematics, the indefinite product operator is the inverse operator of Q(f(x)) = f(x+1)f(x) . It is like a discrete
version of the indefinite product integral. Some authors use term discrete multiplicative integration[1]
Thus
Q(∏x
f(x)) = f(x) .
More explicitly, if∏x f(x) = F (x) , then
F (x+ 1)
F (x)= f(x) .
If F(x) is a solution of this functional equation for a given f(x), then so is CF(x) for any constant C. Therefore eachindefinite product actually represents a family of functions, differing by a multiplicative constant.
1.1 Period rule
If T is a period of function f(x) then
∏x
f(Tx) = Cf(Tx)x−1
1.2 Connection to indefinite sum
Indefinite product can be expressed in terms of indefinite sum:
∏x
f(x) = exp(∑
x
ln f(x))
1.3 Alternative usage
Some authors use the phrase “indefinite product” in a slightly different but related way to describe a product in whichthe numerical value of the upper limit is not given.[2] e.g.
1
2 CHAPTER 1. INDEFINITE PRODUCT
n∏k=1
f(k)
1.4 Rules∏x
f(x)g(x) =∏x
f(x)∏x
g(x)
∏x
f(x)a =
(∏x
f(x)
)a∏x
af(x) = a∑
x f(x)
1.5 List of indefinite products
This is a list of indefinite products∏x f(x) . Not all functions have an indefinite product which can be expressed in
elementary functions.
∏x
a = Cax
∏x
x = C Γ(x)
∏x
x+ 1
x= Cx
∏x
x+ a
x=C Γ(x+ a)
Γ(x)∏x
xa = C Γ(x)a
∏x
ax = CaxΓ(x)
∏x
ax = Cax2 (x−1)
∏x
a1x = Ca
Γ′(x)Γ(x)
∏x
xx = C eζ′(−1,x)−ζ′(−1) = C eψ
(−2)(z)+ z2−z2 − z
2 ln(2π) = C K(x)
(see K-function)
∏x
Γ(x) =C Γ(x)x−1
K(x) = C Γ(x)x−1ez2 ln(2π)− z2−z
2 −ψ(−2)(z) = C G(x)
(see Barnes G-function)
∏x
sexpa(x) =C (sexpa(x))′sexpa(x)(ln a)x
1.6. SEE ALSO 3
(see super-exponential function)∏x
x+ a = C Γ(x+ a)
∏x
ax+ b = C axΓ
(x+
b
a
)∏x
ax2 + bx = C axΓ(x)Γ
(x+
b
a
)∏x
x2 + 1 = C Γ(x− i)Γ(x+ i)
∏x
x+1
x=C Γ(x− i)Γ(x+ i)
Γ(x)∏x
cscx sin(x+ 1) = C sinx
∏x
secx cos(x+ 1) = C cosx
∏x
cotx tan(x+ 1) = C tanx
∏x
tanx cot(x+ 1) = C cotx
1.6 See also• Indefinite sum
• Product integral
• List of derivatives and integrals in alternative calculi
1.7 References[1] N. Aliev, N. Azizi and M. Jahanshahi (2007) “Invariant functions for discrete derivatives and their applications to solve
non-homogenous linear and non-linear difference equations”.
[2] Algorithms for Nonlinear Higher Order Difference Equations, Manuel Kauers
1.8 Further reading• http://reference.wolfram.com/mathematica/ref/Product.html -Indefinite products with Mathematica
• http://www.math.rwth-aachen.de/MapleAnswers/660.html - bug in Maple V to Maple 8 handling of indefiniteproduct
• Markus Müller. How to Add a Non-Integer Number of Terms, and How to Produce Unusual Infinite Summa-tions
• Markus Mueller, Dierk Schleicher. Fractional Sums and Euler-like Identities
Chapter 2
List of derivatives and integrals inalternative calculi
There are many alternatives to the classical calculus of Newton and Leibniz; for example, each of the infinitely manynon-Newtonian calculi.[1] Occasionally an alternative calculus is more suited than the classical calculus for expressinga given scientific or mathematical idea.[2][3][4]
The table below is intended to assist people working with the alternative calculus called the “geometric calculus” (orits discrete analog). Interested readers are encouraged to improve the table by inserting citations for verification, andby inserting more functions and more calculi.
2.1 Table
In the following table ψ(x) = Γ′(x)Γ(x) is the digamma function,K(x) = eζ
′(−1,x)−ζ′(−1) = ez−z2
2 + z2 ln(2π)−ψ(−2)(z)
is the K-function, (!x) = Γ(x+1,−1)e is subfactorial, Ba(x) = −aζ(−a+ 1, x) are the generalized to real numbers
Bernoulli polynomials.
2.2 See also
• Indefinite product
• Multiplicative calculus
• Product integral
2.3 References[1] M. Grossman and R. Katz, Non-Newtonian Calculus, ISBN 0-912938-01-3, Lee Press, 1972.
[2] Agamirza E. Bashirov, Emine Misirli Kurpinar, and Ali Ozyapici. “Multiplicative calculus and its applications”, Journalof Mathematical Analysis and Applications, 2008.
[3] Diana Andrada Filip and Cyrille Piatecki. “A non-Newtonian examination of the theory of exogenous economic growth”,CNCSIS – UEFISCSU(project number PNII IDEI 2366/2008) and LEO, 2010.
[4] Luc Florack and Hans van Assen.“Multiplicative calculus in biomedical image analysis”, Journal of Mathematical Imagingand Vision, DOI: 10.1007/s10851-011-0275-1, 2011.
[5] H. R. Khatami & M. Jahanshahi & N. Aliev (2004). “An analytical method for some nonlinear difference equationsby discrete multiplicative differentiation”., 5—10 July 2004, Antalya, Turkey – Dynamical Systems and Applications,Proceedings, pp. 455—462
4
2.3. REFERENCES 5
[6] M. Jahanshahi, N. Aliev and H. R. Khatami (2004). “An analytic-numerical method for solving difference equations withvariable coefficients by discrete multiplicative integration”., 5—10 July 2004, Antalya, Turkey – Dynamical Systems andApplications, Proceedings, pp. 425—435
Chapter 3
Multiplicative calculus
In mathematics, a multiplicative calculus is a system with two multiplicative operators, called a “multiplicativederivative” and a “multiplicative integral”, which are inversely related in amanner analogous to the inverse relationshipbetween the derivative and integral in the classical calculus of Newton and Leibniz. The multiplicative calculi providealternatives to the classical calculus, which has an additive derivative and an additive integral.There are infinitely many non-Newtonian multiplicative calculi, including the geometric calculus and the bigeometriccalculus discussed below.[1] These calculi all have a derivative and/or integral that is not a linear operator.The geometric calculus is useful in biomedical image analysis.[2][3][4][5]
3.1 Multiplicative derivatives
3.1.1 Geometric calculus
The classical derivative is
f ′(x) = limh→0
f(x+ h)− f(x)
h
The geometric derivative is
f∗(x) = limh→0
(f(x+ h)
f(x)
) 1h
(For the geometric derivative, it is assumed that all values of f are positive numbers.)This simplifies[6] to
f∗(x) = ef′(x)f(x)
for functions where the statement is meaningful. Notice that the exponent in the preceding expression represents thewell-known logarithmic derivative.In the geometric calculus, the exponential functions are the functions having a constant derivative.[1] Furthermore,just as the arithmetic average (of functions) is the 'natural' average in the classical calculus, the well-known geometricaverage is the 'natural' average in the geometric calculus.[1]
3.1.2 Bigeometric calculus
A similar definition to the geometric derivative is the bigeometric derivative
6
3.2. MULTIPLICATIVE INTEGRALS 7
f∗(x) = limh→0
(f((1 + h)x)
f(x)
) 1h
= limk→1
(f(kx)
f(x)
) 1ln(k)
(For the bigeometric derivative, it is assumed that all arguments and all values of f are positive numbers.)This simplifies[7] to
f∗(x) = exf′(x)f(x) .
for functions where the statement is meaningful. Notice that the exponent in the preceding expression represents thewell-known elasticity concept, which is widely used in economics.In the bigeometric calculus, the power functions are the functions having a constant derivative.[1] Furthermore, thebigeometric derivative is scale invariant (or scale free), i.e., it is invariant under all changes of scale (or unit) infunction arguments and values.
3.2 Multiplicative integrals
Each multiplicative derivative has an associated multiplicative integral. For example, the geometric derivative andthe bigeometric derivative are inversely-related to the geometric integral and the bigeometric integral, respectively.Of course, each multiplicative integral is a multiplicative operator, but some product integrals are not multiplicativeoperators. (See Product integral#Basic definitions.)
3.3 Discrete calculus
Just as differential equations have a discrete analog in difference equations with the forward difference operatorreplacing the derivative, so too there is the forward ratio operator f(x + 1)/f(x) and recurrence relations can beformulated using this operator.[8][9][10] See also Indefinite product.
3.4 Complex analysis• Multiplicative versions of derivatives and integrals from complex analysis behave quite differently from theusual operators.[11][12][13][14][15]
3.5 History
Between 1967 and 1988, Jane Grossman, Michael Grossman, and Robert Katz produced a number of publicationson a subject created in 1967 by the latter two, called “non-Newtonian calculus.” The geometric calculus[16] and thebigeometric calculus[17] are among the infinitely many non-Newtonian calculi that are multiplicative.[1] (Infinitelymany non-Newtonian calculi are not multiplicative.)In 1972, Michael Grossman and Robert Katz completed their book Non-Newtonian Calculus. It includes discussionsof nine specific non-Newtonian calculi, the general theory of non-Newtonian calculus, and heuristic guides for appli-cation. Subsequently, with Jane Grossman, they wrote several other books/articles on non-Newtonian calculus, andon related matters such as “weighted calculus”,[18] “meta-calculus”,[19] and averages/means.[20][21]
On page 82 of Non-Newtonian Calculus, published in 1972, Michael Grossman and Robert Katz wrote:
“However, since we have nowhere seen a discussion of even one specific non-Newtonian calculus, andsince we have not found a notion that encompasses the *-average, we are inclined to the view that thenon-Newtonian calculi have not been known and recognized heretofore. But only the mathematicalcommunity can decide that.”
8 CHAPTER 3. MULTIPLICATIVE CALCULUS
3.6 General theory of non-Newtonian calculus
(This section is based on six sources.[1][2][6][22][23][24])
3.6.1 Construction: an outline
The construction of an arbitrary non-Newtonian calculus involves the real number system and an ordered pair * ofarbitrary complete ordered fields.Let R denote the set of all real numbers, and let A and B denote the respective realms of the two arbitrary completeordered fields.Assume that both A and B are subsets of R. (However, we are not assuming that the two arbitrary complete orderedfields are subfields of the real number system.) Consider an arbitrary function f with arguments in A and values in B.By using the natural operations, natural orderings, and natural topologies for A and B, one can define the following(and other) concepts of the *-calculus: the *-limit of f at an argument a, f is *-continuous at a, f is *-continuous ona closed interval, the *-derivative of f at a, the *-average of a *-continuous function f on a closed interval, and the*-integral of a *-continuous function f on a closed interval.Many, if not most, *-calculi are markedly different from the classical calculus, but the structure of each *-calculus issimilar to that of the classical calculus. For example, each *-calculus has two Fundamental Theorems showing thatthe *-derivative and the *-integral are inversely related; and for each *-calculus, there is a special class of functionshaving a constant *-derivative. Furthermore, the classical calculus is one of the infinitely many *-calculi.A non-Newtonian calculus is defined to be any *-calculus other than the classical calculus.
3.6.2 Relationships to classical calculus
The *-derivative, *-average, and *-integral can be expressed in terms of their classical counterparts (and vice versa).(However, as indicated in the Reception-section below, there are situations in which a specific non-Newtonian calculusmay be more suitable than the classical calculus.[2][6][23][24][25][26])Again, consider an arbitrary function f with arguments in A and values in B. Let α and β be the ordered-field isomor-phisms from R onto A and B, respectively. Let α−1 and β−1 be their respective inverses.Let D denote the classical derivative, and let D* denote the *-derivative. Finally, for each number t such that α(t) isin the domain of f, let F(t) = β−1(f(α(t))).Theorem 1. For each number a in A, [D*f](a) exists if and only if [DF](α−1(a)) exists, and if they do exist, then[D*f](a) = β([DF](α−1(a))).Theorem 2. Assume f is *-continuous on a closed interval (contained in A) from r to s, where r and s are in A. ThenF is classically continuous on the closed interval (contained in R) from α−1(r) to α−1(s), and M* = β(M), where M*is the *-average of f from r to s, and M is the classical (i.e., arithmetic) average of F from α−1(r) to α−1(s).Theorem 3. Assume f is *-continuous on a closed interval (contained in A) from r to s, where r and s are in A. ThenS* = β(S), where S* is the *-integral of f from r to s, and S is the classical integral of F from α−1(r) to α−1(s).
3.6.3 Examples
Let I be the identity function on R. Let j be the function on R such that j(x) = 1/x for each nonzero number x, andj(0) = 0. And let k be the function on R such that k(x) = √x for each nonnegative number x, and k(x) = -√(-x) foreach negative number x.Example 1. If α = I = β, then the *-calculus is the classical calculus.Example 2. If α = I and β = exp, then the *-calculus is the geometric calculus.Example 3. If α = exp = β, then the *-calculus is the bigeometric calculus.Example 4. If α = exp and β = I, then the *-calculus is the so-called anageometric calculus.Example 5. If α = I and β = j, then the *-calculus is the so-called harmonic calculus.
3.7. RECEPTION 9
Example 6. If α = j = β, then the *-calculus is the so-called biharmonic calculus.Example 7. If α = j and β = I, then the *-calculus is the so-called anaharmonic calculus.Example 8. If α = I and β = k, then the *-calculus is the so-called quadratic calculus.Example 9. If α = k = β, then the *-calculus is the so-called biquadratic calculus.Example 10. If α = k and β = I, then the *-calculus is the so-called anaquadratic calculus.
3.7 Reception• The First Nonlinear System of Differential And Integral Calculus,[16] a book about the geometric calculus, wasreviewed in Mathematical Reviews in 1980 by Ralph P. Boas, Jr. He included the following assertion: “It isnot yet clear whether the new calculus [geometric calculus] provides enough additional insight to justify its useon a large scale”.
• Bigeometric Calculus: A System with a Scale-Free Derivative[17] was reviewed in Mathematical Reviews in 1984by Ralph P. Boas, Jr. He included the following assertion: “It seems plausible that people who need to studyfunctions from this point of view might well be able to formulate problems more clearly by using bigeometriccalculus instead of classical calculus”.
• Non-Newtonian Calculus,[1] a book including detailed discussions about the geometric calculus and the bige-ometric calculus (both of which are non-Newtonian calculi), was reviewed by David Pearce MacAdam in theJournal of the Optical Society of America.[27] He included the following assertion: “The greatest value ofthese non-Newtonian calculi may prove to be their ability to yield simpler physical laws than the Newtoniancalculus.”
• Non-Newtonian Calculus, a book including detailed discussions about the geometric calculus and the bigeomet-ric calculus (both of which are non-Newtonian calculi), was reviewed by H. Gollmann (Graz, Austria) in thejournal Internationale Mathematische Nachrichten.[28] He included the following assertion: “The possibilitiesopened up by the new [non-Newtonian] calculi seem to be immense.” (German: “Die durch die neuen Kalkuleerschlossenen Möglichkeiten scheinen unermesslich.”)
• Non-Newtonian Calculus, a book including detailed discussions about the geometric calculus and the bigeomet-ric calculus (both of which are non-Newtonian calculi), was reviewed by Ivor Grattan-Guinness in MiddlesexMath Notes.[29] He included the following assertions: “There is enough here [in Non-Newtonian Calculus]to indicate that non-Newtonian calculi ... have considerable potential as alternative approaches to traditionalproblems. This very original piece of mathematics will surely expose a number of missed opportunities in thehistory of the subject.”
• Non-Newtonian calculus was used by James R. Meginniss (Claremont Graduate School and Harvey MuddCollege) to create a theory of probability that is adapted to human behavior and decision making.[23]
• Seminars concerning non-Newtonian calculus and the dynamics of random fractal structures were conducted byWojbor Woycznski (Case Western Reserve University) at The Ohio State University[25] on 22 April 2011, andat Cleveland State University[30] on 2 May 2012. In the abstracts for the seminars he asserted: “Many naturalphenomena, from microscopic bacteria growth, through macroscopic turbulence, to the large scale structure ofthe Universe, display a fractal character. For studying the time evolution of such “rough” objects, the classical,“smooth” Newtonian calculus is not enough.”
• A seminar concerning fractional calculus, random fractals, and non-Newtonian calculus was conducted byWo-jbor Woycznski (Case Western Reserve University) at Case Western Reserve University on 3 April 2013.[31]In the abstract for the seminar he asserted: “Random fractals, a quintessentially 20th century idea, arise asnatural models of various physical, biological (think your mother’s favorite cauliflower dish), and economic(think Wall Street, or the Horseshoe Casino) phenomena, and they can be characterized in terms of the math-ematical concept of fractional dimension. Surprisingly, their time evolution can be analyzed by employing anon-Newtonian calculus utilizing integration and differentiation of fractional order.”
10 CHAPTER 3. MULTIPLICATIVE CALCULUS
• The geometric calculus was used by Agamirza E. Bashirov (Eastern Mediterranean University in Cyprus),together with Emine Misirli Kurpinar and Ali Ozyapici (both of Ege University in Turkey), in an article ondifferential equations and calculus of variations.[6] In that article, they state: “We think that multiplicativecalculus can especially be useful as a mathematical tool for economics and finance ... In the present paper ouraim is to bring multiplicative calculus to the attention of researchers ... and to demonstrate its usefulness.” (The“multiplicative calculus” referred to here is the geometric calculus.)
• The geometric calculus was used by Agamirza E. Bashirov, Emine Misirli, Yucel Tandogdu, and Ali Ozyapiciin an article on modelling with multiplicative differential equations.[26] In that article they state: “In this studyit becomes evident that the multiplicative calculus methodology has some advantages over additive calculus inmodeling some processes in areas such as actuarial science, finance, economics, biology, demographics, etc.”(The “multiplicative calculus” referred to here is the geometric calculus.)
• The geometric calculus was used by Diana Andrada Filip (Babes-Bolyai University of Cluj-Napoca in Roma-nia) and Cyrille Piatecki (Orléans University in France) to re-postulate and analyse the neoclassical exogenousgrowth model in economics.[24] In that article they state: “In this paper, we have tried to present how a non-Newtonian calculus could be applied to repostulate and analyse the neoclassical [Solow-Swan] exogenousgrowth model [in economics]. ... In fact, one must acknowledge that it’s only under the effort of Gross-man & Katz (1972)[1] ... that such a non-Newtonian calculus emerged to give a natural answer to many growthphenomena. ... We must underscore that to discover that there was a non-Newtonian way to look to differentialequations has been a great surprise for us. It opens the question to know if there are major fields of economicanalysis which can be profoundly re-thought in the light of this discovery.”
• A discussion concerning the advantages of using the geometric calculus in economic analysis is presented inan article by Diana Andrada Filip (Babes-Bolyai University of Cluj-Napoca in Romania) and Cyrille Piatecki(Orléans University in France).[32] In that article they state: “The double entry bookkeeping promoted by LucaPacioli in the fifteenth century could be considered a strong argument in behalf of the multiplicative calculus,which can be developed from the Grossman and Katz non-Newtonian calculus concept.” (The “multiplicativecalculus” referred to here is the geometric calculus.)
• The geometric calculus was used by Luc Florack and Hans van Assen (both of the Eindhoven Universityof Technology) in the study of biomedical image analysis.[2][3][4] In their article “Multiplicative calculus inbiomedical image analysis” they state: “We advocate the use of an alternative calculus in biomedical imageanalysis, known as multiplicative (a.k.a. non-Newtonian) calculus. ... The purpose of this article is to providea condensed review of multiplicative calculus and to illustrate its potential use in biomedical image analysis”(The “multiplicative calculus” referred to here is the geometric calculus.) In Professor Florack’s article “Regu-larization of positive definite matrix fields based on multiplicative calculus” he states: “Multiplicative calculusprovides a natural framework in problems involving positive images and positivity preserving operators. In in-creasingly important, complex imaging frameworks, such as diffusion tensor imaging, it complements standardcalculus in a nontrivial way. The purpose of this article is to illustrate the basics of multiplicative calculus andits application to the regularization of positive definite matrix fields.” (The “multiplicative calculus” referredto here is the geometric calculus.)
• The geometric calculus and the bigeometric calculus were among the topics covered in a course on non-Newtonian calculus conducted in the summer-term of 2012 by Joachim Weickert, Laurent Hoeltgen, andother faculty from the Mathematical Image Analysis Group of Saarland University in Germany. Among theother topics covered were applications to digital image processing, rates of return, and growth processes.[5]
• A multiplicative calculus was used in the study of contour detection in images with multiplicative noise byMarcoMora, Fernando Córdova-Lepe, and Rodrigo Del-Valle (all of Universidad Católica del Maule in Chile).In that article they state: “This work presents a new operator of non-Newtonian type which [has] shown [to] bemore efficient in contour detection [in images with multiplicative noise] than the traditional operators. ... In ourview, the work proposed in (Grossman and Katz, 1972) stands as a foundation, for its clarity of purpose.”[33]
• The geometric calculus was used by Emine Misirli and Yusuf Gurefe (both of Ege University in Turkey)in their lecture “The new numerical algorithms for solving multiplicative differential equations”.[34] In thatpresentation they stated: “While one problem can be easily expressed using one calculus, the same problemcan not be expressed as easily [using another].”
3.7. RECEPTION 11
• The bigeometric derivative was used to reformulate the Volterra product integral.[35] (Please see Product inte-gral#Basic definitions.)
• The geometric calculus and the bigeometric calculus were used by Mustafa Riza (Eastern Mediterranean Uni-versity in Cyprus), together with Ali Ozyapici and Emine Misirli (both of Ege University in Turkey), in anarticle on differential equations and finite difference methods.[36]
• A multiplicative type of calculus for complex-valued functions of a complex variable was developed and usedby Ali Uzer (Fatih University in Turkey).[11][12]
• Complex multiplicative calculus was developed by Agamirza E. Bashirov and Mustafa Riza (both of EasternMediterranean University in Cyprus).[13][14][15]
• The geometric calculus was used by Agamirza E. Bashirov (Eastern Mediterranean University in Cyprus) inan article on line integrals and double multiplicative integrals.[37]
• The geometric calculus was used by Emine Misirli and Yusuf Gurefe (both of Ege University in Turkey) in anarticle on the numerical solution of multiplicative differential equations.[38]
• The geometric calculus was used by James D. Englehardt (University of Miami) and Ruochen Li (Shenzhen,China) in an article on pathogen counts in treated water.[39]
• Weighted non-Newtonian calculus[18] is cited by Ziyue Liu and Wensheng Guo (both of the University ofPennsylvania) in their Supplement to their article “Data driven adaptive spline smoothing".[40]
• Weighted geometric calculus[18] was used by David Baqaee (Harvard University) in an article on an axiomaticfoundation for intertemporal decision making.[41]
• Weighted non-Newtonian calculus[18] is cited by P. Arun Raj Kumar and S. Selvakumar (both of the NationalInstitute of Technology, Tiruchirappalli in India) in their article “Detection of distributed denial of serviceattacks using an ensemble of adaptive and hybrid neuro-fuzzy systems”.[42]
• Weighted non-Newtonian calculus[18] is cited byRiswan Efendi and Zuhaimy Ismail (both ofUniversiti TeknologiMalaysia) together with Mustafa Mat Deris (Universiti Tun Hussein Onn Malaysia) in their article “Improvedweight fuzzy time series as used in the exchange rates forecasting of US dollar to ringgit Malaysia”.[43]
• Weighted non-Newtonian calculus[18] is cited by Jie Zhang, Li Li, Luying Peng, Yingxian Sun, Jue Li (the firstfour from Tongji University School of Medicine in Shanghai, China; and the latter from The First Hospital ofChina Medical University, Shenyang, China) in their article “An Efficient Weighted Graph Strategy to IdentifyDifferentiation Associated Genes in Embryonic Stem Cells”.[44]
• Weighted non-Newtonian calculus[18] is cited by ZHENG Xu and LI Jian-Zhong (both of the School of Com-puter Science and Technology, Harbin Institute of Technology, Harbin, China) in their article “Approximateaggregation algorithm for weighted data in wireless sensor networks”.[45]
• The bigeometric calculus was used in an article on multiplicative differential equations by Dorota Aniszewska(Wroclaw University of Technology).[35]
• The bigeometric calculus was used in an article on chaos in multiplicative dynamical systems by DorotaAniszewska and Marek Rybaczuk (both from the Wroclaw University of Technology in Poland).[46]
• The bigeometric calculus was used in an article on multiplicative Lorenz systems by Dorota Aniszewska andMarek Rybaczuk (both from Wroclaw University of Technology).[47]
• The bigeometric calculus was used in an article on multiplicative dynamical systems by Dorota Aniszewskaand Marek Rybaczuk (both from Wroclaw University of Technology).[48]
12 CHAPTER 3. MULTIPLICATIVE CALCULUS
• The bigeometric calculus was used in an article on fractals and material science byM. Rybaczuk and P. Stoppel(both from Wroclaw University of Technology).[49]
• The bigeometric calculus was used in an article on fractal dimension and dimensional spaces by Marek Ry-baczuka (Wroclaw University of Technology in Poland), Alicja Kedziab (Medical Academy of Wroclaw inPoland), and Witold Zielinskia (Wroclaw University of Technology).[50]
• The geometric calculus and the bigeometric calculus are useful in the study of dimensional spaces. In dimensionalspaces (in a similar way to physical quantities) you can multiply and divide quantities which have different di-mensions but you cannot add and subtract quantities with different dimensions. This means that the classicaladditive derivative is undefined because the difference f(x+deltax)-f(x) has no value. However in dimensionalspaces, the geometric derivative and the bigeometric derivative remain well-defined. Multiplicative dynamicalsystems can become chaotic even when the corresponding classical additive system does not because the ad-ditive and multiplicative derivatives become inequivalent if the variables involved also have a varying fractaldimension.[7][35][47][48][51]
• The geometric calculus was used by S. L. Blyumin (Lipetsk State Technical University in Russia) in an articleon information technology.[52]
• The bigeometric derivative was used by Fernando Córdova-Lepe (Universidad Católica del Maule in Chile) inan article on the theory of elasticity in economics.[53]
• The geometric calculus was applied to functional analysis by Cengiz Türkmen and Feyzi Başar (both from FatihUniversity in Turkey).[54]
• The mathematics department of Eastern Mediterranean University in Cyprus has established a research groupfor the purpose of studying and applying multiplicative calculus.[55]
• The bigeometric calculus was used by Ahmet Faruk Çakmak in his lecture at the 2011 International Conferenceon Applied Analysis and Algebra at Yıldız Technical University in Istanbul, Turkey.[56]
• The geometric calculus was used by Gunnar Sparr sv:Gunnar Sparr (Lund Institute of Technology, in Sweden)in an article on computer vision.[57] (The “multiplicative derivative” referred to in the article is the geometricderivative.)
• The geometric calculus was used by Uğur Kadak (Gazi University in Turkey) and Yusef Gurefe (Bozok Uni-versity in Turkey) in their presentation at the 2012 Analysis and Applied Mathematics Seminar Series of FatihUniversity in Istanbul, Turkey.[58]
• The geometric integral is useful in stochastics. (See Product integral#Basic definitions.)
• The geometric calculus was used by Jarno van Roosmalen (Eindhoven University of Technology in the Nether-lands) in an article on statistics and data analysis .[59]
• The geometric calculus is cited by Manfred Peschel and Werner Mende (both of the German Academy ofSciences Berlin) in a book on the phenomena of growth and structure-building.[60]
• The geometric calculus is the subject of an article by Dick Stanley in the journal Primus.[61] The same issue ofPrimus contains a paper by Duff Campbell: “Multiplicative calculus and student projects”.[62]
• The geometric calculus was the subject of a seminar by Michael Coco of Lynchburg College.[63]
• The geometric calculus is the subject of an article by Michael E. Spivey of the University of Puget Sound.[64]
• The geometric calculus is the subject of an article by Alex B. Twist and Michael E. Spivey of the University ofPuget Sound.[65]
3.7. RECEPTION 13
• In 2008, the article “Multiplicative calculus and its applications”,[6] concerning applications of the geometriccalculus, was published in the Journal of Mathematical Analysis and Applications. The article was submittedby Steven G. Krantz and written by Agamirza E. Bashirov, Emine Misirli Kurpinar, and Ali Ozyapici. Thefollowing is an excerpt from a review[66] of that article by Gerard Lebourg:
“What happens to the old calculus when you restrict its application to positive functions and replace thedifferential ratio [f(x+h)−f(x)]
h with the multiplicative one[f(x+h)f(x)
]1/h? Answer: the usual derivative
f ′(x) is replaced with f⋆(x) = [exp(ln[f(x)])]′ . So you are left with some avatar of the classicalcalculus to unfold. The authors of this original paper do play this game. Their stated purpose is topromote this new kind of multiplicative calculus.” (Note that f⋆(x) = [exp(ln[f(x)])]′ should readf⋆(x) = exp[(ln ◦f)′(x)] .[6])
• The article “Multiplicative calculus and its applications” (see preceding item) was reviewed by Stefan G. Samko(University of Algarve, Portugal) in Zentralblatt MATH:[67]
“In this expository article the authors develop the basics of the so called multiplicative calculus, underwhich the definition of derivatives and integrals is given in terms of the operations of multiplication anddivision in contrast to addition and subtraction in the usual definitions. Such an approach was suggestedin a book of M. Grossman and R. Katz [“Non-Newtonian Calculus”. Pigeon Cove, Mass.: Lee Press(1972; Zbl 0228.26002)]. Transforming multiplication to addition by logarithms, it is easy to see that forinstance a multiplicative derivative equals to exp[(lnf)′]. The authors give also some applications wherethey consider the usage of the language of multiplicative calculus as more useful than the usage of theusual calculus.”
• Bigeometric Calculus: A System with a Scale-Free Derivative[17] was reviewed in Mathematics Magazine in1984. The review was preceded by the following statement: “Articles and books are selected for this sectionto call attention to interesting mathematical exposition that occurs outside the mainstream of the mathematicsliterature.” The review included the following assertion: “This book compares [the classical and bigeometriccalculi], shows their relationship, and suggests applications for which the latter might be more appropriate.”
• The geometric calculus and the bigeometric calculus were used by Hatice Aktöre (Eastern MediterraneanUniversity in Cyprus) in an article on multiplicative Runge-Kutta Methods.[68]
• Non-Newtonian Calculus,[1] a book including detailed discussions about the geometric calculus and the bi-geometric calculus (both of which are non-Newtonian calculi), is used in the 2006 report “Stern Review onthe Economics of Climate Change”, according to a 2012 critique of that report (called “What is Wrong withStern?") by former UK Cabinet Minister Peter Lilley and economist Richard Tol. The report “Stern Reviewon the Economics of Climate Change” was commissioned by the UK government and was written by a teamled by Nicholas Stern (former Chief Economist at the World Bank).[69][70]
• Non-Newtonian Calculus, a book including detailed discussions about the geometric calculus and the bigeo-metric calculus (both of which are non-Newtonian calculi), is cited by Ivor Grattan-Guinness in his book TheRainbow of Mathematics: A History of the Mathematical Sciences .[71]
• Non-Newtonian Calculus, a book including detailed discussions about the geometric calculus and the bigeo-metric calculus (both of which are non-Newtonian calculi), is used in an article on sequence spaces by AhmetFaruk Cakmak (Yıldız Technical University in Turkey) and Feyzi Basar (Fatih University in Turkey).[72] Theabstract of the article begins with the statement: “As alternatives to classical calculus, Grossman and Katz(Non-Newtonian Calculus, 1972) introduced the non-Newtonian calculi consisting of the branches of geomet-ric, anageometric, and bigeometric calculus, etc.”
• The non-Newtonian averages (of functions)[20] were used to construct a family of means (of two positivenumbers).[20][21] Included among those means are some well-known ones such as the arithmetic mean, thegeometric mean, the harmonic mean, the power means, the logarithmic mean, the identric mean, and theStolarsky mean. The family of means was used to yield simple proofs of some familiar inequalities.[21] Publi-cations about that family are cited in six articles.[73][74][75][76][77][78]
14 CHAPTER 3. MULTIPLICATIVE CALCULUS
• Non-Newtonian calculus was used by Z. Avazzadeh, Z. Beygi Rizi, G. B. Loghmani, and F. M. Maalek Ghaini(the first three from Yazd University in Iran, and the last from Islamic Azad University in Iran) to devise anumerical method for solving nonlinear Volterra integro-differential equations.[79]
• Application of non-Newtonian calculus to function spaces was made by Ahmet Faruk Cakmak (Yıldız Tech-nical University in Turkey) and Feyzi Basar (Fatih University in Turkey) in their lecture at the 2012 confer-ence The Algerian-Turkish International Days on Mathematics, at University of Badji Mokhtar at Annaba, inAlgeria.[80]
• Application of non-Newtonian calculus to “continuous and bounded functions over the field of non-Newtonian/geometriccomplex numbers” was made by Zafer Cakir (Gumushane University, Turkey).[81][82]
• Non-Newtonian calculus is one of the topics of discussion at the 2013 conference Algerian-Turkish Interna-tional Days on Mathematics at Fatih University in Istanbul, Turkey.[83]
• A seminar involving non-Newtonian calculus was conducted by Jared Burns at the University of Pittsburgh on13 December 2012.[84]
• Non-Newtonian Calculus is cited in Gordon Mackay’s book Comparative Metamathematics. (The eighteenprevious editions of Comparative Metamathematics are entitled The True Nature of Mathematics.)[85]
• Non-Newtonian calculus is cited in a book on popular-culture by Paul Dickson.[86]
• Geometric arithmetic[1] was used by Muttalip Ozavsar and Adem C. Cevikel (both of Yildiz Technical Uni-versity in Turkey) in an article on multiplicative metric spaces and multiplicative contraction mappings.[87]
• Multiplicative calculus was the subject of Christopher Olah’s lecture at the Singularity Summit on 13 Octo-ber 2012.[88] Singularity University's Singularity Summit is a conference on robotics, artificial intelligence,brain-computer interfacing, and other emerging technologies including genomics and regenerative medicine.Christopher Olah is a Thiel Fellow.[89]
• The geometric calculus was the topic of a presentation by Ali Ozyapici and Emine Misirli Kurpinar (both ofEge University in Turkey) at the International ISAAC Congress in August 2007.[90]
• Multiplicative calculus was the topic of a presentation by Ali Ozyapici and EmineMisirli Kurpinar (both of EgeUniversity in Turkey) at the International Congress of the Jangjeon Mathematical Society in August 2008.[91]
• Knowledge of the geometric calculus (“multiplicative calculus”) is a requirement for the master’s degree incomputer-engineering at Inonu University (Malatya, Turkey).[92]
• Non-Newtonian calculus was used in the article “Certain sequence spaces over the non-Newtonian complexfield” by Sebiha Tekin and Feyzi Basar, both of Fatih University in Turkey.[93]
• The geometric calculus is cited by Daniel Karrasch in his article “Hyperbolicity and invariant manifolds forfinite time processes”.[94]
3.8 See also
• List of derivatives and integrals in alternative calculi
• Indefinite product
• Product integral
3.9. REFERENCES 15
3.9 References[1] Michael Grossman and Robert Katz. Non-Newtonian Calculus, ISBN 0912938013, 1972.
[2] Luc Florack and Hans van Assen.“Multiplicative calculus in biomedical image analysis”, Journal of Mathematical Imagingand Vision, DOI: 10.1007/s10851-011-0275-1, 2011.
[3] Luc Florack.“Regularization of positive definite matrix fields based on multiplicative calculus”, Reference 9, Scale Spaceand Variational Methods in Computer Vision, Lecture Notes in Computer Science, Volume 6667/2012, pages 786-796,DOI: 10.1007/978-3-642-24785-9_66, Springer, 2012.
[4] Luc Florack.“Regularization of positive definite matrix fields based on multiplicative calculus”, Third International Con-ference on Scale Space and Variational Methods In Computer Vision, Ein-Gedi Resort, Dead Sea, Israel, Lecture Notesin Computer Science: 6667, ISBN 978-3-642-24784-2, Springer, 2012.
[5] Joachim Weickert and Laurent Hoeltgen. University Course: “Analysis beyond Newton and Leibniz”, Saarland Universityin Germany, Mathematical Image Analysis Group, Summer of 2012.
[6] Agamirza E. Bashirov, Emine Misirli Kurpinar, and Ali Ozyapici. “Multiplicative calculus and its applications”, Journalof Mathematical Analysis and Applications, 2008.
[7] Marek Rybaczuk, Alicja Kedzia and Witold Zielinski (2001) The concept of physical and fractal dimension II. The differ-ential calculus in dimensional spaces, Chaos, Solitons, & Fractals Volume 12, Issue 13, October 2001, pages 2537–2552
[8] M. Jahanshahi, N. Aliev and H. R. Khatami (2004). “An analytic-numerical method for solving difference equations withvariable coefficients by discrete multiplicative integration”., 5—10 July 2004, Antalya, Turkey – Dynamical Systems andApplications, Proceedings, pp. 425—435
[9] H. R. Khatami & M. Jahanshahi & N. Aliev (2004). “An analytical method for some nonlinear difference equationsby discrete multiplicative differentiation”., 5—10 July 2004, Antalya, Turkey – Dynamical Systems and Applications,Proceedings, pp. 455—462
[10] N. Aliev, N. Azizi and M. Jahanshahi (2007) “Invariant functions for discrete derivatives and their applications to solvenon-homogenous linear and non-linear difference equations”., InternationalMathematical Forum, 2, 2007, no. 11, 533–542
[11] Ali Uzer.“Multiplicative type complex calculus as an alternative to the classical calculus”, Computers & Mathematics withApplications, DOI:10.1016/j.camwa.2010.08.089, 2010.
[12] Ali Uzer.“Exact solution of conducting half plane problems in terms of a rapidly convergent series and an application ofthe multiplicative calculus”, Turkish Journal of Electrical Engineering & Computer Sciences, DOI: 10.3906/elk-1306-163,2013.
[13] Agamirza E. Bashirov and Mustafa Riza.“On complex multiplicative differentiation”, TWMS Journal of Applied and En-gineering Mathematics, Volume 1, Number 1, pages 75-85, 2011.
[14] Agamirza E. Bashirov and Mustafa Riza.“Complex multiplicative calculus”, arXiv.org, Cornell University Library, arXiv:1103.1462v1, 2011.
[15] Agamirza E. Bashirov and Mustafa Riza.“On Complex Multiplicative Integration”, arXiv.org, Cornell University Library,arXiv:1307.8293, 2013.
[16] Michael Grossman. The First Nonlinear System of Differential And Integral Calculus, ISBN 0977117006, 1979.
[17] Michael Grossman. Bigeometric Calculus: A System with a Scale-Free Derivative, ISBN 0977117030, 1983.
[18] Jane Grossman, Michael Grossman, Robert Katz. The First Systems of Weighted Differential and Integral Calculus, ISBN0977117014, 1980.
[19] Jane Grossman. Meta-Calculus: Differential and Integral, ISBN 0977117022, 1981.
[20] Jane Grossman, Michael Grossman, and Robert Katz. Averages: A New Approach, ISBN 0977117049, 1983.
[21] Michael Grossman, and Robert Katz. “A new approach to means of two positive numbers”, International Journal ofMathematical Education in Science and Technology, Volume 17, Number 2, pages 205-208, Taylor and Francis, 1986..
[22] Michael Grossman.“An introduction to non-Newtonian calculus”, International Journal of Mathematical Education in Sci-ence and Technology, Volume 10, #4 (Oct.-Dec., 1979), 525-528.
[23] James R. Meginniss. “Non-Newtonian calculus applied to probability, utility, and Bayesian analysis”, American StatisticalAssociation: Proceedings of the Business and Economic Statistics Section, 1980.
16 CHAPTER 3. MULTIPLICATIVE CALCULUS
[24] Diana Andrada Filip and Cyrille Piatecki. “A non-Newtonian examination of the theory of exogenous economic growth”,CNCSIS – UEFISCSU(project number PNII IDEI 2366/2008) and LEO, 2010.
[25] WojborWoycznski.“Non-Newtonian calculus for the dynamics of random fractal structures: linear and nonlinear”, seminarat The Ohio State University on 22 April 2011.
[26] Agamirza E. Bashirov, Emine Misirli, Yucel Tandogdu, and Ali Ozyapici.“On modelling with multiplicative differen-tial equations”, Applied Mathematics - A Journal of Chinese Universities, Volume 26, Number 4, pages 425-428, DOI:10.1007/s11766-011-2767-6, Springer, 2011.
[27] David Pearce MacAdam.Journal of the Optical Society of America, The Optical Society, Volume 63, January of 1973.
[28] H. Gollmann.Internationale Mathematische Nachrichten, Volumes 27 - 29, page 44, 1973.
[29] Ivor Grattan-Guinness.Middlesex Math Notes, Middlesex University, London, England, Volume 3, pages 47 - 50, 1977.
[30] WojborWoycznski.“Non-Newtonian calculus for the dynamics of random fractal structures: linear and nonlinear”, seminarat Cleveland State University on 2 May 2012.
[31] WojborWoycznski.“Fractional calculus for random fractals”, seminar at CaseWestern Reserve University on 3 April 2013.
[32] Diana Andrada Filip and Cyrille Piatecki. “In defense of a non-Newtonian economic analysis”, http://www.univ-orleans.fr/leo/infer/PIATECKI.pdf, CNCSIS – UEFISCSU (Babes-Bolyai University of Cluj-Napoca, Romania) and LEO (OrléansUniversity, France), 2013.
[33] Mora, Marco; Córdova-Lepe, Fernando; Del-Valle, Rodrigo. “A non-Newtonian gradient for contour detection in imageswith multiplicative noise”. Pattern Recognition Letters 33 (10): 1245–1256. doi:10.1016/j.patrec.2012.02.012.
[34] Emine Misirli and Yusuf Gurefe.“The new numerical algorithms for solving multiplicative differential equations”, Inter-national Conference of Mathematical Sciences, Maltepe University, Istanbul, Turkey, 04-10 August 2009.
[35] Aniszewska, Dorota (October 2007). “Multiplicative Runge–Kutta methods”. Nonlinear Dynamics 50 (1–2).
[36] Mustafa Riza, Ali Ozyapici, and Emine Misirli. “Multiplicative finite difference methods”, Quarterly of Applied Mathe-matics, 2009.
[37] Agamirza E. Bashirov. “On line integrals and double multiplicative integrals”, TWMS Journal of Applied and EngineeringMathematics, Volume 3, Number 1, pages 103 - 107, 2013.
[38] EmineMisirli andYusufGurefe.“MultiplicativeAdamsBashforth–Moultonmethods”, Numerical Algorithms, doi: 10.1007/s11075-010-9437-2, Volume 55, 2010.
[39] James D. Englehardt and Ruochen Li.“The discrete Weibull distribution: an alternative for correlated counts with confir-mation for microbial counts in water”, Risk Analysis, doi: 10.1111/j.1539-6924.2010.01520.x, 2010.
[40] Ziyue Liu and Wensheng Guo. “Data driven adaptive spline smoothing": Supplement, Statistica Sinica, Volume 20, pages1143-1163, 2010.
[41] David Baqaee. “Intertemporal choice: a Nash bargaining approach”, Reserve Bank of New Zealand, Research: DiscussionPaper Series, ISSN 1177–7567, September 2010.
[42] Raj Kumar, P. Arun; Selvakumar, S. “Detection of distributed denial of service attacks using an ensemble of adaptive andhybrid neuro-fuzzy systems”. Computer Communications 36 (3): 303–319. doi:10.1016/j.comcom.2012.09.010.
[43] Efendi, Riswan; Ismail, Zuhaimy; Mat Deris, Mustafa. “Improved weight fuzzy time series as used in the exchange ratesforecasting of US dollar to ringgit Malaysia”. International Journal of Computational Intelligence and Applications 12 (1):1350005. doi:10.1142/S1469026813500053.
[44] Zhang, P. Jie; Li, Li; Peng, Luying; Sun, Yingxian; Li, Jue. “An Efficient Weighted Graph Strategy to Identify Differen-tiation Associated Genes in Embryonic Stem Cells”. PLoS ONE 8 (4): e62716. doi:10.1371/journal.pone.0062716.
[45] Xu, P. ZHENG; Jian-Zhong, LI (2012). “Approximate aggregation algorithm for weighted data in wireless sensor net-works”. Journal of Software 23: 108–119.
[46] Dorota Aniszewska and Marek Rybaczuk. “Chaos in multiplicative systems”, from pages 9 - 16 in the book ChaoticSystems: Theory and Applications by Christos H. Skiadas and Ioannis Dimotikalis, ISBN 9814299715, World Scientific,2010.
[47] Dorota Aniszewska and Marek Rybaczuk (2005) Analysis of the multiplicative Lorenz system, Chaos, Solitons & FractalsVolume 25, Issue 1, July 2005, pages 79–90
3.9. REFERENCES 17
[48] Aniszewska, Dorota; Rybaczuk, Marek (2008). “Lyapunov type stability and Lyapunov exponent for exemplary multi-plicative dynamical systems”. Nonlinear Dynamics 54 (4): 345–354. doi:10.1007/s11071-008-9333-7..
[49] M. Rybaczuk and P. Stoppel.“The fractal growth of fatigue defects in materials”, International Journal of Fracture 2000;103(1): 71 - 94.
[50] Rybaczuka, Marek; Kedziab, Alicja; Zielinskia, Witold. “The concept of physical and fractal dimension II. The differentialcalculus in dimensional spaces”. Chaos, Solitons 12 (13): 2537–2552. doi:10.1016/S0960-0779(00)00231-9.
[51] M. Rybaczuk and P. Stoppel (2000) “The fractal growth of fatigue defects in materials”, International Journal of Fracture,Volume 103, Number 1 / May, 2000
[52] S. L. Blyumin. “Discreteness versus continuity in information technologies: quantum calculus and its alternatives”, Au-tomation and Remote Control, Volume 72, Number 11, 2402-2407, DOI: 10.1134/S0005117911110142, Springer, 2011.
[53] Fernando Córdova-Lepe. “The multiplicative derivative as a measure of elasticity in economics”, TMAT Revista Lati-noamericana de Ciencias e Ingeniería, Volume 2, Number 3, 2006.
[54] Cengiz Türkmen and Feyzi Başar. “Some basic results on the sets of sequences with geometric calculus”, First InternationalConference on Analysis and Applied Mathematics, American Institute of Physics: Conference Proceedings, Volume 1470,pages 95-98, ISBN 978-0-7354-1077-0 doi:10.1063/1.4747648 2012.
[55] Mathematics Department of Eastern Mediterranean University. Research Group: Multiplicative Calculus, MathematicsDepartment of Eastern Mediterranean University in Cyprus.
[56] Ahmet Faruk Çakmak. “Some new studies on bigeometric calculus”, International Conference on Applied Analysis andAlgebra, Yıldız Technical University, Istanbul, Turkey, 2011.
[57] Gunnar Sparr.“A Common Framework for Kinetic Depth Reconstruction and Motion for Deformable Objects”, LectureNotes in Computer Science, Volume 801, Springer, Proceedings of the Third European Conference on Computer Vision,Stockholm, Sweden, pages 471-482, May of 1994.
[58] Uğur Kadak and Yusef Gurefe. amp;gsessionid=OK, “Construction of metric spaces by using multiplicative calculus onreals”, Analysis and Applied Mathematics Seminar Series, Fatih University, Mathematics Department, Istanbul, Turkey,30 April 2012.
[59] Jarno van Roosmalen. “Multiplicative principal component analysis”, Eindhoven University of Technology, Netherlands,2012.
[60] Manfred Peschel and Werner Mende. The Predator-Prey Model: Do We live in a Volterra World?, page 246, ISBN0387818480, Springer, 1986.
[61] Dick Stanley (1999) “A multiplicative calculus”, Primus vol 9, issue 4.
[62] Duff Campbell (1999). “Multiplicative calculus and student projects”, Primus vol 9, issue 4.
[63] Michael Coco. Multiplicative Calculus, seminar at Virginia Commonwealth University’s Analysis Seminar, April of 2008.
[64] Michael E. Spivey. “A Product Calculus”, University of Puget Sound.
[65] Alex B. Twist and Michael E. Spivey. “L'Hôpital’s Rules and Taylor’s Theorem for Product Calculus”, University of PugetSound, 2010.
[66] Gérard Lebourg, MR 2356052
[67] Stefan G. Samko. Zentralblatt MATH, Zbl 1129.26007, FIZ Karlsruhe, 2012.
[68] Hatice Aktöre. “Multiplicative Runge-Kutta Methods”, Master of Science thesis, Eastern Mediterranean University, De-partment of Mathematics, 2011.
[69] Nicholas Stern.“Stern Review on the Economics of Climate Change”, Cambridge University Press, DRR10368, 2006.
[70] Andrew Orlowski.“Economics: Was Stern 'wrong for the right reasons’ ... or just wrong?", The Register, 4 September2012.
[71] Ivor Grattan-Guinness.The Rainbow of Mathematics: A History of the Mathematical Sciences, pages 332 and 774, ISBN0393320308, W. W. Norton & Company, 2000.
[72] Ahmet Faruk Cakmak and Feyzi Basar.“Some new results on sequence spaces with respect to non-Newtonian calculus”,Journal of Inequalities and Applications, SpringerOpen, 2012:228, doi:10.1186/1029-242X-2012-228, October of 2012.
18 CHAPTER 3. MULTIPLICATIVE CALCULUS
[73] Horst Alzer. “Bestmogliche abschatzungen fur spezielle mittelwerte”, Reference 19; Univ. u Novom Sadu, Zb. Rad.Prirod.-Mat. Fak., Ser. Mat. 23/1; 1993.
[74] V. S. Kalnitsky. “Means generating the conic sections and the third degree polynomials”, Reference 7, Saint PetersburgMathematical Society Preprint 2004-04, 2004.
[75] Methanias Colaço Júnior, Manoel Mendonça, Francisco Rodrigues. “Mining software change history in an industrialenvironment”, Reference 20, XXIII Brazilian Symposium on Software Engineering, 2009.
[76] Nicolas Carels and Diego Frias. “Classifying coding DNA with nucleotide statistics”, Reference 36, Bioinformatics andBiology Insights 2009:3, Libertas Academica, pages 141-154, 2009.
[77] Sunchai Pitakchonlasup, and Assadaporn Sapsomboon. “A comparison of the efficiency of applying association rule dis-covery on software archive using support-confidence model and support-new confidence model”, Reference 13, Interna-tional Journal of Machine Learning and Computing, Volume 2, Number 4, pages 517-520, International Association ofComputer Science and Information Technology Press, August 2012.
[78] Methanias Colaco Rodrigues Junior. “A comparison of the efficiency of applying association rule discovery on soft-ware archive using support-confidence model and support-new confidence model”, “Identificacao E Validacao Do PerfilNeurolinguistic O De Programadores Atraves Da Mineracao De Repositorios De Engenharia De Software”, thesis, Mul-tiinstitutional Program in Computer Science: Federal University of Bahia (Brazil), State University of Feira de Santana(Brazil), and Salvador University (Brazil), IEVDOP neurolinguistic - repositorio.ufba.br, 2011.
[79] Z. Avazzadeh, Z. Beygi Rizi, G. B. Loghmani, and F. M. Maalek Ghaini.“A numerical solution of nonlinear parabolic-typeVolterra partial integro-differential equations using radial basis functions”, Engineering Analysis with Boundary Elements,ISSN 0955-7997, Volume 36, Number 5, pages 881 - 893, Elsevier, 2012.
[80] Ahmet Faruk Cakmak and Feyzi Basar.“Space of continuous functions over the field of non-Newtonian real numbers”,lecture at the conference Algerian-Turkish International Days on Mathematics, University of Badji Mokhtar at Annaba,Algeria, October of 2012.
[81] Zafer Cakir.“Space of continuous and bounded functions over the field of non-Newtonian complex numbers”, lecture at theconference Algerian-Turkish International Days onMathematics, University of Badji Mokhtar at Annaba, Algeria, Octoberof 2012.
[82] Zafer Cakir. “Space of continuous and bounded functions over the field of geometric complex numbers”, Journal ofInequalities and Applications, Volume 2013:363, doi:10.1186/1029-242X-2013-363, ISSN 1029-242X, Springer, 2013.
[83] ATIMTopics.2013Algerian-Turkish International Days onMathematics, FatihUniversity, İstanbul, Turkey, 12–14 Septem-ber 2013.
[84] Jared Burns.“M-Calculi: Multiplying and Means”, graduate seminar at the University of Pittsburgh on 13 December 2012.
[85] Gordon Mackay.Comparative Metamathematics, ISBN 978-0557249572, 2011.
[86] Paul Dickson.The New Official Rules, page 113, ISBN 0201172763, Addison-Wesley Publishing Company, 1989.
[87] Muttalip Ozavsar and Adem Cengiz Cevikel.“Fixed points of multiplicative contraction mappings on multiplicative metricspaces”, arXiv preprint arXiv:1205.5131, 2012.
[88] Christopher Olah.“Exponential trends and multiplicative calculus” 13 October 2012.
[89] Singularity Summit, 13 October 2012.
[90] Ali Ozyapici and Emine Misirli Kurpinar.“Exponential approximation on multiplicative calculus”, International ISAACCongress, page 471, 2007.
[91] Ali Ozyapici and Emine Misirli Kurpinar.“Exponential approximation on multiplicative calculus”, International Congressof the Jangjeon Mathematical Society, page 80, 2008.
[92] Inonu University, Computer-Engineering. Master’s Degree, 2013.
[93] Sebiha Tekin and Feyzi Basar.“Certain sequence spaces over the non-Newtonian complex field”, Hindawi Publishing Cor-poration, 2013.
[94] Daniel Karrasch.“Hyperbolicity and invariant manifolds for finite time processes”, doctoral dissertation, Technical Univer-sity of Dresden, 2012.
3.10. FURTHER READING 19
3.10 Further reading• “Quotientiation, an extension of the differentiation process”, Robert Edouard Moritz
Chapter 4
Product integral
The expression “product integral” is used informally for referring to any product-based counterpart of the usualsum-based integral of classical calculus. The first product integral was developed by the mathematician Vito Volterrain 1887 to solve systems of linear differential equations.[1][2] (Please see “Type II” below.) Other examples of productintegrals are the geometric integral (“Type I” below), the bigeometric integral, and some other integrals of non-Newtonian calculus.[3]
Product integrals have found use in areas from epidemiology (the Kaplan–Meier estimator) to stochastic populationdynamics using multiplication integrals (multigrals), analysis and quantum mechanics. The geometric integral, to-gether with the geometric derivative, is useful in biomedical image analysis.[4]
This article adopts the “product”∏
notation for product integration instead of the “integral”∫(usually modified by
a superimposed “times” symbol or letter P) favoured by Volterra and others. An arbitrary classification of types isalso adopted to impose some order in the field.
4.1 Basic definitions
The classical Riemann integral of a function f : [a, b] → R can be defined by the relation
∫ b
a
f(x) dx = lim∆x→0
∑f(xi)∆x,
where the limit is taken over all partitions of interval [a, b] whose norm approach zero.Roughly speaking, product integrals are similar, but take the limit of a product instead of the limit of a sum. Theycan be thought of as “continuous” versions of “discrete” products.The most popular product integrals are the following:
4.1.1 Type I
b∏a
f(x)dx = lim∆x→0
∏f(xi)
∆x = exp(∫ b
a
ln f(x) dx),
which is called the geometric integral and is a multiplicative operator.This definition of the product integral is the continuous analog of the discrete product operator
∏bi=a (with i, a, b ∈ Z
) and the multiplicative analog to the (normal/standard/additive) integral∫ badx (with x ∈ [a, b] ):
It is very useful in stochastics where the log-likelihood (i.e. the logarithm of a product integral of independent randomvariables) equals the integral of the log of the these (infinitesimally many) random variables:
20
4.2. RESULTS 21
lnb∏a
p(x)dx =
∫ b
a
ln p(x) dx
4.1.2 Type IIb∏a
(1 + f(x) dx) = lim∆x→0
∏(1 + f(xi)∆x)
Under these definitions, a real function is product integrable if and only if it is Riemann integrable. There are othermore general definitions such as the Lebesgue product integral, Riemann–Stieltjes product integral, or Henstock–Kurzweil product integral.The Type II product integral corresponds to Volterra’s original definition.[2][5][6] The following relationship exists forscalar functions f : [a, b] → R :
b∏a
(1 + f(x) dx) = exp(∫ b
a
f(x) dx
),
which is not a multiplicative operator. (So the concepts of product integral and multiplicative integral are not thesame). The Volterra product integral is most useful when applied to matrix-valued functions or functions with valuesin a Banach algebra, where the last equality is no longer true (see the references below).For scalar functions, the derivative in the Volterra system is the logarithmic derivative, and so the Volterra system isnot a multiplicative calculus and is not a non-Newtonian calculus.[2]
4.2 Results• Basic results
b∏a
cdx = cb−a
b∏a
(f(x)k)dx = (b∏a
f(x)dx)k
b∏a
(cf(x))dx = c∫ baf(x)dx
The geometric integral (Type I above) plays a central role in the geometric calculus,[3] which is a multiplicativecalculus.
• The fundamental theorem
b∏a
f ′∗(x)dx =
b∏a
exp(f ′(x)
f(x)dx
)=f(b)
f(a)
22 CHAPTER 4. PRODUCT INTEGRAL
where f ′∗(x) is the geometric derivative.
• Product rule
(fg)∗ = f∗g∗
• Quotient rule
(f/g)∗ = f∗/g∗
• Law of large numbers
n√X1X2 · · ·Xn →
∏x
XdF (x) as n→ ∞
where X is a random variable with probability distribution F(x)).
Compare with the standard Law of Large Numbers:
X1 +X2 + · · ·+Xn
n→∫X dF (x) as n→ ∞
The above are for the Type I product integral. Other types produce other results.
4.3 See also• List of derivatives and integrals in alternative calculi
• Indefinite product
• Multiplicative calculus
• Logarithmic derivative
• Ordered exponential
4.4 References[1] V. Volterra, B. Hostinský, Opérations Infinitésimales Linéaires, Gauthier-Villars, Paris (1938).
[2] A. Slavík, Product integration, its history and applications, ISBN 80-7378-006-2, Matfyzpress, Prague, 2007.
[3] M. Grossman, R. Katz, Non-Newtonian Calculus, ISBN 0-912938-01-3, Lee Press, 1972.
[4] Luc Florack, Hans van Assen. “Multiplicative Calculus in Biomedical Image Analysis”, Journal of Math Imaging andVision, doi:10.1007/s10851-011-0275-1, 2011.
[5] J. D. Dollard, C. N. Friedman, Product integration with applications to differential equations, Addison Wesley PublishingCompany, 1979.
[6] F.R. Gantmacher (1959) The Theory of Matrices, volumes 1 and 2.
4.5. EXTERNAL LINKS 23
• A. E. Bashirov, E. M. Kurpınar, A. Özyapıcı. Multiplicative calculus and its applications, Journal of Mathe-matical Analysis and Applications, 2008.
• W. P. Davis, J. A. Chatfield, Concerning Product Integrals and Exponentials, Proceedings of the AmericanMathematical Society, Vol. 25, No. 4 (Aug., 1970), pp. 743–747, doi:10.2307/2036741.
• J. D. Dollard, C. N. Friedman, Product integrals and the Schrödinger Equation, Journ. Math. Phys. 18#8,1598–1607 (1977).
• J. D. Dollard, C. N. Friedman, Product integration with applications to differential equations, Addison WesleyPublishing Company, 1979.
4.5 External links• Richard Gill, Product Integration
• Richard Gill, Product Integral Symbol
• David Manura, Product Calculus
• Tyler Neylon, Easy bounds for n!
• An Introduction to Multigral (Product) and Dx-less Calculus
• Notes On the Lax equation
• Antonín Slavík, An introduction to product integration
• Antonín Slavík, Henstock–Kurzweil and McShane product integration
24 CHAPTER 4. PRODUCT INTEGRAL
4.6 Text and image sources, contributors, and licenses
4.6.1 Text• Indefinite product Source: https://en.wikipedia.org/wiki/Indefinite_product?oldid=553128577Contributors: Michael Hardy, CmdrObot,CBM, Ozob, Yobot, Charvest, MathFacts and Anonymous: 2
• List of derivatives and integrals in alternative calculi Source: https://en.wikipedia.org/wiki/List_of_derivatives_and_integrals_in_alternative_calculi?oldid=607371991 Contributors: The Anome, SmackBot, PrimeHunter, Gregbard, Calaka, R'n'B, Hqb, Dmcq, Yobot,Sławomir Biały, MathFacts, Helpful Pixie Bot, Anuclanus and Anonymous: 7
• Multiplicative calculus Source: https://en.wikipedia.org/wiki/Multiplicative_calculus?oldid=656449967 Contributors: Edward, MichaelHardy, Dominus, Charles Matthews, Giftlite, Rich Farmbrough, Daranz, BD2412, Rjwilmsi, Srleffler, Arthur Rubin, Xaxafrad, Smack-Bot, Dicklyon, Texas Dervish, Dl2000, CmdrObot, Magioladitis, David Eppstein, M-le-mot-dit, Voiceofreason01, Smithpith, XLinkBot,Delaszk, Ozob, Yobot, Devoutb3nji, Fcordova, Charvest, FrescoBot, Sławomir Biały, Tkuvho, Antonin Slavik, Trappist the monk,Ilario980, Xnn, John of Reading, Δ, Alelbre, BG19bot, C for Koala, BattyBot, Ugurkadak and Anonymous: 16
• Product integral Source: https://en.wikipedia.org/wiki/Product_integral?oldid=682259513 Contributors: The Anome, Michael Hardy,Oleg Alexandrov, Rjwilmsi, RussBot, Arthur Rubin, Daryl Williams, Digana, Kerdek, Gill110951, Largoplazo, LokiClock, TXiKiBoT,Anonymous Dissident, Dmcq, ClueBot, Smithpith, Addbot, Delaszk, Ozob, Yobot, AnomieBOT, Xqbot, Charvest, Sławomir Biały,MathFacts, Tkuvho, Antonin Slavik, Biker333, Keilandreas,Wikfr, Helpful Pixie Bot, Calabe1992, Alelbre, Mark viking andAnonymous:25
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