Dear Jean. Our definition is the first because it depends on the choice of two arbitrary generator functions. But, by using only one generator (i.e. exp generator) we have multiplicative type calculus which has widely known and discuss.  You can visit "https://sites.google.com/site/nonnewtoniancalculus/references" for more information. Best wishes.

I would be interested to know whether these calculi provide different answers to problems previously done by standard methods or provide different (easier?) ways to get the same answers or apply to a different set of problems or different contexts.

Are they related to the differential operators of DE theory or to the subdifferentials of optimization or ... ?

To  Ugur Kadak@:

To the regret, all of this "multivariate"  and associated activities are  worth of no one cent and can be considered not  more than a poor joke, yet not much related to  mathematics. Even more, this pseudo-mathematical activity is looking as a sort of  not enough fresh and not good smelling substitute of real analysis for strange people,  waisting their  abilities and nothing more.

One can strongly support the posted above  by Thomas I. Seidman rhetorical sentence, --

  "these calculi provide different answers to problems previously done by standard methods or provide  different (easier?) ways to get the same answers or apply to a different set of problems or different contexts."  Dixi!

Regards!

Dear Professor Seidman;

First of all it is a great honor for us to see the interest of you and other Professors from different areas within these few days. Especially we are not interested in the algebra of Non-Newtonian type calculi. We are exactly interested in the notion of *-differentation depending on two  generator functions. The definition of classic derivative was generalized by this way.The classical derivative is constant for linear functions, but the bigeometric derivative is constant for power functions. Besides this if we choose the generators alpha and beta as I and exp, respectively then we get multiplicative type calculus and it has widely known i.e image processing and fractal geometry. So this method can be provided answers to problems previously done by standard methods as you say.

In the article Runge_Kutta method, it is shown that depending on the structure of the problem, even the NNRK_2(non-Newtonian runge-kutta 2) method gives favorable results with respect to classical Runge-Kutta method. Based on the selection of different generating functions, the approximations within the same method varies according to the problem under consideration. The numerical results obtained for narrower and larger intervals in the same table and it is acquired as an advantage of the non-Newtonian method. In particular, for large intervals the selection of "exp" function and for small intervals the selection of "q" function as a generator is seen to be more appropriate. Therefore, depending on the selection of the appropriate generating functions, closer approximations can be made in macro and micro systems for real-life situaitons. This approach is new. We know that the stability and convergence must be given immediately so we will try to construct it. 

Newtonian calculus which is a methodology that allows one to have a different look at problems which can be investigated via calculus. It should be clear that the *-calculus is a self-contained system independent of any other system of calculus. Therefore the reader may be surprised to learn that there is a uniform relationship between the corresponding operators of the *-calculus and the classical calculus. I try to tell again we are not interested algebraic structure of NNC we focus only differentiation for now. Maybe, in next future these notions can be taken forward and this self-contained work is intended for a wide audience, including students, engineers, scientists, and mathematicians.

I would like to say your other questions that many authors have developed its analysis for different areas and have set up new approaches according to its classical concepts. Especially Basar et al have constructed the field of non-Newtonian real numbers and the concept of non-Newtonian metric. Also  they have given the triangle and Minkowski's inequalities. Also they have given the sets  of all, bounded, convergent, null and p-absolutely summable sequences, in the sense of non-Newtonian calculus and have shown that each of the set forms a vector space and a complete metric space. Furthermore Kadak et al have constructed the duals of sequence spaces and matrix transformations between these sets. You can find nearly 212 documents about non-Newtonian calculus in a site "https://sites.google.com/site/nonnewtoniancalculus/references" prepared by M. Grossman.

To be able to answer your other questions on the choice of  generating functions for a certain problem the analysis of convergence and stability should be investigated for the theory of differential equations and subdifferentials of optimization. Nowadays we are working on some open problems to find easier ways. At the stage we will be pleased to hear any suggestions. Sincerely.

As I now understand it, the idea here is first to make some invertible nonlinear transformation, then use classical differential calculus and transform back.   [Correct me if I have misunderstood.] In some ways the valuable part of this in a group setting would be the Lie algebra formulation.Of course, this sort of work with transforms, both linear and nonlinear, is often useful, although I would not want to create a plethora of labels for different variants.  More truly novel calculi are Schwartz's theory  of distributions or the Ito calculus in stochastic analysis or the various extensions of differentiation for nonsmooth variational analysis, or max-plus algebras or ...

@Thomas I. Seidmann

Your observation that   "...  In some ways the valuable part of this in a group setting would be the Lie algebra formulation..."  is well lucidating the situation with these  "new calculi", yet it is necessary to point out  that there exists a wide classes of Lie algebraic structures which could not be on the whole imbedded into any  multiplicative Lie group  structure - that is not every  Lie algebra results from some (multiplicative) Lie group object. This confirms once more that the classical logically natural way of using the linear differential structure is  mostly fundamental  in analysis, as it was devised in reality by Masters.

Even in the case of the L. Sczwartz distribution calculus, you mentioned above, - "More truly novel calculi are Schwartz's theory of distributions" - its  main novelity (being first observed more early   by Gunther and  S.L. Sobolev) consisted in the  standard  notion of differentiation yet understood  in a novel weak sense as a linear functional on a suitably chosen  linear test functional space! 

Thereby, one can state slightly generalizing all that about, - there exists no fruitful and analytically reasonable mathematical structure which possesses no its  linear background. In some sense, one can observe that the main stream of mathematical development  was governed  both by finding and creating new linear structures over related mappings - and these always were entailing both new understanding and new useful applications in diverse sciences.

As a very impressive  example one can present a new enough and very important for mathematical physics mathematical structure called   "quantum group" - which in reality has nothing to do with "group" being strongly a tensorial structure on linear spaces!

Thanks and regards!

To Anatolij K. Prykarpatski:

"In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, indeed to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius' calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius."

  - Carl Friedrich Gauss, as quoted in Gauss, Werke, Bd. 8, page 298; and as quoted in Memorabilia Mathematica (or The Philomath's Quotation Book) (1914) by Robert Edouard Moritz, quotation #1215. 

Sincerely yours.

I think the quote from Gauss very much clarifies the situation.  It becomes a matter of individual convenience as to whether to think of an example of a general theory (then necessitating the conversion between settings) or to learn a specific formalism for a specific class of problems.   [One doesn't really need multiplication of integers if one is willing to do repeated addition -- but it can be a great convenience.]

To Ugur Kadak@:

Dear Ugur, thanks so much for the citation,- I like  reasonings of Gauss and other  Masters, yet this blog is for presenting yours own thoughts, not theirs!-:)-:)-:)

Anyway, I was trying to explain you very natural, simple yet very nontrivial things lying in backgrounds of mathematical thinking.... which were  experienced doing math by all these Masters too. If you will catch them, - ok, if you reject, -  it becomes your problem... not that of mathematics!

Be well and have really creative research!

Sincerely, regards!

Dear Professor Prykarpatski

Thanks for your suggestions. We understood you well and also thanks for your positive criticism. We have started this topic for three years and, at the beginning we thought and discussed on your critics for a long time. Especially in the beginning we have same convictions to you. But we saw an ides from H.V.Neumann that "The classical calculus was the first achievement of modern mathematics, and it is dicult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system or mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking" and also, R. Campbell pointed out that "we must recognize the possibility that a system of measurement may be arbitrary otherwise than in the choice of unit, they may be arbitrariness in the choice of the process of addition" and other Masters i.e. Gauss, Leibnitz.. The above convictions are very reasonable for me to working on this area. Besides this we have researched some different  developments on applied mathematics and other with this topic. We have seen that there was many publications and ideas on fractal geometry, dynamical systems,image processing, economy and biology etc. Therefore we have decided that "we must generalized all types and bring a new approach especially about differentiation" by using these calculi. At the begining we have worked on applied mathematics and sequence spaces. Generally, we have taken very useful results. Furthermore some projects on differential equations and many others are continued now. Some of them are convexity, weighted arithmetic(means), fuzzy systems, Lorenzt chaotic systems, and Runge-Kutta four order method, finite difference method, paranormed sequence spaces with some researchers in turkey and other countries(nearly ten persons).

I understand you very well, but I think we have good results in this title next future. Finally, I can exemplify that;

M. Grossman sent a letter to R. Katz on 21 July 2014;

"As you well know, for many years lots of people, especially various pure mathematicians, claimed that our work was useless. But, despite their discouraging and sometimes arrogant comments, we always knew that NNC has considerable potential for application in science, engineering, and mathematics. And we were right!! ".

I agree with Grossman. I hope this topic and its applications can bring some new tools and ideas for science. Sincerely.

Dear Ugur, thanks - yet I guess you are slightly mislead concerning my  in no sense critical comments (being strongly math backgrounds educational, in particular concerning   things, which are related to general of analytical nature   math problems - as you well, I am sure, understand too - that any  such a math problem is not solvable on the whole, if it does not allow some  either local  or global linearization. - Even more, the math analysis as science would not exist,  if there were no possibility to linearize a mapping in some  vicinity of a domain point! )  - As you can also easily observe,  my  clearly formulated  intentions  were completely aimed at  the pure mathematical essence of these "calculi" activity subject to this fundamental linearity property,  and concerned in no sense their , eventually,  possible applications! Why not? - -nobody knows!

All the best and regards!