Brigham both of Stanford University. The First Systems of Weighted Differential and Integral Calculus [9] is cited by the authors indicated below in their article on thermochemistry of ammonium based ionic liquids: Sergey P. Verevkin and Vladimir N. Arun Raj Kumar and S. Each of the following two books is cited in the journal Acta Scientiarum Mathematicarum. Each of the following six books is cited in the journal Industrial Mathematics. Each of the following two books is cited in the journal Economic Books: Current Selections.

Each of the following five books was reviewed by Ralph P. Boas, Jr. Non-Newtonian Calculus [15] was reviewed in the journal Mathematical Reviews in Each of the following three books was reviewed by K. Strubecker in Zentralblatt Math [43]. Each of the following five books was reviewed in ZDM [50].

Each of the following three books was reviewed in the journal Nieuw Tijdschrift Voor Wiskunde. Each of the following two books was reviewed by P. Non-Newtonian Calculus was reviewed by M. Dutta in the Indian Journal of History of Science. Non-Newtonian Calculus was reviewed at amazon. 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. The theory has proved to be most valuable in several research studies in which I am engaged. I predict that non-Newtonian calculus will come to be recognized as the most important mathematical discovery of the Twentieth Century. They constructed a comprehensive family of calculi, including the Newtonian or Leibnizian calculus, the geometric calculus, the bigeometric calculus, and infinitely-many other calculi.

The applications range from rates of return and other growth processes to highly active areas of digital image processing. They demonstrated that the [geometric calculus] differential equations are more suitable than the ordinary differential equations in investigating some problems in various fields.

Furthermore, Bashirov et al. Some real applications of nonlinear exponential signals will be selected to demonstrate the applicability and efficiency of proposed representation. If a meta-model is primarily concerned with learning probabilities, non-parametric distributions, or anything else where the multiplication is the primary operation, then the geometric calculi may be of interest. If working in a domain where the squares are additive, as is common the case when estimating the variance of a sum of independent random variables, then the quadratic calculi may produce meaningful models.

The sigmoidal calculus, as introduced in your book Non-Newtonian Calculus has the potential to be a very useful approach to the problems I want to solve …. Also, please see [37]. But really all lines that are continuous and of a uniform nature are just as simple as one another. Another kind of mind which might form an equally clear mental perception of some property of any one of these curves, as we do of the congruence of a straight line, might believe these curves to be the simplest of all, and from that property of these curves build up the elements of a very different geometry, referring all other curves to that one, just as we compare them to a straight line.

Brigham By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from 30 to years, in some cases even more; and that the whole system seems to function without any direction , without any reference to usefulness, and without any desire to do things which are useful.

## Multiplicative calculus

Schmalz Theories originally perceived as exotic and fanciful are indeed the foundation of our computer age. In one kind, the goal is given first, and then the mind goes from the goal to the means, that is, from the question to the solution. In the other kind, the mind goes from the means to the goal, that is, the mind first discovers a fact and then seeks a use for it. Obviously Gauss had little regard for the reception of the mathematical community for his new ideas. Cantor, unable to tolerate this, had a breakdown and spent his last days in a mental institution.

Like other closed minds they shield their obtuseness behind the curtain of established ways of thinking while they hurl charges of madness against the men who would tear apart the fabric. A false conclusion once arrived at and widely accepted is not easily dislodged and the less it is understood the more tenaciously it is held. This is certainly true of plate tectonics, one of the most important and far-ranging geological theories of all time; when first proposed, it was ridiculed, but steadily accumulating evidence finally prompted its acceptance, with immense consequences for geology, geophysics, oceanography, and paleontology.

And the man who first proposed this theory was a brilliant interdisciplinary scientist, Alfred Wegener. His legacy can been seen in everything from microwave ovens to MX missiles. Daniel Shechtman, 70, a researcher at Technion-Israel Institute of Technology in Haifa, received the award for discovering seemingly impossible crystal structures in frozen gobbets of metal that resembled the beautiful patterns seen in Islamic mosaics.

His theory challenged the common scientific notion of the s that all stars, after burning up their fuel, became faint, planet-sized remnants known as white dwarfs. The distinguished astronomer Sir Arthur Eddington publicly ridiculed his suggestion that stars could collapse into such objects, which are now known as black holes.

That was met with almost universal hostility and ridicule and disbelief by other scientists. I was sometimes very upset. Judah Folkman - founded angiogenesis research, a field of biology which revolutionized biomedical research and clinical drug development.

He created a novel approach to understanding and treating many diseases, including cancer. William B. Coley [ - ] injected streptococcal organisms into a patient with inoperable cancer. He thought that the infection he produced would have the side effect of shrinking the malignant tumor. He was successful, and this was one of the first examples of immunotherapy. Despite this criticism, however, Coley stuck with his ideas, and today we are recognizing their potential value. For over a decade, Dr. Barry Marshall endured the hostility and ridicule of a medical establishment deeply invested in the received wisdom that peptic ulcers were a chronic condition requiring a lifetime of treatment.

But, despite their discouraging and sometimes arrogant comments, we always knew that non-Newtonian calculus has considerable potential for application in science, engineering, and mathematics.

They were a game changer, opening up completely new ways to think about many aspects of the natural world. But for a long time it was not difficult to find professional research mathematicians who stoutly maintained that fractals and chaos were completely useless and that all of the interest in them was pure hype. This attitude persisted into the current century, when fractals had been around for at least twenty-five years and chaos for forty. That this attitude was narrow-minded and unimaginative is easy to establish, because by that time both areas were being routinely used in branches of science ranging from astrophysics to zoology.

To such intellectuals, Mandelbrot was a visibly freakish phenomenon. In France, where Bourbaki ruled the roost for a while, applied mathematics received a sharp slap in the face; more than that: a body blow. After expending the energies of enthusiasts and spending tens of millions in cash, after abusing the patience of teachers, parents, and a goodly proportion of professional mathematicians, the New Math was finally acknowledged to be an abject and unmitigated failure. And the failure was predictable , according to some. Their goals differed, but they all had this in common: that the step was first, the road new, the vision unborrowed, and the response they received — hatred.

The first airplane was considered impossible. The power loom was considered vicious. Anesthesia was considered sinful. But the men of unborrowed vision went ahead. They fought, they suffered and they paid. But they won. Medawar , from his book The Art of the Soluble He said that they were very, very good, but they were just theorem-provers.

They have an extraordinary arsenal of techniques, remember many previous results, and put them together in new ways. So in mathematics there has been historically this more or less sharp distinction between those who are best known for asking questions and those who are best known for proving theorems that others have conjectured. The principal emphasis is on the invention of concepts.

To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and marks real advance in science. Mittal, from the ResearchGate website on 12 November []. For some, such as HIV, heliocentrism, and evolution, pockets of resistance remain decades or centuries after the war is won. Because history shows that the deeper your idea cuts into the heart of a field, the more your peers are likely to challenge you.

Human nature being what it is, what ought to be reasoned discussion may turn personal, even nasty. Also, please see []. A detailed account of the geometric calculus. Wikipedia article Internet. Russian version: S. Statistical Society of Australia, Volumes , Bashirov and Mustafa Riza. This study is used in [89]. Verevkin, Vladimir N. Research Group: Multiplicative Calculus.

Avazzadeh, Z. Beygi Rizi, G. Loghmani, and F. Maalek Ghaini. Jahanshahi, N. Aliev, and H. Khatami, M. Jahanshahi, and N. Aliev, N. Azizi, and M. Vic Dannon. Bronstein, Michael M. Abbas, B.

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Ali, and I. Abodayeh, A. Pitea, W. Shatanawi, and T. Altshuller and his colleagues in the former U. Kreidik and George Shpenkov. Jahanshahi and N. Reading Michael Grossman. Michael Grossman. Robert Katz. Heath and Company, Robert Edouard Moritz. Unfortunately, some critics of non-Newtonian calculus have fallaciously argued that it follows from that fact that the non-Newtonian calculi are useless. Indeed, if that argument is valid, then each of the absurd arguments indicated below would also be valid: The operation of multiplication of two positive integers is useless because of the fact that it can be expressed in terms of repeated addition.

The operation of multiplication of two positive numbers is useless because of the fact that it can be expressed in terms of addition by using logarithms. The operation of division of two positive numbers is useless because of the fact that it can be expressed in terms of subtraction by using logarithms. The logarithmic derivative is useless because of the fact that it can be expressed in terms of the classical derivative. The elasticity concept used in economics, etc. The classical calculus is useless because of the fact that the classical derivative and classical integral can each be expressed in the context of the real number system e.

Such is the case with the invention of general algebra [and] with the differential calculus …. Recollections and Reflections , by Michael Grossman. In this section I present some recollections and reflections about my participation in the creation and development of non-Newtonian calculus NNC.

I alone am responsible for the content. Many memories still seem vivid. I recommend that you read the Brief History section of the website before continuing. Robert Bob Katz was one of my mathematics professors when I was a student at Tufts University to We eventually became good friends. Our common interests include mathematics, music, and hiking. Heath, , and then in teaching from it.

Incidentally we were both graduate students at the Yale University mathematics department, although Bob preceded me by about twenty years. A remarkable person, Bob is brilliant, amazingly imaginative, hardworking, caring, and generous. He is a superb mathematics writer, and a magnificent teacher - the best teacher I ever had. I have been privileged to be his student, colleague, and friend.

We began our work on NNC on 14 July We happened to come across a simple algebraic identity in a statistics book that led us to create the first non-Newtonian calculus, the geometric calculus. This is explained on page 85 of our book Non-Newtonian Calculus. Shortly afterwards, we created infinitely more, but not all, non-Newtonian calculi. We were thrilled! By August of , we had produced a table, which we called the CHART, on which we compared the concepts of the classical calculus with the corresponding concepts of the geometric calculus and the corresponding concepts of our most general up to that time non-Newtonian calculus.

Bob and his wonderful wife Rosalie were in the front seats. We were driving from their home in Boston to Poughkeepsie, New York, where they had some family matters to take care of. I must have been concentrating intensely on my work because I have no memory of the city of Poughkeepsie or the sights we passed during that trip. At that time Bob was a mathematics editor at a major publishing company based in Boston. I lived with my parents and sister Dotty in Somerville, Massachusetts.

We drove to his house. We thanked her, and drove away feeling cheerful. But unfortunately the professor never responded. So we tried another approach: writing a book that we called Dialogs on Non-Newtonian Calculus. We completed the book in August of Our book contained discussions between a student and a mathematician about the non-Newtonian calculi we had constructed in The dialog format enabled us to present the material in an informal way with lots of questions and answers.

I remember hiking with Bob in the White Mountains of New Hampshire, to celebrate the completion of the book. But months later, the book had been rejected by all the publishers we had contacted. Finally, in , we decided to write a book in the form of a research report. We called it Non-Newtonian Calculus. Interestingly, in August of while writing the book we unexpectedly created infinitely more non-Newtonian calculi, including the bigeometric calculus.

Roughly, we did this by transforming function values and arguments, not just values. Again we were thrilled! We knew that the bigeometric calculus like the geometric calculus and maybe many other non-Newtonian calculi would be useful. Non-Newtonian Calculus was first published in It contains among other things discussions about nine specific non-Newtonian calculi, including the geometric and bigeometric calculi, and the general theory of non-Newtonian calculus. We published the book ourselves in order to maintain control of the contents. We tried to make the book readable for scientists and engineers, as well as for mathematicians.

Well aware that people are reluctant to accept new ideas without good reasons, we worked hard to develop motivations and explanations for each concept. Included in the book are various ideas concerning potential applications, including a chapter with heuristic guides for choosing an appropriate calculus.

We were determined to write the book clearly and concisely, and made a special effort to avoid mistakes. We distributed copies of Non-Newtonian Calculus to individual purchasers, to libraries, and to journals for review. We received enthusiastic encouragement from a few people. The response we received from Dirk J. Struik, the eminent historian and mathematician, was particularly pleasing.

He graciously invited us to visit him at his home in Belmont, Massachusetts. Our visit was enjoyable and memorable. His deep historical perspective is uncommon among research mathematicians. Cautiously optimistic, we patiently waited to see if anybody would discover some application s of NNC. But much to our dismay, it turned out that various pure mathematicians said NNC was useless. Please see Appendix 1. Some of them were rude and arrogant. It became discouraging. The negativity of some pure mathematicians toward NNC surprised us at first, but eventually we became painfully accustomed to it.

In the beginning we were naively unaware that new and unusual ideas, even in science and mathematics, are often ignored, ridiculed, or demeaned by the academic establishment, even in the 20th and 21st centuries. Jane Tang and I got married in She joined Bob and me in our work on NNC, and over a period of several years we came up with various new ideas, and wrote more books and some articles. The meta-calculi can be used for financial-investment analysis, and are discussed in our book Meta-Calculus: Differential and Integral That article provides reasons for using the geometric derivative for studying the growth of organisms.

Our book The First Nonlinear System of Differential and Integral Calculus, published in , contains a detailed treatment of the geometric calculus. And our book Bigeometric Calculus: A System with a Scale-Free Derivative, published in , contains a detailed treatment of the bigeometric calculus. Between and , we wrote five books and several journal articles. These publications received some favorable responses, but also discouraging criticism from some pure mathematicians.

It was an interesting and enjoyable meeting. Professor Grattan-Guinness was good-natured, extremely knowledgeable, and obviously interested in NNC. He was impressed by the potential and originality of our work. For me, one of the most difficult aspects of the NNC project was making sure that we made no mistakes. I had worked hard on the proofs, and then I checked them and rechecked them to make sure they were correct. Even after our last publication, I continued to check the proofs. The books had been reviewed, and undoubtedly examined carefully, by first-rate mathematicians, and no errors were found.

And yet, for years I was plagued with doubt. But ironically it turned out that there was something wrong with me: obsessive-compulsive disorder OCD.

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By the spring of , my checking and rechecking had become all-consuming. It resulted in anxiety and depression, so bad that I could not function normally. It was the first time I had ever heard of OCD, which is an anxiety disorder. Some OCD patients compulsively wash their hands, disturbed that their hands might be dirty and unable to ever convince themselves that their hand-washing was effective.

Other patients compulsively check their door locks, unable to ever convince themselves that their doors are locked and their houses are safe. In my case the obsession is mathematics, and the compulsion is checking my NNC notes to make sure that all the proofs are valid. Many months of behavior therapy with an excellent psychologist, and a daily dose of antidepressant medicine, prescribed by excellent psychiatrists, have enabled me to recover enough to lead a normal life - except that I can not do mathematics anymore. Jane and I gave up our positions in the mathematics department at the University of Massachusetts Lowell, and moved to Florida.

Jane taught herself investing during the many extended periods when I was busy checking my NNC notes; and as a result of her wise investment decisions, we now have a comfortable life.

## Category:Non-Newtonian calculus - Wikipedia

On 19 October , Bob sent me a letter indicating that he felt it would be best for both of us if we suspended our friendship. Anxiety and depression are extraordinarily difficult to combat. Depression is not the same as sadness. My psychiatrists and psychologist have been invaluable.

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Without them I would not have recovered as much as I have, if at all. I am deeply grateful to Drs. Jane and I have been living in Jupiter, Florida, since We enjoy the beaches. And we enjoy the freedom to pursue our interests. Active 8 months ago. Viewed 90 times. Craig Craig 8 8 bronze badges.

Peter Carr Peter Carr 36 2 2 bronze badges. Pap's g calculus is very closely related to non-Newtonian calculus and has been applied this way. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Featured on Meta. Unicorn Meta Zoo 8: What does leadership look like in our communities?

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