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# Category: Reference

## HIGCSE Mathematics Module 2 (Cambridge Open Learning Project

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Product representation produces an accurate representation of signals, especially where exponentials occur. The distinguished astronomer Sir Arthur Eddington publicly ridiculed his suggestion that stars could collapse into such objects, which are now known as black holes. ... Interestingly enough, the great mathematician, astronomer, and physicist Carl Friedrich Gauss (1777-1855) discussed the usefulness of new calculi a long time ago: "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.

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No more than a working knowledge of [classical] calculus is assumed." - David Pearce MacAdam, Cape Cod Community College, USA; from his 1973 review of Non-Newtonian Calculus [100]. "After a long period of silence in the field of non-Newtonian calculus introduced by Grossman and Katz [Non-Newtonian Calculus] in 1972, the field experienced a revival with the mathematically comprehensive description of the geometric calculus by Bashirov et al. [2], which initiated a kick-start of numerous publications in this field." - Mustafa Riza (Eastern Mediterranean University in North Cyprus) and Bugce Eminaga (Girne American University in Cyprus); from their 2015 article "Bigeometric Calculus and Runge Kutta Method" [215]. "Grossman and Katz have shown that it is possible to create infinitely many calculi independently.

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For studying the time evolution of such "rough" objects, the classical, "smooth" Newtonian calculus is not enough." Each of the following two books was reviewed by Leo Barsotti in the journal Boletim da Sociedade Paranaense de Matemática. [54] 1) The First Nonlinear System of Differential and Integral Calculus: Volume 2, page 32, 1981. 2) The First Systems of Weighted Differential and Integral Calculus: Volume 2, pages 32-33, 1981. The purpose of this article is to illustrate the basics of multiplicative calculus and its application to the regularization of positive definite matrix fields." (The expression "multiplicative calculus" refers here to the geometric calculus.) - Luc Florack, Eindhoven University of Technology in The Netherlands; from his 2012 article "Regularization of positive definite matrix fields based on multiplicative calculus" [96]. "Our work is an example of how one can benefit from physically refined modelling in conjunction with multiplicative calculi.

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Fraction Pal is an awesome way for students to practice operations with fractions. And biomedical imaging is used so extensively that some of its nomenclature has become commonplace: "X-ray", "ultrasound", "MRI", "CAT-scan", "PET-scan". In particular, when the acquisition system yields a multiplicative interaction model involving a non-constant signal, then geometric representation frameworks such as that presented in this paper are expected to be more relevant than additive frameworks." [212] The "log-transform" technique for converting arithmetic expressions of a multiplicative nature to arithmetic expressions of an additive nature is utilized in the article "Wavelet operators and multiplicative observation models—application" by Abdourrahmane Mahamane Atto (University Savoie-Mont-Blanc in France), Emmanuel Trouve (University Savoie-Mont-Blanc), Jean Marie Nicolas (Telecom ParisTech in France), and Thu Trang Lê (University Savoie-Mont-Blanc). [299] (The "log-transform" technique is demonstrated in discussions about least-squares methods in Non-Newtonian Calculus [15], The First Nonlinear System of Differential and Integral Calculus [11], and Bigeometric Calculus: A System with a Scale-Free Derivative [10].) From the Abstract to the article: "This paper first provides statistical properties of wavelet operators when the observation model can be seen as the product of a deterministic piecewise regular function (signal) and a stationary random field (noise).

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Keynote Speech at the 27th International Conference on Technology in Collegiate Mathematics (ICTCM), Pearson PLC, YouTube, YouTube, March of 2015. [225] Ali Ozyapici and Bülent Bilgehan. "Finite product representation via multiplicative calculus and its applications to exponential signal processing", Numerical Algorithms, ISSN: 1017-1398 (Print), ISSN: 1572-9265 (Online), DOI 10.1007/s11075-015-0004-8, Springer, May of 2015. [226] Ugur Kadak, Murat Kirişci, and Ahmet Faruk Cakmak. "On the classical paranormed sequence spaces and related duals over the non-Newtonian complex field", Journal of Function Spaces, Hindawi Publishing Company, May of 2015. [227] James D.

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In addition to fractal geometry and biomedical image analysis, I've come across many other areas of NNC application, some of which we anticipated years ago: growth/decay analysis (e. g., in economics and biology), dynamical systems and chaos theory, finance (e.g., rates of return), the theory of elasticity in economics, marketing, wave theory in physics, the economics of climate change, signal processing, atmospheric temperature, information technology, pathogen counts in treated water, actuarial science, tumor therapy in medicine, materials science/engineering, demographics, differential equations (including multiplicative Lorenz systems and Runge-Kutta methods), calculus of variations, finite-difference methods, averages of functions, means of two positive numbers, weighted calculus, meta-calculus, approximation theory, least-squares methods, multivariable calculus, complex analysis, functional analysis, probability theory, utility theory, Bayesian analysis, stochastics, and decision making.

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Arrow, Nobel-Laureate, Stanford University, USA; from his correspondence, dated 26 November 1980, with Grossman, Grossman, and Katz. "In 1972 Grossman and Katz [Non-Newtonian Calculus] proposed alternative calculi to the calculus of Newton and Leibnitz. ... In my case the obsession is mathematics, and the compulsion is checking my NNC notes to make sure that all the proofs are valid. What I would like, is a cross reference chart that would cross ref. a . Option q expects a string; option s expects a string without embedded zeros.

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In this paper, the well-known Newton minimization method for one and two variables is developed in the framework of [geometric] and [bigeometric] calculi. They are starting points and their truth values are assumed right at the start. Grift of the Magi (BABF07, 12/19/99) The elementary school is being run by a toy company, with marketers substituting as teachers. Stoppel (both from Wroclaw University of Technology in Poland). [18] The bigeometric calculus was used in the article "Critical growth of fractal patterns in biological systems" by Marek Rybaczuk of Wroclaw University of Technology in Poland. [238] The bigeometric calculus was used in an article on fractal dimension and dimensional spaces by Marek Rybaczuk (Wroclaw University of Technology in Poland), Alicja Kedziab (Medical Academy of Wroclaw in Poland), and Witold Zielinskia (Wroclaw University of Technology in Poland). [132] The bigeometric calculus was used in the book Measurements, Dimensions, Invariant Models and Fractals by Wacław Kasprzak, Bertold Lysik, Marek Rybaczuk (all from Wroclaw University of Technology in Poland). [203] The bigeometric calculus was used in the article "Physical stability and critical effects in models of fractal defects evolution based on single fractal approximation" by Dorota Aniszewska and Marek Rybaczuk (both from Wroclaw University of Technology in Poland). [228] The bigeometric calculus was used in an article [126] on chaos in multiplicative dynamical systems by Dorota Aniszewska and Marek Rybaczuk, both from the Wroclaw University of Technology in Poland.

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Then they skipped off to summer vacation. From the former article: "Moreover, we advocate to replace the traditional Euler-Lagrange formalism by a multiplicative variant." Furthermore, Bashirov et al. [" Multiplicative calculus and its applications"] illustrated the usefulness of [the geometric] calculus with some interesting applications." - Feng Gu (Hangzhou Normal University in China) and Yeol-Je Cho (Gyeongsang National University in Korea); from their 2015 article [249]. "Bigeometric calculus - a modelling tool" - Mustafa Riza (Eastern Mediterranean University in North Cyprus) and Bugce Eminaga (Girne American University in Cyprus); from their 2014 article "Bigeometric calculus - a modelling tool" [178], which includes a new mathematical model for studying tumor therapy with oncolytic virus. "Bigeometric Runge-Kutta method is, at least for a particular set of initial value problems, superior with respect to accuracy and computation-time to the ordinary Runge-Kutta method." - Mustafa Riza (Eastern Mediterranean University in North Cyprus) and Bugce Eminaga (Girne American University in Cyprus); from their 2015 article "Bigeometric Calculus and Runge Kutta Method" [215], which includes new mathematical models (of the growth of cells, genes, bacteria, and viruses) for studying such things as tumor therapy with oncolytic virus and cell-cycle-specific cancer-chemotherapy. "While one problem can be easily expressed in one calculus, the same problem can not be expressed as easily [in another]." - Emine Misirli and Yusuf Gurefe, both of Ege University in Turkey; from their 2009 lecture [123]. "If non-Newtonian calculus is employed together with classical calculus in the formulations, then many of the complicated phenomena in physics or engineering may be analyzed more easily." - Ahmet Faruk Çakmak (Yıldız Technical University in Turkey) and Feyzi Basar (Fatih University in Turkey); from their 2014 article "Certain spaces of functions over the field of non-Newtonian complex numbers". [161] In this paper, the multiplicative least square method is introduced and is applied to integrals for the finite product representation of the positive functions.

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Non-Newtonian Calculus, ISBN 0912938013, Lee Press, 1972. [16] James R. For very young babies, this is a fun game because it elicits an element of surprise when someone appears and then disappears. The well-known logarithmic derivative is useless because of the fact that it can be expressed in terms of the classical derivative. Generally, as shown above, a local variable is declared holding a table, functions are added to this table, and the table is returned at the end of the module code.