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We must underscore that to discover that there was a non-Newtonian way to look to differential equations has been a great surprise for us. In our dynamic system as so far described, there is no reason the “distance” between the red cell and blue cell couldn’t be defined as simply abs(Xdif) + abs(Ydif), in this case 8+6 = 14. The error bounds obtained are sharper that the ones obtained by other methods. The non-Newtonian calculi most often used are the geometric and bigeometric calculi, which, of course, is not surprising.

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In these coordinates the matrix g(x) satisfies g(0) = I and the lines t ↦ tv are geodesics through 0. In the article, Professors Riza and Eminaga state: "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." [215] The bigeometric calculus was used in "New solution method for electrical systems represented by ordinary differential equation", an article recommended by Piero Malcovati (University of Pavia in Italy) and written by Bülent Bilgehan (Girne American University in Cyprus), Buğçe Eminağa (Girne American University in Cyprus), and Mustafa Riza (Eastern Mediterranean University in Cyprus). [250] From the article's abstract: "The aim is to examine the existing models from bigeometric calculus point of view to obtain accuracy on the results.

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From that article: "This paper introduces the concept of geometric wavelets defined from multiplicative algebras. ... Fujimoto (all from the Institute of Health Biosciences at The University of Tokushima in Japan). [289] From the article: "In this paper, we propose an iterative method as discretization of the differential equation using the geometric multiplicative calculus, and show the effectiveness of our method. ... Thus co-ordinative definitions are meaningful and not verifiable.

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Gauss curvature measures the extent to which a surface fails the local Euclidean geometry test. The purpose of this article is to provide a condensed review of multiplicative calculus and to illustrate its potential use in biomedical image analysis." (The expression "multiplicative calculus" refers here to the geometric calculus.) - Professors Luc Florack and Hans van Assen, both of Eindhoven University of Technology in The Netherlands; from their article "Multiplicative calculus in biomedical image analysis", Journal of Mathematical Imaging and Vision, Springer, 2012. "Multiplicative calculus provides a natural framework in problems involving positive images and positivity preserving operators.

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Let L be the symmedian point of a triangle ABC. Like a converse, an inverse does not necessarily have the same truth value as the original conditional. Even with a computer, it needed months of experimentation. Perhaps it is also in the spirit of this paper that when doing classical differential, we submerge ourselves in lengthy calculations. But after making several edits, I decided to stop participating in Wikipedia. Blyumin of the Lipetsk State Technical University in Russia. [23] An article concerning minimization methods based on the geometric and bigeometric calculi was written by Ali Ozyapici (Girne American University in Cyprus/Turkey), Mustafa Riza (Eastern Mediterranean University in North Cyprus), Bulent Bilgehan (Girne American University), and Agamirza E.

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Some features aren't working yet; we're restoring them ASAP. Volume is like area expanded out into 3 dimensions. If you can find a small piece of the surface around the given point which only touches the tangent plane at that point, then the surface has positive or zero sectional curvature there. Initially his theory was rejected by peers and professional journals in England. The geodesic curvature at a point is an intrinsic invariant depending only on the metric near the point.

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By use of methods of multiplicative calculus it is proved that an infinitely-many times differentiable function may not be analytic." (The expression "multiplicative calculus" refers here to the geometric calculus.) - Agamirza E. Excerpt: "The possibilities opened up by the [non-Newtonian] calculi seem to be immense." 2) The First Nonlinear System of Differential and Integral Calculus [11]: Volumes 35-36, page 42, 1981. 3) The First Systems of Weighted Differential and Integral Calculus [9]: Volumes 35-36, page 40, 1981. 4) Meta-Calculus: Differential and Integral [7]: Volumes 35-36, page 140, 1981. 5) Bigeometric Calculus: A System with a Scale-Free Derivative [10]: Volumes 37-38, page 266, 1983. 6) Averages: A New Approach [8]: Volumes 37-38, page 266, 1983.

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Furthermore, I will argue that this notion of convergence sits at a good ``spectral gap'' for Bianchi 9 cosmologies: The definition is weak enough to allow for relatively easy results, while beeing strong enough that these results are still meaningful. His prize work, Elements, was the textbook of elementary geometry and logic up to the early twentieth century. Among his teachers were the neo-Kantian philosopher Ernst Cassirer, the mathematician David Hilbert, and the physicists Max Planck, Max Born and Albert Einstein.

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Michael Valenzuela (University of Arizona in the United States); from his 2016 doctoral dissertation "Machine learning, optimization, and anti-training with sacrificial data" (In computer science, machine learning is a branch of artificial intelligence.) [279] "It seems plausible that people who need to study functions from this point of view might well be able to formulate problems more clearly by using [bigeometric] calculus instead of [classical] calculus." - Ralph P.

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Bashirov. "On multiplicative minimization methods", Numerical Algorithms, ISSN: 1017-1398 (Print), ISSN: 1572-9265 (Online), DOI 10.1007/s11075-013-9813-9, Springer, December of 2013. [177] Christopher Olah. This practice proved to be valuable for our classroom presentations and in the writing of Axiomatic Analysis. A final chapter features historical discussions and indications for further reading. However, I do consider the title misleading, the book is not about geometry as we usually consider it in the Euclidean sense.