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Greek astronomy and Greek geometry were both used in order to answer many difficult questions of the time. Another type of geometry, called affine geometry, includes Euclid's parallel postulate but disregards two other postulates concerning circles and angle measurement; the propositions of affine geometry are also valid in the four-dimensional geometry of space-time used in the theory of relativity. Poncelet, who was an officer in the French corps of engineers, learned scraps of Desargues’s work from his teacher Gaspard Monge (1746–1818), who developed his own method of projection for drawings of buildings and machines.

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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. Therefore if you know the size and shape of one you know the size and shape of the others. Simulation tests confirm that the newly introduced models produce accurate results while using substantially less computation and provide support for applying the new model in the field of parametric linear, non-linear signal representation for processing." (The expression "multiplicative calculus" refers here to the geometric calculus.) The geometric calculus was used in the 2015 article "Finite product representation via multiplicative calculus and its applications to exponential signal processing" by Ali Ozyapici (Cyprus International University in Cyprus/Turkey) and Bülent Bilgehan (Girne American University in Cyprus/Turkey). (The expression "multiplicative calculus" refers here to the geometric calculus.) [225] The Abstract: "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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The most important kind of geometry for understanding 3-manifolds is hyperbolic geometry: this is the non-Euclidean geometry of constant negative curvature, where angles in triangles add up to less than 180 degrees. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, but did not realize it. By August of 1967, 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.

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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 addition, solutions of some multiplicative differential equations are obtained by the help of this transform." Some of the representative leading figures in modern geometry are Michael Atiyah, Mikhail Gromov, and William Thurston. To get any sort of good footing into this area, you want a basic understanding of “basic abstract algebra” (groups, rings, fields, modules) plus a bit of information about “higher algebra” (categories, etc.).

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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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We apply a finite volume scheme combining the Godunov numerical flux with the Euler-Maruyama integrator in time. Added some more introductory documentation on the LibSea file format. Topics include analytical, transformational, Euclidean and non-Euclidean geometries, constructions, tiling the plane, and topology. Later, on 28 September 2003, I set up another NNC website. Animal groups often form striking aggregation patterns. I said in printed review that book by Mazur is the best book ever published on combinatorics, or something like that.

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The universe does not have a structure of immutable Euclidean space woven by an independent time; it is described as a space-time distorted by the presence of matter and energy. According to Poincaré, all geometric systems deal with the same properties of space, although each of them employs its own language, whose syntax is defined by the set of axioms. It was the first time I had ever heard of OCD, which is an anxiety disorder.

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The bigeometric calculus was used in an article on fractals and multiplicative dynamical systems by Dorota Aniszewska and Marek Rybaczuk (both from Wroclaw University of Technology in Poland). [131] In that article they state: "Describing the evolution of defects [in materials] treated as fractals implies usage of the multiplicative derivative, because the ordinary [classical] additive derivative of a function depending on fractal dimension or measure does not exist. ...

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The greatest flowering of the field occurred with Jean-Victor Poncelet (1788-1867). The goal of this paper is chaos examination in multiplicative dynamical systems described with the multiplicative derivative." (The expression "multiplicative derivative" refers here to the bigeometric derivative.) - Dorota Aniszewska and Marek Rybaczuk, both from Wroclaw University of Technology in Poland; from their 2008 article "Lyapunov type stability and Lyapunov exponent for exemplary multiplicative dynamical systems". [131] " ... evolution of fractal characteristics will be examined with the help of dynamical system theory or more precisely in terms of multiplicative calculus." (The expression "multiplicative calculus" refers here to the bigeometric calculus.) - Dorota Aniszewska and Marek Rybaczuk, both from Wroclaw University of Technology in Poland; from their 2009 article "Fractal characteristics of defects evolution in parallel fibre reinforced composite in quasi-static process of fracture". [184] "We advocate the use of an alternative calculus in biomedical image analysis, known as multiplicative (a.k.a. non-Newtonian) calculus. ...

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Remarkably, it turns out that the geometric calculus is a useful tool for biomedical image analysis. "We advocate the use of an alternative calculus in biomedical image analysis, known as multiplicative (a.k.a. non-Newtonian) calculus. ... Coley [1862 - 1936] injected streptococcal organisms into a patient with inoperable cancer. 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."