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# Category: Non-Euclidean Geometries

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I lived with my parents and sister Dotty in Somerville, Massachusetts.) We drove to his house. Bashirov (Eastern Mediterranean University in North Cyprus), Emine Misirli (Ege University in Turkey), Yucel Tandogdu (Eastern Mediterranean University in North Cyprus) Ali Ozyapici, Lefke European University in Turkey); from their 2011 article [94]. "In [1967] Michael Grossman and Robert Katz gave definitions of a new kind of derivative and integral ... and thus established a new calculus, called multiplicative calculus [the geometric calculus]. ...

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Later in life, Gauss also claimed to have investigated a kind of non-Euclidean geometry using curved space but, unwilling to court controversy, he decided not to pursue or publish any of these avant-garde ideas. This smooth and match estimator (SME) bypasses numerical integration of the system altogether and reduces the amount of computational time drastically compared to ordinary least squares. This pioneering work initiated numerous studies." [293] In the abstracts to his two seminars both called "Non-Newtonian calculus for the dynamics of random fractal structures", Wojbor Woyczynski (Case Western Reserve University) asserted: "Many natural phenomena, from microscopic bacteria growth, through macroscopic turbulence, to the large scale structure of the Universe, display a fractal character.

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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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I just could not convince myself that there was nothing wrong with our work. Space and time are relational concepts and we can study their properties because of the existence of physical objects, for example, clocks, that realize relationships between space-time entities. As numerical minimization methods have a wide range of applications in science and engineering, the idea of the design of minimization methods based on [geometric] and [bigeometric] calculi is self-evident.

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Within this interpretation, Euclid’s fifth postulate was an empirical finding; non-Euclidean geometries did not apply to the real world. These are the flat Riemannian manifolds and non-trivial ones exist in all dimensions. In August of 2014, my sister Dotty (Dave and Kenny's mother) and I visited Bob and Rosalie at their home in Rockport. Each of the satellites contains an atomic clock. Functional central limit theorems under dependence. In an abstract to another seminar he asserted: "Random fractals, a quintessentially 20th century idea, arise as natural models of various physical, biological (think your mother's favorite cauliflower dish), and economic (think Wall Street, or the Horseshoe Casino) phenomena, and they can be characterized in terms of the mathematical concept of fractional dimension.

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Digital Public Library of America for online and downloadable books. Meta-Calculus: Differential and Integral, ISBN 0977117022, 1981. [7] Michael Grossman. It turns out that infinitely many non-Newtonian calculi are multiplicative calculi. (But, infinitely many non-Newtonian calculi are not multiplicative calculi.) Because there are many multiplicative calculi, the expression "the multiplicative calculus" should be avoided, and no one specific calculus should be named "multiplicative calculus".

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Building on several libraries, it uses modern numerical methods, such as adaptive mesh refinement, and is both highly parallelized and easily extensible. Sadr al-Din (sometimes known as "Pseudo-Tusi"). These approaches allow them to focus on different fractal structures, including morphogenesis of fractals at elastic-inelastic transitions in solids, composites and soils, as well as materials that have anomalous heat conduction properties and fractal patterns that are seen in biological materials." - Martin Ostoja-Starzewski, University of Illinois at Urbana-Champaign, USA; from the 2013 media-upload "The inner workings of fractal materials", University of Illinois at Urbana-Champaign. [163] "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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It is hoped that the scientific community will soon reach accord with regard to names for these two calculi. From the outside, the building simply appears to be impossible. Selvakumar (both of the National Institute of Technology, Tiruchirappalli, in India) in their article "Detection of distributed denial of service attacks using an ensemble of adaptive and hybrid neuro-fuzzy systems". [147] The First Systems of Weighted Differential and Integral Calculus [9] is cited by Riswan Efendi and Zuhaimy Ismail (both of Universiti Teknologi Malaysia) together with Mustafa Mat Deris (Universiti Tun Husein Onn Malaysia) in their article "Improved weight fuzzy time series as used in the exchange rates forecasting of US dollar to ringgit Malaysia". [142] The First Systems of Weighted Differential and Integral Calculus [9] is cited by Riswan Efendi and Zuhaimy Ismail (both of Universiti Teknologi Malaysia) together with Mustafa Mat Deris (Universiti Tun Husein Onn Malaysia) in their article "A new linguistic out-sample approach of fuzzy time series for daily forecasting of Malaysian electricity load demand". [246] The First Systems of Weighted Differential and Integral Calculus [9] is cited by ZHENG Xu and LI Jian-Zhong (School of Computer Science and Technology, Harbin Institute of Technology, Harbin, China) in their work on wireless sensor networks in computer science. [125] The First Systems of Weighted Differential and Integral Calculus [9] is cited by Hong-Kyu Kim (Korea University, Seoul), Mirim Lee (Korea University, Seoul), Kwang-Ryeol Lee (Korea Institute of Science and Technology, Seoul), and Jae-Chul Lee (Korea University, Seoul) in their article "How can a minor element added to a binary amorphous alloy simultaneously improve the plasticity and glass-forming ability?". [248] The First Systems of Weighted Differential and Integral Calculus [9] is cited by Jie Zhang, Li Li, Luying Peng, Yingxian Sun, Jue Li (the first four from Tongji University School of Medicine in Shanghai, China; and the latter from The First Hospital of China Medical University, Shenyang, China) in their article "An Efficient Weighted Graph Strategy to Identify Differentiation Associated Genes in Embryonic Stem Cells". [156] The First Systems of Weighted Differential and Integral Calculus [9] is cited by Christoffel Wilhelmus Janse Rensburg (North-West University at Potchefstroom, in South Africa) in his master-of-science dissertation (in computer science and information systems) "The relationship between process maturity models and the use and effectiveness of systems development methodologies". [247] The First Systems of Weighted Differential and Integral Calculus [9] is cited in the article "Ideality equation" by Alex Lyubomirskiy (GEN3 Partners). [273] The First Systems of Weighted Differential and Integral Calculus [9] is cited in the article "Service ratio-optimal, content coherence-aware data push systems" by Christos Liaskos (Foundation of Research and Technology in Hellas, Crete, Greece) and Ageliki Tsioliaridou (Democritus University of Thrace, Xanthi, Greece). [280] The First Systems of Weighted Differential and Integral Calculus [9] is cited in the article "Developing seismic vulnerability curves for typical Iranian buildings" by Mehdi Sadeghi (Islamic Azad University in Iran), Mohsen Ghafory-Ashtiany (International Institute of Earthquake Engineering and Seismology in Iran), and Naghmeh Pakdel-Lahiji (Islamic Azad University in Iran). [284] The First Systems of Weighted Differential and Integral Calculus [9] is cited in the journal Praxis der Mathematik. [79] Meta-Calculus: Differential and Integral [7] is cited in the journal Indian Journal of Theoretical Physics. [80] Each of the following two books is cited in the journal Publicationes Mathematicae. [56] 1) Non-Newtonian Calculus: Volume 19, page 351, 1972. 2) Bigeometric Calculus: A System with a Scale-Free Derivative: Volume 32, page 282, 1985.

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Prey (2006) is based aboard a cybernetic moon size space ship where things like gravity and even space-time are not consistent. Surprisingly, their time evolution can be analyzed by employing a non-Newtonian calculus ..." - Wojbor Woyczynski, Case Western Reserve University, USA; from an abstract to his 2013 seminar [146]. "The goal of this paper is chaos examination in multiplicative dynamical systems described with 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 [the bigeometric] calculus."

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By classical results [Merton 1969] both in discrete and continuous time, the optimal portfolio strategy in a friction-less market is is given by keeping the proportion between the wealth invested in the stock and the total portfolio wealth constant. Blyumin. "Binary arithmetic operations - Functional Equations" (Russian:"БИНАРНЫЕ АРИФМЕТИЧЕСКИЕ ОПЕРАЦИИ"), http://www.stu.lipetsk.ru/files/materials/2415/2008_01_010.pdf, Mathematical Modelling, 38, UDC 512.534.1, (Russian: МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ, 38, УДК,512.534.1), News of Universities Chernozemya, Number 1 (11), Lipetsk State Technical University, 2008. [181] Diana Andrada Filip and Cyrille Piatecki. "An overview on non-Newtonian calculus and its potential applications to economics", halshs.archives-ouvertes.fr/docs/00/94/57/88/PDF/nncam.pdf, Applied Mathematics - A Journal of Chinese Universities, Volume 28, China Society for Industrial and Applied Mathematics, Springer, 2014. [182] Xiaoju He, Meimei Song, and Danping Chen. "Common fixed points for weak commutative mappings on a multiplicative metric space", Fixed Point Theory and Applications, http://www.fixedpointtheoryandapplications.com/content/2014/1/48, Springer, 2014. [183] Ugur Kadak. "Determination of the Kothe-Toeplitz duals over the non-Newtonian complex field", National Center for Biotechnology Information, PMC4085728, Scientific World Journal, doi:10.1155/2014/438924, 2014. [184] Dorota Aniszewska and Marek Rybaczuk. "Fractal characteristics of defects evolution in parallel fibre reinforced composite in quasi-static process of fracture", Theoretical and Applied Fracture Mechanics, Elsevier, 2009. [185] Ali Uzer. "Improvement of the diffraction coefficient of GTD [geometrical theory of diffraction] by using multiplicative calculus", IEEE Transactions on Antennas and Propagation, Volume 62, Issue 7, ISSN 0018-926X, DOI 10.1109/TAP.2014.2319852, IEEE Antennas and Propagation Society, July of 2014. [186] Uğur Kadak, Feyzi Başar & Hakan Efe. “Construction of the duals of classical sets of sequences and related matrix transformations with non-Newtonian calculus”, Algerian Turkish International Days on Mathematics, Fatih University in Turkey, 2013. [187] Ugur Kadak. "Non-Newtonian analysis and its applications", doctoral thesis, Gazi University in Turkey, http://websitem.gazi.edu.tr/site/ugurkadak/posts/view/id/81101, 2011. [188] Khiord Boruah. "Difference sequence spaces and non-Newtonian calculus", National Conference on Recent Trends of Mathematics and its Applications, RTMA-2014, Rajiv Gandhi University in India, May of 2014. [189] Ugur Kadak and Hakan Efe. "Matrix transformations between certain sequence spaces over the non-Newtonian complex field", National Center for Biotechnology Information, PMC4090579, Scientific World Journal, doi:10.1155/2014/705818, 2014. [190] Ugur Kadak and Hakan Efe. "The construction of Hilbert spaces over the non-Newtonian field", International Journal of Analysis, http://www.researchgate.net/publication/262919697_The_construction_of_Hilbert_spaces_over_the_non-Newtonian_field, Hindawi Publishing Corporation, 2014. [191] Ali Ozyapici. "Multiplicative calculus and its applications", doctoral thesis, Ege University in Turkey, http://www.yarbis.yildiz.edu.tr/common/uploads/242578d8ce/TezGosterdoktora.pdf, 2009. [192] David Godes. "Product policy in markets with word-of-mouth communication", Social Science Research Network, SSRN-id2275617, 2013. [193] Duff Campbell. "Multiplicative calculus and student projects", Primus, Volume 9, Issue 4, 1999. [194] Paolo Perrone. "A gauge theoretic approach to quantum physics", INFN-Istituto Nazionale di Fisica Nucleare (Italian National Institute for Nuclear Physics), www.infn.it/thesis/PDF/getfile.php?filename=8193--laurea.pdf, 2013. [195] Luc Florack. "Neuro and cardio imaging", BIRS Workshop, 11w5018, Banff International Research Station for Mathematical Innovation and Discovery, Banff (Alberta, Canada), 2011. [196] H.