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A New Approach with Software Implementation to Extend the Pythagorean Theorem in Multi-Dimensions
Proceedings of the International Conference on Applied Innovations in ITThe Pythagorean theorem, which states that "for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the lengths of the two legs," applies only within a single plane (i.e., in two dimensions).
Abd Ali Hussein +5 more
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Non-Euclidean Pythagorean triples, a problem of Euler, and rational points on K3 surfaces
, 2006 We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem ...
Robin Hartshorne, Ronald, Van Luijk
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Pythagorean Theorem Concept Image in Junior High School: An Analysis in The Online-Based Learning
, 2021 Mohamad Gilar Jatisunda +2 more
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Norm groups and class fields of formally real quasilocal fields
, 2008 This paper establishes a relationship between finite extensions and norm groups of formally real quasilocal fields, which yields a generally nonabelian local class field theory, including analogues to the fundamental correspondence, the local reciprocity
Chipchakov, I. D.
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The Pythagorean Theorem: I. The finite case [PDF]
Proceedings of the National Academy of Sciences, 2002 The Pythagorean Theorem and variants of it are studied. The variations evolve to a formulation in terms of noncommutative, conditional expectations on von Neumann algebras that displays the theorem as the basic result of noncommutative, metric, Euclidean Geometry.
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A Puzzle: How to Prove the Pythagorean Theorem in Antenna Analysis? [Letter to the Editor] [PDF]
, 2013 Tomasz Długosz
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A Feature Extraction Method Using Improved Multi-Scale Entropy for Rolling Bearing Fault Diagnosis
Entropy, 2018 A feature extraction method named improved multi-scale entropy (IMSE) is proposed for rolling bearing fault diagnosis. This method could overcome information leakage in calculating the similarity of machinery systems, which is based on Pythagorean ...
Bin Ju +5 more
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Generalized Pythagorean Theorem and Generalized Derivative [PDF]
, 2019 Zhou Zhongwang
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Geometry of q-Exponential Family of Probability Distributions
Entropy, 2011 The Gibbs distribution of statistical physics is an exponential family of probability distributions, which has a mathematical basis of duality in the form of the Legendre transformation.
Shun-ichi Amari, Atsumi Ohara
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