The Experts below are selected from a list of 1104 Experts worldwide ranked by ideXlab platform
Pierre Martinetti - One of the best experts on this subject based on the ideXlab platform.
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On Pythagoras Theorem for Products of Spectral Triples
Letters in Mathematical Physics, 2013Co-Authors: Francesco D’andrea, Pierre MartinettiAbstract:We discuss a version of Pythagoras Theorem in noncommutative geometry. Usual Pythagoras Theorem can be formulated in terms of Connes’ distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non-pure states and arbitrary spectral triples. We show that Pythagoras Theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and we provide non-unital counter-examples inspired by K -homology.
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On Pythagoras' Theorem for products of spectral triples
Letters in Mathematical Physics, 2012Co-Authors: Francesco D'andrea, Pierre MartinettiAbstract:We discuss a version of Pythagoras Theorem in noncommutative geometry. Usual Pythagoras Theorem can be formulated in terms of Connes' distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non pure states and arbitrary spectral triples. We show that Pythagoras Theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and provide non-unital counter-examples inspired by K-homology.
Francesco D'andrea - One of the best experts on this subject based on the ideXlab platform.
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Pythagoras Theorem in Noncommutative Geometry
arXiv: Mathematical Physics, 2015Co-Authors: Francesco D'andreaAbstract:After a review of the results in arXiv:1203.3184 [math-ph] about Pythagorean inequalities for products of spectral triples, I will present some new results and discuss classes of spectral triples and states for which equality holds.
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On Pythagoras' Theorem for products of spectral triples
Letters in Mathematical Physics, 2012Co-Authors: Francesco D'andrea, Pierre MartinettiAbstract:We discuss a version of Pythagoras Theorem in noncommutative geometry. Usual Pythagoras Theorem can be formulated in terms of Connes' distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non pure states and arbitrary spectral triples. We show that Pythagoras Theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and provide non-unital counter-examples inspired by K-homology.
Francesco D’andrea - One of the best experts on this subject based on the ideXlab platform.
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On Pythagoras Theorem for Products of Spectral Triples
Letters in Mathematical Physics, 2013Co-Authors: Francesco D’andrea, Pierre MartinettiAbstract:We discuss a version of Pythagoras Theorem in noncommutative geometry. Usual Pythagoras Theorem can be formulated in terms of Connes’ distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non-pure states and arbitrary spectral triples. We show that Pythagoras Theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and we provide non-unital counter-examples inspired by K -homology.
Norman J. Wildberger - One of the best experts on this subject based on the ideXlab platform.
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Rational Trigonometry in Higher Dimensions and a Diagonal Rule for 2-planes in Four-dimensional space
KoG, 2017Co-Authors: Norman J. WildbergerAbstract:We extend rational trigonometry to higher dimensions by introducing rational invariants between $k$-subspaces of $n$-dimensional space to give an alternative to the canonical or principal angles studied by Jordan and many others, and their angular variants. We study in particular the cross, spread and det-cross of $2$-subspaces of four-dimensional space, and show that Pythagoras Theorem, or the Diagonal Rule, has a natural generalization forsuch $2$-subspaces.
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Chromogeometry and Relativistic Conics
2009Co-Authors: Norman J. WildbergerAbstract:If Q1 = Q(A2,A3), Q2 = Q(A1,A3) and Q3 = Q(A1,A2) are the quadrances of a triangle A1A2A3, then Pythagoras’ Theorem and its converse can together be stated as: A1A3 is perpendicular to A2A3 precisely when Q1 + Q2 = Q3. Figure 1 shows an example where Q1 = 5, Q2 = 20 and Q3 = 25. As indicated for the large square, these areas may also be calculated by subdivision and (translational) rearrangement, followed by counting cells. There is a sister Theorem—the Triple quad formula—that Euclid did not know, but which is fundamental for rational trigonometry, introduced in [2]: A1A3 is parallel to A2A3 precisely when
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The ancient Greeks present: Rational Trigonometry
arXiv: Metric Geometry, 2008Co-Authors: Norman J. WildbergerAbstract:Pythagoras' Theorem, the area of a triangle as one half the base times the height, and Heron's formula are amongst the most important and useful results of ancient Greek geometry. Here we look at all three in a new and improved light, using quadrance not distance. This leads to a simpler and more elegant trigonometry, in which angle is replaced by spread, and which extends to arbitrary fields and more general quadratic forms.
Ramkesh Sharma - One of the best experts on this subject based on the ideXlab platform.
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Applications of the Pythagoras Theorem
International Journal of Research, 2014Co-Authors: Shubham Chauhan, Tushar Jain, Ramkesh SharmaAbstract:Investigate the history of Pythagoras and the Pythagorean Theorem. We analyzed the information on the Pythagorean Theorem including not only the meaning and application of the Theorem, but also the proofs.
