The Experts below are selected from a list of 306 Experts worldwide ranked by ideXlab platform
Roman V Belavkin - One of the best experts on this subject based on the ideXlab platform.
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law of cosines and shannon Pythagorean Theorem for quantum information
International Conference on Geometric Science of Information, 2013Co-Authors: Roman V BelavkinAbstract:The concept of information distance in non-commutative setting is re-considered. Additive information, such as Kullback-Leibler divergence, is defined using convex functional with gradient having the property of homomorphism between multiplicative and additive subgroups. We review several geometric properties, such as the logarithmic law of cosines, Pythagorean Theorem and a lower bound given by squared Euclidean distance. We also prove a special case of Pythagorean Theorem for Shannon information, which finds applications in information-theoretic variational problems.
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GSI - Law of Cosines and Shannon-Pythagorean Theorem for Quantum Information
Lecture Notes in Computer Science, 2013Co-Authors: Roman V BelavkinAbstract:The concept of information distance in non-commutative setting is re-considered. Additive information, such as Kullback-Leibler divergence, is defined using convex functional with gradient having the property of homomorphism between multiplicative and additive subgroups. We review several geometric properties, such as the logarithmic law of cosines, Pythagorean Theorem and a lower bound given by squared Euclidean distance. We also prove a special case of Pythagorean Theorem for Shannon information, which finds applications in information-theoretic variational problems.
Temur Z Kalanov - One of the best experts on this subject based on the ideXlab platform.
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The Logical Analysis of the Pythagorean Theorem and of the Problem of Irrational Numbers
Journal of Mathematics and Physics, 2013Co-Authors: Temur Z KalanovAbstract:The critical analysis of the Pythagorean Theorem and of the problem of irrational numbers is proposed. Methodological basis for the analysis is the unity of formal logic and of rational dialectics. It is proved that: 1) the Pythagorean Theorem does not represent an absolute scientific truth; this Theorem represents a conventional (conditional) theoretical proposition because, in some cases, the Theorem contradicts the formal-logical laws and leads to the appearance of irrational numbers; 2) the standard theoretical proposition on the existence of incommensurable segments is a mathematical fiction, a logical error, a consequence of violation of the two formal-logical laws: the law of identity of geometrical forms and the law of lack of contradiction of geometrical forms; 3) the concept of irrational numbers represents a logical error: this concept is a result of violation of the dialectical unity of the qualitative aspect (i.e. form) and quantitative aspect (i.e. content: length, area) of geometric objects. Irrational number is image of calculation process and, therefore, do not exist on the number scale. There are only rational numbers.
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critical analysis of the mathematical formalism of theoretical physics ii Pythagorean Theorem
viXra, 2013Co-Authors: Temur Z KalanovAbstract:The critical analysis of the Pythagorean Theorem and of the problem of irrational numbers is proposed. Methodological basis for the analysis is the unity of formal logic and of rational dialectics. It is shown that: 1) the Pythagorean Theorem represents a conventional (conditional) theoretical proposition because, in some cases, the Theorem contradicts the formal-logical laws and leads to the appearance of irrational numbers; 2) the standard theoretical proposition on the existence of incommensurable segments is a mathematical fiction, a consequence of violation of the two formal-logical laws: the law of identity of geometrical forms and the law of lack of contradiction of geometrical forms; 3) the concept of irrational numbers is the result of violation of the dialectical unity of the qualitative aspect (i.e. form) and quantitative aspect (i.e. content: length, area) of geometric objects. Irrational numbers represent a calculation process and, therefore, do not exist on the number scale. There are only rational numbers.
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The Critical Analysis of the Pythagorean Theorem and of the Problem of Irrational Numbers
viXra, 2013Co-Authors: Temur Z KalanovAbstract:@@The critical analysis of the Pythagorean Theorem and of the problem of irrational numbers is proposed. Methodological basis for the analysis is the unity of formal logic and of rational dialectics. It is shown that: 1) the Pythagorean Theorem represents a conventional (conditional) theoretical proposition because, in some cases, the Theorem contradicts the formal-logical laws and leads to the appearance of irrational numbers; 2) the standard theoretical proposition on the existence of incommensurable segments is a mathematical fiction, a consequence of violation of the two formal-logical laws: the law of identity of geometrical forms and the law of lack of contradiction of geometrical forms; 3) the concept of irrational numbers is the result of violation of the dialectical unity of the qualitative aspect (i.e. form) and quantitative aspect (i.e. content: length, area) of geometric objects. Irrational numbers represent a calculation process and, therefore, do not exist on the number scale. There are only rational numbers.
Yusheng Wang - One of the best experts on this subject based on the ideXlab platform.
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Pythagorean Theorem & curvature with lower or upper bound
arXiv: Metric Geometry, 2019Co-Authors: Hongwei Sun, Yusheng WangAbstract:In this paper, we give a comparison version of Pythagorean Theorem to judge the lower or upper bound of the curvature of Alexandrov spaces (including Riemannian manifolds).
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Pythagorean Theorem curvature with lower or upper bound
arXiv: Metric Geometry, 2019Co-Authors: Hongwei Sun, Yusheng WangAbstract:In this paper, we give a comparison version of Pythagorean Theorem to judge the lower or upper bound of the curvature of Alexandrov spaces (including Riemannian manifolds).
Paolo Maraner - One of the best experts on this subject based on the ideXlab platform.
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A Spherical Pythagorean Theorem
The Mathematical Intelligencer, 2010Co-Authors: Paolo MaranerAbstract:for a spherical right triangle with hypotenuse c and legs a and b, is generally presented as the ‘spherical Pythagorean Theorem’. Still, it has to be remarked that this formula does not have an immediate meaning in terms of areas of simple geometrical figures, as the Pythagorean Theorem does. There is no diagram that can be drawn on the surface of the sphere to illustrate the statement in the spirit of ancient Greek geometry. In this note I reconsider the issue of extending the geometrical Pythagorean Theorem to non-Euclidean geometries (with emphasis on the more intuitive spherical geometry). In apparent contradiction with the statement that the Pythagorean proposition is equivalent to Euclid’s parallel postulate, I show that such an extension not only exists, but also yields a deeper insight into the classical Theorem. The subject matter being familiar, I can dispense with preliminaries and start right in with Euclid’s Elements [1].
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A spherical Pythagorean Theorem
The Mathematical Intelligencer, 2010Co-Authors: Paolo MaranerAbstract:for a spherical right triangle with hypotenuse c and legs a and b, is generally presented as the ‘spherical Pythagorean Theorem’. Still, it has to be remarked that this formula does not have an immediate meaning in terms of areas of simple geometrical figures, as the Pythagorean Theorem does. There is no diagram that can be drawn on the surface of the sphere to illustrate the statement in the spirit of ancient Greek geometry. In this note I reconsider the issue of extending the geometrical Pythagorean Theorem to non-Euclidean geometries (with emphasis on the more intuitive spherical geometry). In apparent contradiction with the statement that the Pythagorean proposition is equivalent to Euclid’s parallel postulate, I show that such an extension not only exists, but also yields a deeper insight into the classical Theorem. The subject matter being familiar, I can dispense with preliminaries and start right in with Euclid’s Elements [1].
Hongwei Sun - One of the best experts on this subject based on the ideXlab platform.
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Pythagorean Theorem & curvature with lower or upper bound
arXiv: Metric Geometry, 2019Co-Authors: Hongwei Sun, Yusheng WangAbstract:In this paper, we give a comparison version of Pythagorean Theorem to judge the lower or upper bound of the curvature of Alexandrov spaces (including Riemannian manifolds).
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Pythagorean Theorem curvature with lower or upper bound
arXiv: Metric Geometry, 2019Co-Authors: Hongwei Sun, Yusheng WangAbstract:In this paper, we give a comparison version of Pythagorean Theorem to judge the lower or upper bound of the curvature of Alexandrov spaces (including Riemannian manifolds).
