Version 1
: Received: 28 October 2016 / Approved: 31 October 2016 / Online: 31 October 2016 (02:39:45 CET)
How to cite:
Amiri, M. Tensor Properties of Joint Probability Densities on Riemannian Parametric Manifold. Preprints2016, 2016100131. https://doi.org/10.20944/preprints201610.0131.v1
Amiri, M. Tensor Properties of Joint Probability Densities on Riemannian Parametric Manifold. Preprints 2016, 2016100131. https://doi.org/10.20944/preprints201610.0131.v1
Amiri, M. Tensor Properties of Joint Probability Densities on Riemannian Parametric Manifold. Preprints2016, 2016100131. https://doi.org/10.20944/preprints201610.0131.v1
APA Style
Amiri, M. (2016). Tensor Properties of Joint Probability Densities on Riemannian Parametric Manifold. Preprints. https://doi.org/10.20944/preprints201610.0131.v1
Chicago/Turabian Style
Amiri, M. 2016 "Tensor Properties of Joint Probability Densities on Riemannian Parametric Manifold" Preprints. https://doi.org/10.20944/preprints201610.0131.v1
Abstract
We show the tensor properties of joint probabilities on a Riemannian parametric manifold. Initially we develop a binary data matrix for parameter measurements of a large number of particles confining in a closed system in order to retrieve the joint probability densities of related parameters. By introducing a new generalized inner product as a multilinear operation on dual basis of parametric space, we extract the set of joint probabilities and prove them to meet contravariant tensor properties in a general Riemannian parametric space. We show these contravariant tensors reduce to classical ordinary partial derivatives definition in ordinary Euclidean parametric space. Finally we prove by a theorem that symmetrized iterative contravariant derivative of cumulative probability function on Riemannian manifold gives the set of joint probabilities in those manifold. We bring some examples for compatibility with physical tensors.
Keywords
joint probability density; tensor properties; probability on Riemannian manifold; covariant derivative; contravariant derivative
Subject
Computer Science and Mathematics, Geometry and Topology
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
