Version 1
: Received: 17 October 2023 / Approved: 18 October 2023 / Online: 19 October 2023 (02:44:59 CEST)
Version 2
: Received: 24 October 2023 / Approved: 25 October 2023 / Online: 25 October 2023 (08:32:39 CEST)
How to cite:
Amiri, M. Joint Probability Densities as Symmetric Covariant Tensor Densities on Riemannian Manifold. Preprints2023, 2023101154. https://doi.org/10.20944/preprints202310.1154.v2
Amiri, M. Joint Probability Densities as Symmetric Covariant Tensor Densities on Riemannian Manifold. Preprints 2023, 2023101154. https://doi.org/10.20944/preprints202310.1154.v2
Amiri, M. Joint Probability Densities as Symmetric Covariant Tensor Densities on Riemannian Manifold. Preprints2023, 2023101154. https://doi.org/10.20944/preprints202310.1154.v2
APA Style
Amiri, M. (2023). Joint Probability Densities as Symmetric Covariant Tensor Densities on Riemannian Manifold. Preprints. https://doi.org/10.20944/preprints202310.1154.v2
Chicago/Turabian Style
Amiri, M. 2023 "Joint Probability Densities as Symmetric Covariant Tensor Densities on Riemannian Manifold" Preprints. https://doi.org/10.20944/preprints202310.1154.v2
Abstract
This paper presents the tensor properties of joint probability densities on a Riemannian manifold. Initially we develop a binary data matrix to record variables' values of a large number of particles confining in a closed system at a certain time in order to retrieve the joint probability densities of related variables. By introducing the particle oriented coordinate and generalized inner product as a multi-linear operation on the basis of this coordinate, we extract the set of joint probabilities and prove them to meet covariant tensor properties on a general Riemannian space of variables. Based on the Taylor expansion of scalar field in Riemannian manifolds, it has been shown that the symmetrized iterative covariant derivatives of cumulative probability function defined on Riemannian manifold also gives the set of related joint probability densities equivalent to the aforementioned multi-linear method. We show these covariant tensors reduce to classical ordinary partial derivatives in ordinary Euclidean space with Cartesian coordinates and gives the formal definition of joint probabilities by partial derivatives of cumulative distribution function. The equivalence between symmetrized covariant derivative and generalized inner product has been concluded. Some examples of well-known physical tensors clarify that many deterministic physical variables are presented as tensor densities with an interpretation similar to probability densities.
Keywords
Joint probability density; Covariant Tensor; Riemannian manifold; Symmetrized Covariant derivative; Taylor expansion
Subject
Computer Science and Mathematics, Applied Mathematics
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.
Commenter: Manouchehr Amiri
Commenter's Conflict of Interests: Author