arXiv subjects
Publications and source records attributed to Ram N. Mohapatra.
In this paper, we introduce the generalized Cauchy dual $w(T) = T(T^{*}T)^{\dagger}$ of a closed operator $T$ with the closed range between Hilbert spaces and present intriguing findings that characterize the Cauchy dual of $T$. Additionally, we establish the result $w(T^{n}) = (w(T))^{n}$, for all $n \in \mathbb{N}$, where $T$ is a quasinormal EP operator.
The theory and computation of tensors with different tensor products play increasingly important roles in scientific computing and machine learning. Different products aim to preserve different algebraic properties from the matrix algebra, and the choice of tensor product determines the algorithms that can be directly applied. This study introduced a novel full-rank decomposition and $M$-$\mc{QDR}$ decomposition for third-order tensors based on $M$-product. Then, we designed algorithms for computing these two decompositions along with the Moore-Penrose inverse, and outer inverse of the tensors. In support of these theoretical results, a few numerical examples were discussed. In addition, we derive exact expressions for the outer inverses of tensors using symbolic tensor (tensors with polynomial entries) computation. We designed efficient algorithms to compute the Moore-Penrose inverse of symbolic tensors. The prowess of the proposed $M$-$\mc{QDR}$ decomposition for third-order tensors is applied to compress lossy color images.
In this paper, we discuss the Hyers-Ulam stability of closable (unbounded) operators with several interesting examples. We also present results pertaining to the Hyers-Ulam stability of the sum and product of closable operators to have the Hyers-Ulam stability and the necessary and sufficient conditions of the Schur complement and the quadratic complement of $2 \times 2$ block matrix $\mathcal A$ in order to have the Hyers-Ulam stability.
In this paper, we introduce new representation and characterization of the weighted core inverse of matrices. Several properties of these inverses and their interconnections with other generalized inverses are also explored. Through one-sided core and dual-core inverses, we have shown the existence of a generalized weighted Moore-Penrose inverse of matrices. Further, by applying a new representation and using the properties of the weighted core inverse of a matrix, we discuss some new results related to the reverse order law for these inverses.