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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., Comment: 23

The outer inverse of tensors plays increasingly significant roles in computational mathematics, numerical analysis, and other generalized inverses of tensors. In this paper, we compute outer inverses with prescribed ranges and kernels of a given tensor through tensor QR decomposition and hyperpower iterative method under the M-product structure, which is a family of tensor-tensor products, generalization of the t-product and c-product, allows us to suit the physical interpretations across those different modes. We discuss a theoretical analysis of the nineteen-order convergence of the proposed tensor-based iterative method. Further, we design effective tensor-based algorithms for computing outer inverses using M-QR decomposition and hyperpower iterative method. The theoretical results are validated with numerical examples demonstrating the appropriateness of the proposed methods., Comment: 23 pages

This paper introduces notions of the Drazin and the core-EP inverses on tensors via M-product. We propose a few properties of the Drazin and core-EP inverses of tensors, as well as effective tensor-based algorithms for calculating these inverses. In addition, definitions of composite generalized inverses are presented in the framework of the M-product, including CMP, DMP, and MPD inverse of tensors. Tensor-based higher-order Gauss-Seidel and Gauss-Jacobi iterative methods are designed. Algorithms for these two iterative methods to solve multilinear equations are developed. Certain multilinear systems are solved using the Drazin inverse, core-EP inverses, and composite generalized inverses such as CMP, DMP, and MPD inverse. A tensor M-product-based regularization technique is applied to solve the color image deblurring., Comment: 27

The main objective of this paper is to introduce unique representations and characterizations for the weighted core inverse of matrices. We also investigate various properties of these inverses and their relationships with other generalized inverses. Proposed representations of the matrix-weighted core inverse will help us to discuss some results associated with the reverse order law for these inverses. Furthermore, this paper introduces an extension of the concepts of generalized bilateral inverse and $\{1,2,3,1^k\}$-inverse and their respective dual for complex rectangular matrices. Furthermore, we establish characterizations of EP-ness and the condition when both $W$-weighted $\{1,2,3\}$ and $W$-weighted $\{1,2,3,1^k\}$ inverses coincide. Then, a W-weighted index-MP, W-weighted MP-index, and W-weighted MP-index-MP matrices for rectangular complex matrices is introduced. In addition, we define the dual inverses for both weighted bilateral inverses and $\{1,2,3,1^k\}$-inverse. Characteristics that lead to self-duality in weighted bilateral inverses are also examined., Comment: 24 Pages

Mosic and Djordjevic introduced the notation of the gDMP inverse for Hilbert space operators in [J. Spectr. Theory, 8(2):555-573, 2018] by considering generalized Drazin inverse with the Moore-Penrose inverse. This paper introduces two new classes of inverses: GD1 (generalized Drazin and inner) inverse and 1GD (inner and generalized Drazin) inverse for Hilbert space operators. The existence and uniqueness of the GD1 (also 1GD) inverse are discussed, along with some properties through core-quasinilpotent decomposition and closed range decomposition operator. We further establish a few explicit representations of the GD1 inverse and their interconnections with generalized Drazin inverse. In addition, we discuss a few properties of GD1 (also 1GD) inverse through binary relation., Comment: 15 pages

Limited urban parking space combined with urbanization has necessitated the development of smart parking systems that can communicate the availability of parking slots to the end users. Towards this, various deep learning based solutions using convolutional neural networks have been proposed for parking space occupation detection. Though these approaches are robust to partial obstructions and lighting conditions, their performance is found to degrade in the presence of haze conditions. Looking in this direction, this paper investigates the use of dehazing networks that improves the performance of parking space occupancy classifier under hazy conditions. Additionally, training procedures are proposed for dehazing networks to maximize the performance of the system on both hazy and non-hazy conditions. The proposed system is deployable as part of existing smart parking systems where limited number of cameras are used to monitor hundreds of parking spaces. To validate our approach, we have developed a custom hazy parking system dataset from real-world task-driven test set of RESIDE-\b{eta} dataset. The proposed approach is tested against existing state-of-the-art parking space detectors on CNRPark-EXT and hazy parking system datasets. Experimental results indicate that there is a significant accuracy improvement of the proposed approach on the hazy parking system dataset., Comment: 20 pages

The notion of weighted $(b,c)$-inverse of an element in rings were introduced, very recently [Comm. Algebra, 48 (4) (2020): 1423-1438]. In this paper, we further elaborate on this theory by establishing a few characterizations of this inverse and their relationships with other $(v, w)$-weighted $(b,c)$-inverses. We introduce some necessary and sufficient conditions for the existence of the hybrid $(v, w)$-weighted $(b,c)$-inverse and annihilator $(v, w)$-weighted $(b,c)$-inverse of elements in rings. In addition to this, we explore a few sufficient conditions for the reverse-order law of the annihilator $(v, w)$-weighted $(b,c)$-inverses., Comment: 19

Generalized inverses of tensors play increasingly important roles in computational mathematics and numerical analysis. It is appropriate to develop the theory of generalized inverses of tensors within the algebraic structure of a ring. In this paper, we study different generalized inverses of tensors over a commutative ring and a non-commutative ring. Several numerical examples are provided in support of the theoretical results. We also propose algorithms for computing the inner inverses, the Moore-Penrose inverse, and weighted Moore-Penrose inverse of tensors over a non-commutative ring. The prowess of some of the results is demonstrated by applying these ideas to solve an image deblurring problem., Comment: 23

In this paper, we introduce the notion of weak core and central weak core inverse in a {\it proper $*$-ring}. We further elaborate on these two classes by producing a few representations and characterizations of the weak core and central weak core invertible elements. We investigated additive properties and a few explicit expressions for these two classes of inverses through other generalized inverses. In addition, numerical examples are provided to validate claims on weak core inverses. Following {\it proper $*$-ring} and their interconnections with Clifford algebra, we also present examples of the group inverse and the weak core inverse of a non-zero non-invertible quaternion $\mathbb{H}_s$., Comment: 22 pages, 1 figure

In this paper, we introduce the notation of $E$-weighted core-EP and $F$-weighted dual core-EP inverse of matrices. We then obtain a few explicit expressions for the weighted core-EP inverse of matrices through other generalized inverses. Further, we discuss the existence of generalized weighted Moore-Penrose inverse and additive properties of the weighted core-EP inverse of matrices. In addition to these, we propose the star weighted core-EP and weighted core-EP star class of matrices for solving the system of matrix equations. We further elaborate on this theory by producing a few representation and characterization of star weighted core-EP and weighted core-EP star classes of matrices., Comment: 20 Pages

The notion of the weighted core inverse in a ring with involution was introduced, recently [Mosic et al. Comm. Algebra, 2018; 46(6); 2332-2345]. In this paper, we explore new representation and characterization of the weighted core inverse of sum and difference of two weighted core invertible elements in an unital ring with involution under different conditions. Further, we discuss reverse order laws and mixed-type reverse order laws for the weighted core invertible elements in a ring., Comment: 20 Pages

We employ both random forests and LSTM networks (more precisely CuDNNLSTM) as training methodologies to analyze their effectiveness in forecasting out-of-sample directional movements of constituent stocks of the S&P 500 from January 1993 till December 2018 for intraday trading. We introduce a multi-feature setting consisting not only of the returns with respect to the closing prices, but also with respect to the opening prices and intraday returns. As trading strategy, we use Krauss et al. (2017) and Fischer & Krauss (2018) as benchmark. On each trading day, we buy the 10 stocks with the highest probability and sell short the 10 stocks with the lowest probability to outperform the market in terms of intraday returns -- all with equal monetary weight. Our empirical results show that the multi-feature setting provides a daily return, prior to transaction costs, of 0.64% using LSTM networks, and 0.54% using random forests. Hence we outperform the single-feature setting in Fischer & Krauss (2018) and Krauss et al. (2017) consisting only of the daily returns with respect to the closing prices, having corresponding daily returns of 0.41% and of 0.39% with respect to LSTM and random forests, respectively.

In this paper, the concept of matrix splitting is introduced to solve a large sparse ill-posed linear system via Tikhonov's regularization. In the regularization process, we convert the ill-posed system to a well-posed system. The convergence of such a well-posed system is discussed by using different types of matrix splittings. Comparison analysis of both systems are studied by operating certain types of weak splittings. Further, we have extended the double splitting of [Song J and Song Y, Calcolo 48(3), 245-260, 2011] to double weak splitting of type II for nonsingular symmetric matrices. In addition to that, some more comparison results are presented with the help of such weak double splittings of type I and type II.

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., Comment: 18 pages

Stock prediction is a topic undergoing intense study for many years. Finance experts and mathematicians have been working on a way to predict the future stock price so as to decide to buy the stock or sell it to make profit. Stock experts or economists, usually analyze on the previous stock values using technical indicators, sentiment analysis etc to predict the future stock price. In recent years, many researches have extensively used machine learning for predicting the stock behaviour. In this paper we propose data driven deep learning approach to predict the future stock value with the previous price with the feature extraction property of convolutional neural network and to use Neural Arithmetic Logic Units with it., Comment: Accepted at ICACDS 2019 - Springer CCIS

The notion of the Drazin inverse of an even-order tensor with the Einstein product was introduced, very recently [J. Ji and Y. Wei. Comput. Math. Appl., 75(9), (2018), pp. 3402-3413]. In this article, we further elaborate this theory by producing a few characterizations of the Drazin inverse and the W-weighted Drazin inverse of tensors. In addition to these, we compute the Drazin inverse of tensors using different types of generalized inverses and full rank decomposition of tensors. We also address the solution to the multilinear systems using the Drazin inverse and iterative (higher order Gauss-Seidel) method of tensors. Besides this, the convergence analysis of the iterative technique is also investigated within the framework of the Einstein product., Comment: 30

Applications of the theory and computations of boolean matrices are of fundamental importance to study a variety of discrete structural models. But the increasing ability of data collection systems to store huge volumes of multidimensional data, the boolean matrix representation of data analysis is not enough to represent all the information content of the multiway data in different fields. From this perspective, it is appropriate to develop an infrastructure that supports reasoning about the theory and computations. In this paper, we discuss the generalized inverses of the Boolean tensors with the Einstein product. Further, we elaborate on this theory by producing a few characterizations of different generalized inverses and several equivalence results on boolean tensors. In addition to these, we define the rank of a boolean tensor through space decomposition., Comment: 29 Pages

Iterative methods based on matrix splittings are useful in solving large sparse linear systems. In this direction, proper splittings and its several extensions are used to deal with singular and rectangular linear systems. In this article, we introduce a new iteration scheme called three-step alternating iterations using proper splittings and group inverses to find an approximate solution of singular linear systems, iteratively. A preconditioned alternating iterative scheme is also proposed to relax some sufficient conditions and to obtain faster convergence as well. We then show that our scheme converges faster than the existing one. The theoretical findings are then validated numerically.

In this paper, we propose an improved version of the power index related to the Banzhaf power index for weighted voting systems. This index now takes into account the mutual persuasion power matrix(PPM) existing among the voters. This improved index is calculated for European Union voting by basing the PPM on immigration data among the EU countries. We also provide better approximation bounds for the Monte Carlo approximation method for computing power indices., Comment: 15 pages, 3 tables