Columns Of Matrix Research Articles

Effect of Markov Chain and System of Linear Equations on the Analysis of Nigerian Current Account Net

The idea behind Markov chain model and system of equations cannot be emphasized in terms of modeling physical systems like the Nigerian Current Account formation; since each finite state communicates for suitable decisions making. So this paper studied stochastic analysis of Markov chain on NCA data (2004-2022). The NCA data were transformed into 3-steps transition probability matrix solution to cover the number of independent years. From the transition matrix of stochastic analysis showed that in 2004-2012 has the highest probability of reducing in payments by 72%, the year 2005-2013 has the highest probability of reducing by 66% and finally in 2014-2022 has the highest probability of no-change in payments of goods and services by 3.3%. The future NCA data changes were known by introducing the concepts of percentages in each interval of years as column vectors where system of linear equations were developed and solutions were obtained to guide Nigerian economy on various levels of imports and exports of goods and services. Lastly, other statistical disparities were also considered and well discussed accordingly. These results are informative to Nigerian economy to effectively take vital decisions in their investment plans.

Superpixels meet essential spectra for fast Raman hyperspectral microimaging.

In the context of spectral unmixing, essential information corresponds to the most linearly dissimilar rows and/or columns of a two-way data matrix which are indispensable to reproduce the full data matrix in a convex linear way. Essential information has recently been shown accessible on-the-fly via a decomposition of the measured spectra in the Fourier domain and has opened new perspectives for fast Raman hyperspectral microimaging. In addition, when some spatial prior is available about the sample, such as the existence of homogeneous objects in the image, further acceleration for the data acquisition procedure can be achieved by using superpixels. The expected gain in acquisition time is shown to be around three order of magnitude on simulated and real data with very limited distortions of the estimated spectrum of each object composing the images.

Constructing and Machine Learning Calabi‐Yau Five‐Folds

AbstractMotivated by their role in M‐theory, F‐theory, and S‐theory compactifications, all possible complete intersections Calabi‐Yau five‐folds in a product of four or less complex projective spaces are constructed, with up to four constraints. A total of 27 068 spaces are obtained, which are not related by permutations of rows and columns of the configuration matrix, and determine the Euler number for all of them. Excluding the 3909 product manifolds among those, the cohomological data for 12 433 cases are calculated, i.e., 53.7% of the non‐product spaces, obtaining 2375 different Hodge diamonds. The dataset containing all the above information is available here. The distributions of the invariants are presented, and a comparison with the lower‐dimensional analogues is discussed. Supervised machine learning is performed on the cohomological data, via classifier, and regressor (both fully connected and convolutional) neural networks. h1, 1 can be learnt very efficiently, with very high R2 score and an accuracy of 96% is found, i.e., 96% of the predictions exactly match the correct values. For , very high R2 scores are also found, but the accuracy is lower, due to the large ranges of possible values.

Post-processing for Bayesian analysis of reduced rank regression models with orthonormality restrictions

Orthonormality constraints are common in reduced rank models. They imply that matrix-variate parameters are given as orthonormal column vectors. However, these orthonormality restrictions do not provide identification for all parameters. For this setup, we show how the remaining identification issue can be handled in a Bayesian analysis via post-processing the sampling output according to an appropriately specified loss function. This extends the possibilities for Bayesian inference in reduced rank regression models with a part of the parameter space restricted to the Stiefel manifold. Besides inference, we also discuss model selection in terms of posterior predictive assessment. We illustrate the proposed approach with a simulation study and an empirical application.

Reconstruction of Supersonic Flowfield Using Physical Neural Network Based on Channel Interaction

The evolution of shock waves in a combustor is a complex process comprising multiple time series, which results in a long computational cycle for unsteady numerical simulations. Current data-driven methods integrating physical information provide novel opportunities to rapidly and accurately predict the flowfield in the combustor. In this study, we construct a dataset under Mach numbers from 2 to 5 based on a self-designed scramjet combustor model and propose a physical convolutional neural network flowfield reconstruction method based on channel interaction (PICNNCI). This method embeds Euler equations that characterize the flow characteristics of inviscid fluids as physical constraints to solve the low-accuracy problem of traditional convolutional neural networks arising due to insufficient prior information. The model first performs dimensional transformations on temporal and spatial information input and converts a one-dimensional column vector into a two-dimensional vector to meet the convolution operation rules. Then, convolution is used for feature extraction to output the predicted flowfield, including velocity, pressure, and density fields. Finally, the predicted flowfield is optimized using the Euler equation and compared with the results obtained by numerical calculation and traditional neural network methods. The average peak signal-to-noise ratio of PICNNCI reaches 34.01 dB, and the linear correlation reaches 0.994.

An innovative orthogonal matrix based on nonlinear chaotic system for compressive sensing

Compressed sensing exploits the signal’s sparsity by non-uniform sampling to achieve high-quality signal reconstruction at low sampling rates. This work aims to show the efficient performance of chaotic binary orthogonal matrices (CBOM) in compressed sensing. The nonlinear, high dimensional, irregular, and high complexity properties of chaotic systems can provide more diverse and efficient ways of sampling and reconstructing signals. The CBOM construction method proposed in this paper is divided into two steps, in the first step, the real-valued sequence of a one-dimensional chaotic map is binarised using the proposed Threshold-Matching Symbol Algorithm (TMSA) to obtain a chaotic binary sequence (CBS). The i.i.d properties of the CBS were proved using the Perron–Frobenius operator and the properties of the joint probability. The binarized CBS conditionally preserves the pseudo-random of chaotic sequences, as evidenced by the derivation of the well-distribution measure and the k-order correlation measure. In the second step, the binarised sequence CBS was split and orthogonalized to construct CBOM, which satisfies low storage and correlation. We prove that CBOM obeys the Restricted Isometric Condition (RIP). The orthogonalization of the matrix will further reduce matrix column correlation and improve the quality of the reconstruction. Numerical simulation results show that the proposed matrix has considerable sampling efficiency, comparable to Gaussian and partial Hadamard matrices, close to the theoretical limit. Meanwhile, the generation and reconstruction time of the proposed matrix is smaller than other matrices. Our framework covers partial one-dimensional chaotic maps, including Chebyshev, Tent, Logistic, and so on. We can easily apply this paradigm to various fields.

APPLICATION OF SINGLE STEP EULER AND TRAPEZOID METHODS OF SOLVING TRANSIENT DISTRIBUTION IN MARKOV CHAIN

The computation of state probability distributions at an arbitrary point in time, which in the case of a discrete-time Markov chain means finding the distribution at some arbitrary time step n denoted π^((n) ), a row vector whose i^th component is the probability that the Markov chain is in state i at time step n, and this is the iterative solution methods for transient distribution in Markov chain. In this study, the solutions of transient distribution in Markov chain using Euler and trapezoid methods have been investigated, in order to provide some insight into the solutions of transient distribution in Markov chain, which produce a significantly more accurate response in less time for some types of situations and also tries to get to the end result as quickly as possible with the solution conform with a specified number of well-defined stages is computed for large space Markov chain. Matrices operations, such as the product and matrix inversion are performed and Markov chain laws, theorems, formulas with MATLAB software are utilized. For illustrative examples, the transient distribution vector's π_((i+1) ),i=0,1,2,…, ; is computed for both Euler and Trapezoid methods and their corresponding error when compared with different method. The effect of decreasing the step size using the same infinitesimal generator is examined, also, by taking π_((0) )=(■(1&0&0)) and the length of the interval of integration to be τ = 1, it is observed that the accuracy achieved with the explicit Euler method has a direct relationship with the step size. It is also concluded that the much more accurate results are obtained with the trapezoid method.

Exploring a novel tubular-type modular reactor for solar-driven thermochemical energy storage

Thermochemical energy storage (TCES) has gained extensive attention as a potential solution to address the mismatch between solar thermal energy production and demand. In this study, a novel tubular-type modular TCES reactor is introduced. COMSOL modelling of the system is developed and experimentally validated using a laboratory-scale TCES system. Both types of reactors show similar temperature increases, intensifying with higher inlet relative humidity. Their maximum temperature lifts exceeding 26 °C at 90 % RH. Tubular designs offer better axial flexural strength and dispersion of TCES composite materials compared to plate structures. This property of tubular structures beneficial reducing bed thickness and pressure drop and enhancing equivalent thermal efficiency. Simulations show tubular-type modular reactors reduce pressure drop by 4–5 times compared to plate-type modular reactors, increasing equivalent thermal efficiency by nearly 7% points. Increasing the number of reactor beds and inner tube radius improves equivalent thermal efficiency due to reduced bed thickness and pressure drop. As the number of matrix rows and columns in the reactor bed increases from 2 to 10, bed thickness decreases from 0.058 m to 0.012 m, reducing pressure drop from 845.53 Pa to 38 Pa and increasing equivalent thermal efficiency from 78.82 % to 96.61 %.

Convex Subspace Clustering by Adaptive Block Diagonal Representation.

Subspace clustering is a class of extensively studied clustering methods where the spectral-type approaches are its important subclass. Its key first step is to desire learning a representation coefficient matrix with block diagonal structure. To realize this step, many methods were successively proposed by imposing different structure priors on the coefficient matrix. These impositions can be roughly divided into two categories, i.e., indirect and direct. The former introduces the priors such as sparsity and low rankness to indirectly or implicitly learn the block diagonal structure. However, the desired block diagonalty cannot necessarily be guaranteed for noisy data. While the latter directly or explicitly imposes the block diagonal structure prior such as block diagonal representation (BDR) to ensure so-desired block diagonalty even if the data is noisy but at the expense of losing the convexity that the former's objective possesses. For compensating their respective shortcomings, in this article, we follow the direct line to propose adaptive BDR (ABDR) which explicitly pursues block diagonalty without sacrificing the convexity of the indirect one. Specifically, inspired by Convex BiClustering, ABDR coercively fuses both columns and rows of the coefficient matrix via a specially designed convex regularizer, thus naturally enjoying their merits and adaptively obtaining the number of blocks. Finally, experimental results on synthetic and real benchmarks demonstrate the superiority of ABDR to the state-of-the-arts (SOTAs).

Selected Columns of the Density Matrix in an Atomic Orbital Basis I: An Intrinsic and Non-iterative Orbital Localization Scheme for the Occupied Space.

In this work, we extend the selected columns of the density matrix (SCDM) methodology [J. Chem. Theory Comput. 2015, 11, 1463-1469]─a non-iterative and real-space procedure for generating localized occupied orbitals for condensed-phase systems─to the construction of local molecular orbitals (LMOs) in systems described using non-orthogonal atomic orbital (AO) basis sets. In particular, we introduce three different theoretical and algorithmic variants of SCDM (referred to as SCDM-M, SCDM-L, and SCDM-G) that can be used in conjunction with the AO basis sets used in standard quantum chemistry codebases. The SCDM-M and SCDM-L variants are based on a pivoted QR factorization of the Mulliken and Löwdin representations of the density matrix and are tantamount to selecting a well-conditioned set of projected atomic orbitals (PAOs) and projected (symmetrically-) orthogonalized atomic orbitals, respectively, as proto-LMOs that can be orthogonalized to exactly span the occupied space. The SCDM-G variant is based on a real-space (grid) representation of the wavefunction, and therefore has the added flexibility of considering a large number of grid points (or δ functions) when selecting a set of well-conditioned proto-LMOs. A detailed comparative analysis across molecular systems of varying size, dimensionality, and saturation level reveals that the LMOs generated by these three non-iterative/direct SCDM variants are robust, comparable in orbital locality to those produced with the iterative Boys or Pipek-Mezey (PM) localization schemes, and completely agnostic toward any single orbital locality metric. Although all three SCDM variants are based on the density matrix, we find that the character of the generated LMOs can differ significantly between SCDM-M, SCDM-L, and SCDM-G. In this regard, only the grid-based SCDM-G procedure (like PM) generates LMOs that qualitatively preserve σ-π symmetry (in systems such as s-trans alkenes), and are well-aligned with chemical (i.e., Lewis structure) intuition. While the direct and standalone use of SCDM-generated LMOs should suffice for most chemical applications, our findings also suggest that the use of these orbitals as an unbiased and cost-effective (initial) guess also has the potential to improve the convergence of iterative orbital localization schemes, in particular for large-scale and/or pathological molecular systems.

Red cell antibodies or noise? A case series on reactivity against the ingredients in column matrix

Abstract Noise in the immunohematological investigations can be described as a false reactivity of red blood cells (RBCs) in serologic testing that is not related to the interaction of RBC antigens and antibodies that the test system is intended to detect. These false-positive reactions can cause confusion during the cross-matching and RBC antibody screening and may result in delays in patient transfusion. Although these antibodies are predominantly clinically insignificant, proper laboratory work-up is indicated to avoid misidentification of a clinically significant antibody as a noise. In this report, we describe the three rare cases where the reactivity was found against the ingredients of the column matrix (glass beads). It is imperative that such reactivity is recognized and resolved during the investigation of blood group discrepancies, positive RBC antibody screens and in cases of positive cross-matches.

Observability Analysis and Restoration for Distribution Grids: A Robust Greedy Numerical Algorithm

This paper presents a robust and efficient algorithm for analyzing and restoring the observability of distribution grids. A linearized load flow model is used to characterize three-and multi-phase distribution grids under balanced and unbalanced conditions. Assuming that metering devices are installed to measure nodal magnitudes at some buses, the rank of the matrix associated with the non-metered magnitudes is determined by using a greedy strategy combined with a Gram-Schmidt orthonormalization procedure. Unlike other numerical-based techniques, the proposed algorithm avoids common errors caused by denominators close to zero. As an additional salient feature, a mathematically sound stopping criterion is provided to guarantee the robustness of the proposed algorithm. Furthermore, for unobservable grids, the algorithm identifies the linearly dependent rows in the aforementioned matrix, which correspond to the buses that should be metered to restore grid observability. The effective performance of the proposed algorithm is illustrated using several medium-and low-voltage benchmarks with up to 906 buses.

Lower Bounds for Column Matrix Approximations

Lower Bounds for Column Matrix Approximations

LOWER BOUNDS FOR COLUMN MATRIX APPROXIMATIONS

We show a connection between lower and upper bounds on the column matrix approximation accuracy and the bounds on the norms of the pseudoinverses of the submatrices of orthogonal matrices. This connection is exploited to derive lower bounds for column approximations accuracy in spectral and Frobenius norms.

Reversibly selective encryption for medical images based on coupled chaotic maps and steganography

The security and confidentiality of medical images are of utmost importance due to frequent issues such as leakage, theft, and tampering during transmission and storage, which seriously impact patient privacy. Traditional encryption techniques applied to entire images have proven to be ineffective in guaranteeing timely encryption and preserving the privacy of organ regions separated from the background. In response, this study proposes a specialized and efficient local image encryption algorithm for the medical field. The proposed encryption algorithm focuses on the regions of interest (ROI) within massive medical images. Initially, the Laplacian of Gaussian operator and the outer boundary tracking algorithm are employed to extract the binary image and achieve ROI edge extraction. Subsequently, the image is divided into ROI and ROB (regions outside ROI). The ROI is transformed into a row vector and rearranged using the Lorenz hyperchaotic system. The rearranged sequence is XOR with the random sequence generated by the Henon chaotic map. Next, the encrypted sequence is arranged according to the location of the ROI region and recombined with the unencrypted ROB to obtain the complete encrypted image. Finally, the least significant bit algorithm controlled by the key is used to embed binary image into the encrypted image to ensure lossless decryption of the medical images. Experimental verification demonstrates that the proposed selective encryption algorithm for massive medical images offers relatively ideal security and higher encryption efficiency. This algorithm addresses the privacy concerns and challenges faced in the medical field and contributes to the secure transmission and storage of massive medical images.

Neutrino Mixing Sum Rules and the Littlest Seesaw

In this work, we study the neutrino mixing sum rules arising from discrete symmetries and the class of Littlest Seesaw (LS) neutrino models. These symmetry-based approaches all offer predictions for the cosine of the leptonic CP phase cosδ in terms of the mixing angles, θ13, θ12, θ23, while the LS models also predict the sine of the leptonic CP phase sinδ, as well as making other predictions. In particular, we study the solar neutrino mixing sum rules, arising from charged lepton corrections to tri-bimaximal (TB), bimaximal (BM), golden ratio (GR) and hexagonal (HEX) neutrino mixing, and the atmospheric neutrino mixing sum rules, arising from preserving one of the columns of these types of mixing—for example, the first or second column of the TB mixing matrix (TM1 or TM2)—and we confront them with an up-to-date global fit of the neutrino oscillation data. We show that some mixing sum rules, such as an atmospheric neutrino mixing sum rule arising from a version of neutrino golden ratio mixing (GRa1), are already excluded at 3σ, and we determine the remaining models allowed by the data. We also consider the more predictive LS models (which obey the TM1 sum rules and offer further predictions) based on constrained sequential dominance CSD(n) with n≈3. We compare for the first time the three cases n=2.5, n=3 and n=1+6≈3.45, which are favored by theoretical models, using a new type of analysis to accurately predict the observables θ12, θ23 and δ. We study all the above approaches, solar and atmospheric mixing sum rules and LS models, together so that they may be compared and to give an up-to-date analysis of the predictions of all of these possibilities, when confronted with the most recent global fits.

A local POE-based self-calibration method using position and distance constraints for collaborative robots

Collaborative robots are designed to not only work alongside humans but also adapt to new tasks quickly. To improve their absolute accuracy, self-calibration methods utilizing portable measurement devices are cost-effective solutions. Based on the local frame representation of the Product-of-Exponential (POE) formula, a new modular kinematic error model is derived for collaborative robots, in which all geometric errors in a joint-link assembly, i.e., a dyad, are lumped into a local rigid body motion applied to its nominal joint axis. Its major advantage over the conventional local POE-based kinematic error model is that all columns in the error transformation Jacobian matrix are functions of joint displacements, which is critical to formulate singularity-free self-calibration models based on geometric constraints. A portable and cost-effective self-calibration device with both position and distance constraints is proposed for collaborative robots, which can readily collect a wide range of constraint errors by drag teaching within a collaborative robot’s workspace. Additionally, a two-step self-calibration method is established for both the robot arm and its base calibration. Its effectiveness and robustness are validated through simulations and experiments. The experimental results on Aubo i5 show that its average position error is significantly reduced from 4.11mm to 0.36mm after calibration.

Dynamic mode decomposition of deformation fields in elastic and elastic–plastic solids

Without recourse to constitutive assumptions, without knowledge of material properties, and without solving any of the conservation equations, we construct deformation fields for linear solids using Dynamic Mode Decomposition (DMD). Originally developed for analyzing experimental or computational field variables in fluid mechanics, it takes as input vectors of flow variables assembled as columns of a data matrix, and requires a remarkably few lines of code. It is a natural fit for solid mechanics wherein surface displacements can be used for the analysis. Vectors of surface displacements are arrayed in columns to created a data matrix. Singular Value Decomposition of the time-shifted data matrix affords selection of dominant modes (rank) in the deformation field. The DMD algorithm operates on time-shifted data matrices, obtains a reduced-order model that can reconstruct the deformation field, and also provides the dominant temporal and spatial modes of the deformation. In the case of linear elastic solids, DMD can be used to: reconstruct or predict the displacement states. In elastic–plastic solids, the transition from elastic to plastic results in the eigenvalues of the low-rank data matrix going out of the unit circle (implying an unstable growth mode). Using a combination of finite element analyses and displacement measurements obtained using Digital Image Correlation (DIC), we make the case for using DMD for state-estimation and state prediction in elastic solids and identifying onset of plasticity in elastic–plastic solids.

State of charge estimation for lithium-ion battery based on adaptive extended Kalman filter with improved residual covariance matrix estimator

The accurate estimation of the state of charge (SOC) in lithium-ion batteries plays a pivotal role in battery management systems. This paper proposes an improved adaptive extended Kalman filter (IAEKF) algorithm for more precise SOC estimation. Initially, a novel characteristic parameter is introduced to assess the suitability of the forgetting factor in the adaptive forgetting factor recursive least squares (AFFRLS) algorithm. This evaluation aims to improve the accuracy of recognizing the parameters of the battery model. Additionally, an even function with adjustable parameters is constructed to solve the appropriate value for the forgetting factor. Subsequently, the correlation coefficients of the residual series at different moments are approximated by mathematical transformations that link the inner products of the column vectors in the constructed residual matrix to the calculation of the correlation coefficients. By comparing it with optimized thresholds using particle swarm optimization, one can adjust the sliding window length to enhance the estimation accuracy of IAEKF. The experimental results confirm that IAEKF achieves superior estimation accuracy and robustness compared to other AEKF algorithms, and the accuracy of AFFRLS parameter identification is higher than that of recursive least squares, thereby substantiating the effectiveness of the proposed algorithms.

The Schur decomposition of discrete Sine and Cosine transformations of type IV

Eigendecomposition of discrete sine and cosine transformations of type IV is the main objective of this study. A closed form formula for eigenvectors is derived using the property that a square of the transform matrix is proportional to the identity matrix. Odd and even transform sizes are considered separately. In both cases, it has been demonstrated that the eigenvectors can be obtained by a straightforward modification of the columns of the transformation matrix. Comparing this approach with the related works, it provides a simple explanation of the eigendecomposition with a reliable numerical stability. Additionally, these eigenvectors can be used to obtain the eigendecomposition of the counterpart offset discrete Fourier transform.