Matrix Operations Research Articles

Efficient and Robust Adaptive Beamforming Based on Coprime Array Interpolation

Unlike uniform linear arrays (ULAs), coprime arrays require fewer physical sensors yet provide higher degrees of freedom (DOF) and larger array apertures. However, due to the existence of “holes” in the differential co-array, the target detection performance deteriorates, especially in adaptive beamforming. To address these challenges, this paper proposes an efficient and robust adaptive beamforming algorithm leveraging coprime array interpolation. The algorithm eliminates unwanted signals and uses the Gauss–Legendre quadrature method to reconstruct an Interference-plus-Noise Covariance Matrix (INCM), thereby obtaining the beamforming coefficients. Unlike previous techniques, we utilize a virtual interpolated ULA to expand the aperture, enabling the acquisition of a high-dimensional covariance matrix. Additionally, a projection matrix is constructed to eliminate unwanted signals from the received data, greatly enhancing the accuracy of INCM reconstruction. To address the high computational complexity of integral operations used in most INCM reconstruction algorithms, we propose an approximation based on the Gauss–Legendre quadrature, which reduces the computational load while maintaining accuracy. This algorithm avoids the array aperture loss caused by using only the ULA segment in the difference co-array and improves the accuracy of INCM reconstruction. Simulation and experimental results show that the performance of the proposed algorithm is superior to the compared beamformers and is closer to the optimal beamformer in various scenarios.

Tool Condition Monitoring in the Milling Process Using Deep Learning and Reinforcement Learning

Tool condition monitoring (TCM) is crucial in the machining process to confirm product quality as well as process efficiency and minimize downtime. Traditional methods for TCM, while effective to a degree, often fall short in real-time adaptability and predictive accuracy. This research work aims to advance the state-of-the-art methods in predictive maintenance for TCM and improve tool performance and reliability during the milling process. The present work investigates the application of Deep Learning (DL) and Reinforcement Learning (RL) techniques to monitor tool conditions in milling operations. DL models, including Long Short-Term Memory (LSTM) networks, Feed Forward Neural Networks (FFNN), and RL models, including Q-learning and SARSA, are employed to classify tool conditions from the vibration sensor. The performance of the selected DL and RL algorithms is evaluated through performance metrics like confusion matrix, recall, precision, F1 score, and Receiver Operating Characteristics (ROC) curves. The results revealed that RL based on SARSA outperformed other algorithms. The overall classification accuracies for LSTM, FFNN, Q-learning, and SARSA were 94.85%, 98.16%, 98.50%, and 98.66%, respectively. In regard to predicting tool conditions accurately and thereby enhancing overall process efficiency, SARSA showed the best performance, followed by Q-learning, FFNN, and LSTM. This work contributes to the advancement of TCM systems, highlighting the potential of DL and RL techniques to revolutionize manufacturing processes in the era of Industry 5.0.

A Novel Operational Matrix Method for Solving the Fractional Delay Integro-Differential Equations with a Weakly Singular Kernel

A Novel Operational Matrix Method for Solving the Fractional Delay Integro-Differential Equations with a Weakly Singular Kernel

Improved Berezin radius inequalities for certain operator matrices

Improved Berezin radius inequalities for certain operator matrices

Some new families of asymmetric nested orthogonal arrays with any strength

We propose a general method for constructing asymmetric nested orthogonal arrays (NOAs) with any strength via a newly defined matrix operation. The NOAs obtained can recursively produce other new NOAs. A lot of selective new asymmetric NOAs are tabulated.

Total gluon helicity from lattice without effective theory matching

We propose two approaches for extracting the total gluon helicity contribution to proton spin from lattice QCD, one from local operator matrix elements in a fixed gauge accessible on lattice with feasible renormalization, and the other from gauge-invariant nonlocal gluon correlators. Neither of these approaches requires a matching procedure when converted to the MS¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\ extrm{MS}} $$\\end{document} scheme. Our proposal resolves a long-standing inconsistency in the literature regarding lattice calculations of the total gluon helicity, and has the potential to greatly facilitate these calculations.

Semi-Supervised Learning for Multi-View Data Classification and Visualization

Data visualization has several advantages, such as representing vast amounts of data and visually demonstrating patterns within it. Manifold learning methods help us estimate lower-dimensional representations of data, thereby enabling more effective visualizations. In data analysis, relying on a single view can often lead to misleading conclusions due to its limited perspective. Hence, leveraging multiple views simultaneously and interactively can mitigate this risk and enhance performance by exploiting diverse information sources. Additionally, incorporating different views concurrently during the graph construction process using interactive visualization approach has improved overall performance. In this paper, we introduce a novel algorithm for joint consistent graph construction and label estimation. Our method simultaneously constructs a unified graph and predicts the labels of unlabeled samples. Furthermore, the proposed approach estimates a projection matrix that enables the prediction of labels for unseen samples. Moreover, it incorporates the information in the label space to further enhance the accuracy. In addition, it merges the information in different views along with the labels to construct a consensus graph. Experimental results conducted on various image databases demonstrate the superiority of our fusion approach compared to using a single view or other fusion algorithms. This highlights the effectiveness of leveraging multiple views and simultaneously constructing a unified graph for improved performance in data classification and visualization tasks in semi-supervised contexts.

On the norm of the Hilbert matrix operator on weighted Bergman spaces

It is known that the norm of the Hilbert matrix operator on weighted Bergman spaces Aαp was conjectured by Karapetrović to be πsin⁡(α+2)πp when α>−1 and p>α+2. The conjecture has been confirmed by Božin and Karapetrović in the case α=0. In this paper we prove the conjecture for the cases both α=1 and 0<α≤147. Moreover, we also show that the conjecture is valid when −1<α<0 and p≥2(α+2).

A new method for solving fractional optimal control problems

In this article, we present a novel approach that utilizes the operational matrix method to solve fractional optimal control problems. We address the challenge of missing initial conditions for the co-state function by incorporating the linear shooting method. Our objective is to provide a method that not only enhances computational efficiency and reduces cost but also improves accuracy and usability. With our new approach, we can calculate the coefficients of the expansion solution without the need to solve an algebraic system. These coefficients can be generated explicitly or through an iterative process. We provide a proof of the uniform convergence of the approximate series of functions to the unique solution of the optimal control problem. To demonstrate the effectiveness of our proposed method, we solve several numerical examples and compare the results with those obtained by other researchers. Additionally, we extend the application of this new method to solve problems with multiple states, thereby expanding its scope to a wider range of problem domains. This allows us to address diverse scenarios using the same efficient and accurate approach. Through our research, we contribute to the development of a powerful and versatile method for solving fractional optimal control problems, offering significant advantages over existing techniques.

BAFusion: Bidirectional Attention Fusion for 3D Object Detection Based on LiDAR and Camera.

3D object detection is a challenging and promising task for autonomous driving and robotics, benefiting significantly from multi-sensor fusion, such as LiDAR and cameras. Conventional methods for sensor fusion rely on a projection matrix to align the features from LiDAR and cameras. However, these methods often suffer from inadequate flexibility and robustness, leading to lower alignment accuracy under complex environmental conditions. Addressing these challenges, in this paper, we propose a novel Bidirectional Attention Fusion module, named BAFusion, which effectively fuses the information from LiDAR and cameras using cross-attention. Unlike the conventional methods, our BAFusion module can adaptively learn the cross-modal attention weights, making the approach more flexible and robust. Moreover, drawing inspiration from advanced attention optimization techniques in 2D vision, we developed the Cross Focused Linear Attention Fusion Layer (CFLAF Layer) and integrated it into our BAFusion pipeline. This layer optimizes the computational complexity of attention mechanisms and facilitates advanced interactions between image and point cloud data, showcasing a novel approach to addressing the challenges of cross-modal attention calculations. We evaluated our method on the KITTI dataset using various baseline networks, such as PointPillars, SECOND, and Part-A2, and demonstrated consistent improvements in 3D object detection performance over these baselines, especially for smaller objects like cyclists and pedestrians. Our approach achieves competitive results on the KITTI benchmark.

Dejdumrong Collocation Approach and Operational Matrix for a Class of Second-Order Delay IVPs: Error Analysis and Applications

Dejdumrong Collocation Approach and Operational Matrix for a Class of Second-Order Delay IVPs: Error Analysis and Applications

Exposure difference network for low-light image enhancement

Low-light image enhancement aims to simultaneously improve the brightness and contrast of low-light images and recover the details of the visual content. This is a challenging task that makes typical data-driven methods suffer, especially when faced with severe information loss in extreme low-light conditions. In this work, we approach this task by proposing a novel exposure difference network. The proposed network generates a set of possible exposure corrections derived from the differences between synthesized images under different exposure levels, which are fused and adaptively combined with the raw input for light compensation. By modeling the intermediate exposure differences, our model effectively eliminates the redundancy existing in the synthesized data and offers the flexibility to handle image quality degradation resulting from varying levels of inadequate illumination. To further enhance the naturalness of the output image, we propose a global-aware color calibration module to derive low-frequency global information from inputs, which is further converted into a projection matrix to calibrate the RGB output. Extensive experiments show that our method can achieve competitive light enhancement performance both quantitatively and qualitatively.

An efficient numerical method for two-dimensional fractional integro-differential equations with modified Atangana–Baleanu fractional derivative using operational matrix approach

In the present paper, a numerical method is developed for solving non-homogeneous two-dimensional fractional integro-differential equations with a modified Atangana–Baleanu fractional derivative. The approach is based on the operational matrix method, involving the derivation of operational matrices for the terms of the equation to generate an algebraic system that is solved using mathematical software. First, basic definitions of fractional calculus and block pulse functions are presented. Then, the operational matrices and the method of solution are derived. The existence and uniqueness of the solution of the proposed equation are proved. Examples to test the numerical method are provided, demonstrating the efficiency of the proposed method.

The reduced order model for creep using dynamic mode decomposition

Given the computational challenges associated with finite element methods in analyzing creep behavior in high-temperature components, this work presents an advanced method for modeling thermal creep in complex structures through a reduced order model (ROM) developed via dynamic mode decomposition (DMD). The results indicate that the ROM using DMD can predict long-term structural responses related to creep based on short-term data with high accuracy, enhancing significantly computational efficiency. The application of ROM allows for the rapid estimation of creep effects in complex structures, like monoliths in the MegaPower reactor, thereby improving the design efficiency. The eigenvalues of the creep related ROM all fall in the unit circle on the complex plane, indicating that creep is a stable development process. The DMD with improved time mode coefficient can reconstruct strain-rate related plastic deformation and cyclic visco-plastic deformation process. Additionally, we derive the necessary stress updating algorithm and consistent tangent operator matrix for the 316H unified visco-plastic constitutive equation in 2023 edition ASME code, ensuring accurate modeling of material creep behavior under high temperatures in this paper. This work provides a valuable tool for the safe design and operation of critical components in nuclear engineering and other high-temperature applications.

Generalized $\pi$-Hirano inverses of the sum in Banach algebras

In this paper, we investigate some additive results on g$\pi$-Hirano invertibility in Banach algebras. By applying our results, some new results for operator matrices are obtained. This extends the main results of [H. Zou, T. Li and Y. Wei, arXiv:2302.06080v1].

A fully digital timing background calibration algorithm based on first-order auto-correlation for time-interleaved ADCs

This paper presents a fully digital background calibration method for time-interleaved (TI) analog-to-digital converters (ADCs). The timing detector employs a timing detection method based on an enhanced autocorrelation function, combined with matrix operations, to improve the accuracy of timing mismatch acquisition. The algorithm makes full use of the autocorrelation functions of each channel and proposes a more precise method for obtaining autocorrelation derivatives. Furthermore, the algorithm features a simple structure, low computational complexity, and can support timing calibration for any number of channels without requiring additional channels. The algorithm was validated using an ADC + FPGA combined system, with a 8-channel TI-ADC manufactured in 28 nm technology and a sampling rate of 20 GS/s. Results demonstrate that compared to the uncalibrated state, the algorithm significantly improves the SNDR and SFDR from 27.98 dB and 30.76 dB to 39.38 dB and 42.13 dB, respectively when the inputs are near the Nyquist frequency.

An unbounded operator theory approach to lower frame and Riesz-Fischer sequences

Frames and orthonormal bases are important concepts in functional analysis and linear algebra. They are naturally linked to bounded operators. To describe unbounded operators those sequences might not be well suited. This has already been noted by von Neumann in the 1920ies. But modern frame theory also investigates other sequences, including those that are not naturally linked to bounded operators. The focus of this manuscript will be two such kind of sequences: lower frame and Riesz-Fischer sequences. We will discuss the inter-relation of those sequences. We will fill a hole existing in the literature regarding the classification of these sequences by their synthesis operator. We will use the idea of generalized frame operator and Gram matrix and extend it. We will use that to show properties for canonical duals for lower frame sequences, such as a minimality condition regarding its coefficients. We will also show that other results that are known for frames can be generalized to lower frame sequences. Finally, we show that the converse of a well-known result is true, i.e. that minimal lower frame sequences are equivalent to complete Riesz-Fischer sequences, without any further assumptions.To be able to tackle these tasks, we had to revisit the concept of invertibility (in particular for non-closed operators). In addition, we are able to define a particular adjoint, which is uniquely defined for any operator.

Trisection-fusion and fusion-trisection methods of three-way conflict analysis with Pythagorean fuzzy information

A basic task of Pawlak conflict analysis is to cluster agents based on their attitudes towards various issues. When ratings of agents are imprecise and uncertain, for example, as represented by a Pythagorean fuzzy situation table, single-measure based methods fall short in ensuring the accuracy of agent clusters. Multi-measure based conflict analysis methods are more effective for generating higher-quality agent clusters. The objective of this paper is to propose two new multi-measure based methods under uncertainty represented by Pythagorean fuzzy sets. We introduce three distinct conflict measures regarding an issue by leveraging the maximum positive and negative agents. The first measure is based on support degrees, the second measure incorporates opposition degrees, and the third measure considers both support and opposition degrees. These measures trisect a set of agents into three disjoint coalitions concerning an individual issue. For multiple issues, we propose two models by combing the two fundamental tasks, namely, trisection and fusion. A trisection-fusion model amalgamates a family of trisections generated from multiple issues based on conflict measures regarding single issues. Matrix representations of trisections are provided, and trisection fusions are transformed into a series of matrix operations. A fusion-trisection model trisects the set of agents according to fused conflict measures on multiple issues. To demonstrate the value of the two models, we apply them in assisting decision-makers in Hunan Province, China, to adjust development plans. The experimental results show that multi-measure based models offer decision-makers more comprehensive guidance.

RBF-based partition of unity methods for two-dimensional time-dependent PDEs: Numerical and theoretical aspects

This article discusses standard Radial Basis Function (RBF) partition of unity method and an enhanced version known as the direct RBF partition of unity method. We introduce the framework of these methods for the effective and accurate numerical analysis of certain two-dimensional time-dependent Partial Differential Equations (PDEs). Additionally, we conduct a convergence analysis of these methods and establish error bounds for local approximations. These error bounds are determined based on conditions associated with the eigenvalues of the Laplacian operator matrices generated by the proposed methods. To validate the accuracy and reliability of these approaches, we provide several numerical examples. We assess the consistency between theoretical and numerical results. The localized techniques introduced in this research demonstrate a significant reduction in computational cost and algorithmic complexity, achieved through the direct local procedure. Despite this computational advantage, both versions of the RBF partition of unity method exhibit a similar level of accuracy. Finally, we conduct a comparative analysis between the RBF partition of unity local methods and the RBF-generated finite differences method, offering valuable insights into the strengths and limitations of each approach.

Reasoning with the inverse of 3D cardinal direction relations based on direction matrices

In order to improve the ability of intelligent reasoning and prediction of 3DR27 model which presented in our previous work, enhance the usability of this model, and better fulfill the demands of real applications for spatial database, we focused on the problem of reasoning with the inverse of 3D cardinal direction relations between spatial objects. In order to realize automated reasoning, an algorithm for computing the inverse of 3D cardinal direction relations based on matrices is proposed on the basis of the mapping between the 3D rectangular cardinal direction relations and 3D interval relation matrix. This algorithm improves the power of reasoning for this model by means of the excellent properties of matrix operations. Theorems are provided to prove formally that our algorithm is correct and complete. This study realized the automatic inference and calculation of the inverse of the 3D cardinal direction relations based on 3DR27 model and further improved the ability of spatial reasoning and spatial analysis of spatial database.