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A polynomial program is one in which all assignments are given by polynomial expressions and in which all branching is nondeterministic (as opposed to conditional). Given such a program, an algebraic invariant is one that is defined by polynomial equations over the program variables at each program location.Müller-Olm and Seidl have posed the question of whether one can compute the strongest algebraic invariant of a given polynomial program. In this article, we show that, while strongest algebraic invariants are not computable in general, they can be computed in the special case of affine programs, that is, programs with exclusively linear assignments. For the latter result, our main tool is an algebraic result of independent interest: Given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate. [ABSTRACT FROM AUTHOR]

In the process of remote sensing data processing, cloud data will have a considerable negative impact on data processing. Therefore, a matrix optimization cloud detection algorithm based on multiple receptive fields is proposed in this paper. First, the algorithm adopts a block reshaping model to improve the efficiency of the algorithm, while reducing the need for hardware configuration. Second, the cloud region adaptive sparse attention matrix is used to improve the characteristics of cloud regions with different concentrations. Finally, the multi-receptive field scaling module is used to improve the ability to segment cloud regions at different scales. The experimental results show that the accuracy of the proposed algorithm is 85.5%, and the recall rate is 86.4%. The proposed algorithm can basically accurately screen the location of a cloud region and provide technical support for the subsequent image application processing. [ABSTRACT FROM AUTHOR]

In this paper, a formulation of reduced-order finite element (FE) model is presented for geometrically nonlinear dynamic analysis of viscoelastic structures based on the fractional-order derivative constitutive relation and harmonic balance method. The main focus is to formulate the nonlinear reduced-order models (ROMs) in the time and frequency domain without involving the corresponding full-order FE models, and it is carried out by means of a special factorization of the nonlinear strain–displacement matrix. Furthermore, a methodology for the enrichment of reduction basis (RB) over that obtained from conventional approaches is presented where the proper orthogonal decomposition method is utilized by comprising the correlation matrix as the union of stiffness-normalized reduction basis vectors and the corresponding static derivatives. The results reveal a significantly reduced computational time due to the formulation of the nonlinear ROMs without involving the full-order FE model. A good accuracy of the nonlinear ROMs of viscoelastic structures is also achieved through the present method of enrichment of RB. [ABSTRACT FROM AUTHOR]

This paper presents a novel hierarchical control scheme for solving the data-driven optimal cooperative tracking control problem of heterogeneous multi-agent systems. Considering that followers cannot communicate with the leader, a prescribed-time fully distributed observer is devised to estimate the leader's state for each follower. Then, the data-driven decentralized controller is designed to ensure that the follower's output can track the leader's one. Compared with the existing results, the advantages of the designed distributed observer are that the prescribed convergence time is completely predetermined by the designer, and the design of the observer gain is independent of the global topology information. Besides, the advantages of the designed decentralized controller are that neither the follower's system model nor a known initial stabilizing control policy is required. Finally, simulation results exemplify the advantage of the proposed method. • Compared with the existing methods, the advantage of our proposed prescribed-time fully distributed observer is that the leader's state can be accurately estimated within the designer's predetermined time; • Unlike the existing RL-based algorithms that rely on the initial stabilizing control policy associated with the accurate system matrices, the proposed new RL-based algorithm not only does not require a known initial stabilizing control policy, but also ensures the optimal tracking control in a data-driven manner. • Different from the homotopy-based policy iteration algorithm, an improved counterpart is presented, where the design parameter can be directly designed in the data-driven manner. [ABSTRACT FROM AUTHOR]

This article analyzes a key exchange protocol based on a modified tropical structure proposed by Ahmed et al. in 2023. It is shown that the modified tropical semiring is isomorphic to the 2 × 2 tropical circular matrix semiring. Therefore, matrices in this modified tropical semiring can be represented as tropical matrices, and the key exchange protocol is actually based on the tropical matrix semiring. Tropical irreducible matrices exhibit almost linear periodic property. Efficient algorithms for calculating the linear period and defect of irreducible matrices are designed. Based on the public information of the protocol, the equivalent private key can be computed and then the shared key is easily obtained. The analysis shows that the key exchange protocol based on this modified tropical structure is not secure. [ABSTRACT FROM AUTHOR]

The present paper investigates a new method for computationally solving the time-fractional coupled Korteweg–de Vries equations (TFCKdVEs) with initial boundary conditions (IBCs). The method utilizes a set of generalized shifted Jacobi polynomials (GSJPs) that adhere to the specified initial and boundary conditions (IBCs). Our approach involves constructing operational matrices (OMs) for both ordinary derivatives (ODs) and fractional derivatives (FDs) of the GSJPs we employ. We subsequently employ the collocation spectral method using these OMs. This method successfully converts the TFCKdVEs into a set of algebraic equations, greatly simplifying the task. In order to assess the efficiency and precision of the proposed numerical technique, we utilized it to solve two distinct numerical instances. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the convergence of the multivariate cascade algorithm with an s × s dilation matrix M and a nonnegative finite mask, which generalizes the result obtained recently. Moreover, we present two examples, which demonstrate the power and applicability of our main result. [ABSTRACT FROM AUTHOR]

Finding the highest possible cardinality, A q (n , d ; k) , of the set of k-dimensional subspaces in F q n , also known as codewords, is a fundamental problem in constant dimension codes (CDCs). CDCs find applications in a number of domains, including distributed storage, cryptography, and random linear network coding. The goal of recent research papers has been to establish A q (n , d ; k). We further improved the echelon-Ferrers construction with an algorithm, and enhanced the inserting construction by swapping specified columns of the generator matrix to obtain new lower bounds. [ABSTRACT FROM AUTHOR]

A regular matroid M on a finite set E is represented by a totally unimodular matrix. The set of vectors from Z E orthogonal to rows of the matrix form a regular chain group N. Assume that ψ is a homomorphism from N into a finite additive Abelian group A and let A ψ [ N ] be the set of vectors g from (A − 0) E , such that ∑ e ∈ E g (e) · f (e) = ψ (f) for each f ∈ N (where · is a scalar multiplication). We show that | A ψ [ N ] | can be evaluated by a polynomial function of | A | . In particular, if ψ (f) = 0 for each f ∈ N , then the corresponding assigning polynomial is the classical characteristic polynomial of M. [ABSTRACT FROM AUTHOR]

This paper focuses on the minimum eigenvalue involving the Fan product. By utilizing the Hölder inequality and the classic eigenvalue inclusion theorem, we introduce two novel lower bounds for τ (A 1 ⋆ A 2) , representing the minimum eigenvalue involving the Fan product of two M-matrices A 1 , A 2 . The newly derived lower bounds are then compared with the traditional findings. Numerical tests are presented to illustrate that the new lower bound formulas significantly enhance Johnson and Horn's results in certain scenarios and are more precise than other existing findings. [ABSTRACT FROM AUTHOR]

AHP is a well-known multi-criteria procedure which has been investigated and developed by many researchers and practitioners. Some AHP modifications are designed for decision making under uncertainty. The goal of this paper is to present a new AHP approach which can be useful in the case of uncertain one-shot decisions and independent criteria. The method proposed in the article is based on scenario planning, features characteristic for the Hurwicz rule (i.e. the use of the optimism coefficient) and on a scenario set reduction. The novel procedure gives the possibility to generate a relatively small number of pairwise comparison matrices thanks to the reduction of the initial sets of scenarios. The modified version of AHP may be helpful when the decision maker's knowledge about probabilities of the occurrence of particular scenarios is partial. Such a situation occurs in the case of innovative, innovation and risky projects for which historical data are not known. The idea of the suggested scenario-based AHP is to adjust the final choice not only to the decision makers' preferences (concerning criteria for example), but also to their nature, attitude towards risk, predictions, expectations and fears. [ABSTRACT FROM AUTHOR]

Let GS be the graph of order n and containing σ self-loops. The energy E(GS) of graph GS is defined as E(GS) = n∑i=1 | λi - σ/n |, where λ1, λ2, . . ., λn be the eigenvalues of the adjacency matrix of GS. Two non-isomorphic graphs G1 and G2 of the same order are said to be equienergetic if they have same energy. The proposed research is an effort to expand the concept of equienergetic graphs from simple graphs to graphs having self-loops. In the present work, we have obtained a pair of equienergetic graphs and the energy of complete graphs as well as complete bipartite graphs with self loops. [ABSTRACT FROM AUTHOR]

For a finite field GF( p L ) with prime p and L > 1 , one of the standard representations is L × L matrices over GF(p) so that the arithmetic of GF( p L ) can be realized by the arithmetic among these matrices over GF(p). Based on the matrix representation of GF( p L ), a conventional linear network coding scheme over GF( p L ) can be transformed to an L-dimensional vector LNC scheme over GF(p). Recently, a few real implementations of coding schemes over GF( 2 L ), such as the Reed–Solomon (RS) codes in the ISA-L library and the Cauchy-RS codes in the Longhair library, are built upon the classical result to achieve matrix representation, which focuses more on the structure of every individual matrix but does not shed light on the inherent correlation among matrices which corresponds to different elements. In this paper, we first generalize this classical result from over GF( 2 L) to over GF( p L ) and paraphrase it from the perspective of matrices with different powers to make the inherent correlation among these matrices more transparent. Moreover, motivated by this correlation, we can devise a lookup table to pre-store the matrix representation with a smaller size than the one utilized in current implementations. In addition, this correlation also implies useful theoretical results which can be adopted to further demonstrate the advantages of binary matrix representation in vector LNC. In the following part of this paper, we focus on the study of vector LNC and investigate the applications of matrix representation related to the aspects of random and deterministic vector LNC. [ABSTRACT FROM AUTHOR]

Algebraic expressions and equalities can be constructed arbitrarily in a given algebraic framework according to the operational rules provided, and thus it is a prominent and necessary task in mathematics and applications to construct, classify, and characterize various simple general algebraic expressions and equalities. As an update to this prominent topic in matrix algebra, this article reviews and improves upon the well-known block matrix methodology and matrix rank methodology in the construction and characterization of matrix equalities. We present a collection of fundamental and useful formulas for calculating the ranks of a wide range of block matrices and then derive from these rank formulas various valuable consequences. In particular, we present several groups of equivalent conditions in the characterizations of the Hermitian matrix, the skew-Hermitian matrix, the normal matrix, etc. [ABSTRACT FROM AUTHOR]

In this study, we obtained results for the computation of eigen-pairs, singular value decomposition, pseudoinverse, and the least squares problem for elliptic quaternion matrices. Moreover, we established algorithms based on these results and provided illustrative numerical experiments to substantiate the accuracy of our conclusions. In the experiments, it was observed that the p-value in the algebra of elliptic quaternions directly affects the performance of the problem under consideration. Selecting the optimal p-value for problem-solving and the elliptic behavior of many physical systems make this number system advantageous in applied sciences. [ABSTRACT FROM AUTHOR]

In this article, the regular Schröder–Catalan matrix is constructed and acquired by benefiting Schröder and Catalan numbers. After that, two sequence spaces are introduced, described as the domain of Schröder–Catalan matrix. Additionally, some algebraic and topological properties of the spaces in question, such as completeness, inclusion relations, basis and duals, are examined. In the last two sections, the necessary and sufficient conditions of some matrix classes and compact operators related aforementioned spaces are presented. [ABSTRACT FROM AUTHOR]

A bicluster consists of a subset of rows and columns of a given matrix, whose intersection defines the region (bicluster) of values of precisely defined condition. Through the decades, a variety of biclustering techniques have been successfully developed. Recently, it was proved that many possible patterns defined in two-dimensional data could be found with the application of Boolean reasoning. The provided theorems showed that any existing pattern in the data could be unequivocally encoded as an implicant of a proper Boolean function. Moreover, a prime implicant of that function encoded the inclusion-maximal (non-extendable) pattern. On the other hand, the definition of some two-dimensional patterns may be easily extended to three-dimensional patterns (triclusters) as well as to any number of dimensions (n-clusters). This paper presents a new approach for searching for three- and higher-dimensional simple patterns in continuous data with Boolean reasoning. Providing the definition of the Boolean function for this tasks, it is shown that the similar correspondence—implicants encode patterns, and prime implicants encode inclusion-maximal patterns—has a strong mathematical background: the proofs of appropriate theorems are also presented in this paper. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce the concept of idempotent-aided factorization (I.-A. factorization) of a regular element of a semigroup, which can be understood as a semigroup-theoretical extension of full-rank factorization of matrices over a field. I.-A. factorization of a regular element d is defined by means of an idempotent e from its Green's D -class as decomposition into the product d = u v , so that the element u belongs to the Green's R -class of the element d and the Green's L -class of the idempotent e, while the element v belongs to the Green's L -class of the element d and the Green's R -class of the idempotent e. The main result of the paper is a theorem which states that each regular element of a semigroup possesses an I.-A. factorization with respect to each idempotent from its Green's D -class. In addition, we prove that when one of the factors is given, then the other factor is uniquely determined. I.-A. factorizations are then used to provide new existence conditions and characterizations of group inverses and (b , c) -inverses in a semigroup. In our further research, these factorizations will be applied to matrices with entries in a field, and efficient algorithms for realization of such factorizations will be provided. [ABSTRACT FROM AUTHOR]

An improved transfer matrix method is developed with modular matrices for rectangular electric and magnetic vector potentials, as well as the δ-function like potentials. As an example, the spin-dependent conductance in two-dimensional electron gas under modulation of two magnetic barriers are recalculated and compared with those illustrated in (Wang et al. Chin. Phys. B 19, 037301(2010)). The previous results are found inaccurate, their transfer matrix was derived for slowly varying magnetic vector potential, which is inappropriate for δ-function like Zeeman interaction potentials. Furthermore, the last transfer matrix of eq. (4) is also found incorrect, and the Zeeman interaction is wrongly expressed and mistakenly enhanced about (1/0.067 ≈ 15) times. Our improved method here is general and is demonstrated very fast forevaluating various spin-dependent transport properties in complex hybrid magnetic–electric structures. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the non-confluence property of a class of stochastic differential equations with Markovian switching driven by fractional Brownian motion with Hurst parameter H ∈ (1 / 2 , 1) . By using the generalized Itô formula and stopping time techniques, we obtain some sufficient conditions ensuring the non-confluence property for the considered equations. Additionally, we present two important corollaries on the non-confluence property by the Poisson equation and M-matrix, respectively, which can verify the non-confluence property more effectively than the general condition. Finally, we provide an example to illustrate the practical usefulness of our theoretical results. [ABSTRACT FROM AUTHOR]

The primary objective of this article is to comprehensively investigate and elucidate the characteristics associated with both T-ordering and minus-ordering within the realm of picture fuzzy matrices. Furthermore, we aim to explore and elucidate various properties pertaining to T-ordering and minusordering when applied to picture fuzzy matrices, leveraging the framework of generalized inverses for a more comprehensive understanding of these ordering techniques. Additionally, the article explores a novel concept termed reverse T ordering and reverse Minus ordering and Left-T and Right-T Partial Orderings on Picture Fuzzy Matrix, introducing and elucidating these concepts through illustrative examples. Finally, we apply picture fuzzy to a decision-making case study. [ABSTRACT FROM AUTHOR]

We propose a single-pixel non-imaging target recognition scheme which that exploits the singular values of target objects. By choosing the first few singular values and the corresponding unitary matrices in the singular value decomposition of all the targets, we form the measurement matrices to be projected onto the target in a single-pixel non-imaging scheme. One can quickly and accurately recognize the target images after directly recording the single-pixel signals. From the simulation and experimental results, we found that the accuracy of target recognition was high when the first three singular values were used. The efficiency of target recognition was improved by randomly rearranging the orders of the row vectors in the measurement matrix. Therefore, our research results offer a novel perspective for recognizing non-imaging targets. [ABSTRACT FROM AUTHOR]

We give a criterion based on reflection symmetries in the spirit of Jitomirskaya--Simon to show absence of point spectrum for (split-step) quantum walks and Cantero--Moral--Velázquez (CMV) matrices. To accomplish this, we use some ideas from a recent paper by the authors and collaborators to implement suitable reflection symmetries for such operators. We give several applications. For instance, we deduce arithmetic delocalization in the phase for the unitary almost-Mathieu operator and singular continuous spectrum for generic CMV matrices generated by the Thue--Morse subshift. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive assessment procedure for evaluating Ensemble-based Machine Learning algorithms in the context of exoplanet classification. Each of the algorithm hyperparameter values were tuned. Deployments were carried out using the cross-validation method. Performance metrics, including accuracy, sensitivity, specificity, precision, and F1 score, were evaluated using confusion matrices generated from each implementation. Machine Learning (ML) algorithms were trained and used to identify exoplanet data. Most of the current research deals with traditional ML algorithms for this purpose. The Ensemble algorithm is another type of ML technique that combines the prediction performance of two or more algorithms to obtain an improved final prediction. Few studies have applied Ensemble algorithms to predict exoplanets. To the best of our knowledge, no paper that has exclusively assessed Ensemble algorithms exists, highlighting a significant gap in the literature about the potential of Ensemble methods. Five Ensemble algorithms were evaluated in this paper: Adaboost, Random Forest, Stacking, Random Subspace Method, and Extremely Randomized Trees. They achieved an average performance of more than 80% in all metrics. The results underscore the substantial benefits of fine tuning hyperparameters to enhance predictive performance. The Stacking algorithm achieved a higher performance than the other algorithms. This aspect is discussed in this paper. The results of this work show that it is worth increasing the use of Ensemble algorithms to improve exoplanet identification. [ABSTRACT FROM AUTHOR]

Solving the Perspective-n-Point (PnP) problem is difficult in low-power systems due to the high computing workload. To handle this challenge, we present an originally designed FPGA implementation of a PnP solver based on Vivado HLS. A matrix operation library and a matrix decomposition library based on QR decomposition have been developed, upon which the EPnP algorithm has been implemented. To enhance the operational speed of the system, we employed pipeline optimization techniques and adjusted the computational process to shorten the calculation time. The experimental results show that when the number of input data points is 300, the proposed system achieves a processing speed of 45.2 fps with a power consumption of 1.7 W and reaches a peak-signal-to-noise ratio of over 70 dB. Our system consumes only 3.9% of the power consumption per calculation compared to desktop-level processors. The proposed system significantly reduces the power consumption required for the PnP solution and is suitable for application in low-power systems. [ABSTRACT FROM AUTHOR]

A Hermitian matrix can be parametrized by a set consisting of its determinant and the eigenvalues of its submatrices. We established a group of equations which connect these variables with the mixing parameters of diagonalization. These equations are simple in structure and manifestly invariant in form under the symmetry operations of dilatation, translation, rephasing, and permutation. When applied to the problem of neutrino oscillation in matter, they produced two new "matter invariants" which are confirmed by available data. [ABSTRACT FROM AUTHOR]

The electrophoretic deposition of a mixture of copper-manganese spinel oxide matrix with perovskite oxide dispersion phase and subsequent two-step sintering was investigated as a means of obtaining a new spinel/perovskite composite coating for application in metallic interconnects for solid oxide cell (SOC) technologies. The effect of a number of factors was examined, including electrophoretic deposition parameters such as solvent type, time, voltage, and dispersant content as well as different heat treatment conditions on the quality of the matrix of the coating material. A solution of ethanol-acetone (25/75 v/v) and 2.0 g L−1 of iodine dispersant with a deposition time of 60 s and a voltage of 60 V provided the thickest, most uniform coating as deposited. SEM observations showed that a dense coating of Cu–Mn spinel was obtained after 8 h of reduction at 1000 °C followed by 6 h of oxidation at 850 °C. Selected preparation parameters were applied to deposit composite coatings with a Cu–Mn spinel matrix and the addition of LNF, LSC and LSCF perovskites. [Display omitted] • EOH25/75ACT was determined to be the most suitable mixture for the deposition. • Mass gain of the deposit generally correlates with particle mobility. • Reduction in Ar–10%H 2 at 1000 °C and re-synthesis in air at 850 °C is optimal. • Morphology of composite coatings depend on type of perovskite additive. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel four-dimensional (4D) parameter estimation method to localize the mixed far-field (FF) and near-field (NF) targets using bistatic MIMO arrays and higher-order singular value decomposition (HOSVD). The estimated four parameters include the angle-of-departure (AOD), angle-of-arrival (AOA), range-of-departure (ROD), and range-of-arrival (ROA). In the method, we store array data in a tensor form to preserve the inherent multidimensional properties of the array data. First, the observation data are arranged into a third-order tensor and its covariance tensor is calculated. Then, the HOSVD of the covariance tensor is performed. From the left singular vector matrices of the corresponding module expansion of the covariance tensor, the subspaces with respect to transmit and receive arrays are obtained, respectively. The AOD and AOA of the mixed FF and NF targets are estimated with signal-subspace, and the ROD and ROA of the NF targets are achieved using noise-subspace. Finally, the estimated four parameters are matched via a pairing method. The Cramér–Rao lower bound (CRLB) of the mixed target parameters is also derived. The numerical simulations demonstrate the superiority of the tensor-based method. [ABSTRACT FROM AUTHOR]

Sparse general matrix–matrix multiplication (SpGEMM) is a crucial and complex computational task in many practical applications. Improving the performance of SpGEMM on SIMT processors like modern GPUs is challenging due to the unpredictable sparsity of sparse matrices. Although existing GPU solutions have made progress in improving performance through advanced algorithm design, they ignore some optimizations related to specific processor architectures. This can result in a partially inefficient implementation of their algorithms. This paper focuses on optimizing four inefficient parts of the NSparse algorithm on DCU (a GPU-like accelerator). The optimizations include: 1) setting parameters to improve the load balance of the second matrix by extracting maximum row information at runtime; 2) reducing overhead of binning operations by making full use of registers and shared memory effectively; 3) improving numerical SpGEMM performance by adjusting its calculation mode; and 4) enhancing global load balance through finer-grained grouping and kernel configurations. Experiment results demonstrate that when compared to five state-of-the-art SpGEMM algorithms (bhSparse, KokkosKernels, NSparse, rocSparse, and spECK), our optimized method achieves an average of 7.99x (up to 18.2x), 8.01x (up to 20.83x), 2.37x (up to 6.16x), 1.82x (up to 4.20x), and 1.63x (up to 5.01x) speedups on 29 sparse matrices with different sparse structures, respectively. [ABSTRACT FROM AUTHOR]

Obtaining discriminative joint features is crucial for skeleton-based human action recognition. Current models mainly focus on the research of skeleton topology encoding. However, their predefined topology is the same and fixed for all action samples, making it challenging to obtain discriminative joint features. Although some studies have considered the complex non-natural connection relationships between joints, the existing methods cannot fully capture this complexity by using high-order adjacency matrices or adding trainable parameters and instead increase the computation parameters. Therefore, this study constructs a novel adaptive channel-enhanced graph convolution (ACE-GCN) model for human action recognition. The model generates similar and affinity attention maps by encoding channel attention in the input features. These maps are complementarily applied to the input feature map and graph topology, which can realize the refinement of joint features and construct an adaptive and non-shared channel-based adjacency matrix. This method of constructing the adjacency matrix improves the model's capacity to capture intricate non-natural connections between joints, prevents the accumulation of unnecessary information, and minimizes the number of computational parameters. In addition, integrating the Edgeconv module into a multi-branch aggregation improves the model's ability to aggregate different scale and temporal features. Ultimately, comprehensive experiments were carried out on NTU-RGB+D 60 and NTU-RGB+D 120, which are two substantial datasets. On the NTU RGB+D 60 dataset, the accuracy of human action recognition was 92% (X-Sub) and 96.3% (X-View). The model achieved an accuracy of 96.6% on the NW-UCLA dataset. The experimental results confirm that the ACE-GCN exhibits superior recognition accuracy and lower computing complexity compared to current methodologies. [ABSTRACT FROM AUTHOR]

AbstractDetermining the number of factors in high-dimensional factor modeling is essential but challenging, especially when the data are heavy-tailed. In this paper, we introduce a new estimator based on the spectral properties of Spearman sample correlation matrix under the high-dimensional setting, where both dimension and sample size tend to infinity proportionally. Our estimator is robust against heavy tails in either the common factors or idiosyncratic errors. The consistency of our estimator is established under mild conditions. Numerical experiments demonstrate the superiority of our estimator compared to existing methods. [ABSTRACT FROM AUTHOR]

Multiple kernel k‐means clustering (MKKC) can efficiently incorporate multiple base kernels to generate an optimal kernel. Many existing MKKC methods all need two‐step operation: learning clustering indicator matrix and performing clustering on it. However, the optimal clustering results of two steps are not equivalent to those of original problem. To address this issue, in this paper we propose a novel method named one‐step multiple kernel k‐means clustering based on block diagonal representation (OS‐MKKC‐BD). By imposing a block diagonal constraint on the product of indicator matrix and its transpose, this method can encourage the indicator matrix to be block diagonal. Then the indicator matrix can produce explicit clustering indicator, so as to implement one‐step clustering, which avoids the disadvantage of two‐step operation. Furthermore, a simple kernel weighting strategy is used to obtain an optimal kernel, which boosts the quality of optimal kernel. In addition, a three‐step iterative algorithm is designed to solve the corresponding optimization problem, where the Riemann conjugate gradient iterative method is used to solve the optimization problem of the indicator matrix. Finally, by extensive experiments on eleven real data sets and comparison of clustering results with 10 MKC methods, it is concluded that OS‐MKKC‐BD is effective. [ABSTRACT FROM AUTHOR]

With the increasing demand for the refined management of residential loads, the study of the non-invasive load monitoring (NILM) technologies has attracted much attention in recent years. This paper proposes a novel method of residential load identification based on load feature matrix and improved neural networks. Firstly, it constructs a unified scale bitmap format gray image consisted of multiple load feature matrix including: V-I characteristic curve, 1-16 harmonic currents, 1- cycle steady-state current waveform, maximum and minimum current values, active and reactive power. Secondly, it adopts a convolutional layer to extract image features and performs further feature extraction through a convolutional block attention module (CBAM). Thirdly, the feature matrix is converted and input to a bidirectional long short-term memory (BiLSTM) for training and identification. Furthermore, the identification results are optimized with dynamic time warping (DTW). The effectiveness of the proposed method is verified by the commonly used PLAID database. [ABSTRACT FROM AUTHOR]

In this paper we define a matrix sequence called the Pell matrix sequence whose elements consist of Pell numbers. Using a positive parameter k, we generalize the Pell matrix sequence to a k-Pell matrix sequence and using two parameters s, t we generalize them to (s, t)-Pell matrix sequences. We give the basic properties of these matrix sequences. Then, using these properties we obtain exponential representations of the Pell matrix sequence and its generalizations in different ways. [ABSTRACT FROM AUTHOR]

This paper explores geometric circulant matrices whose entries are Chebyshev polynomials of the first or second kind. Motivated by our previous research on r − circulant matrices and Chebyshev polynomials, we focus on calculating the Frobenius norm and deriving estimates for the spectral norm bounds of these matrices. Our analysis reveals that this approach yields notably improved results compared to previous methods. To validate the practical significance of our research, we apply it to existing studies on geometric circulant matrices involving the generalized k − Horadam numbers. The obtained results confirm the effectiveness and utility of our proposed approach. [ABSTRACT FROM AUTHOR]

In this article, we study the properties of 4-by-4 metric matrices and characterize their dependence and independence by M 4 × 4 = (M 4 × 4 − D M 4 × 4) ∪ D M 4 × 4 , where D M 4 × 4 is the set of all dependent metric matrices. D M 4 × 4 is further characterized by D M 4 × 4 = D M 1 4 × 4 ∪ D M 2 4 × 4 , where D M 2 4 × 4 is characterized by D M 2 4 × 4 = D M 21 4 × 4 ∪ D M 22 4 × 4 . These characterizations provide some insightful findings that go beyond the Euclidean distance or Euclidean distance matrix and link the distance functions to vector spaces, which offers some theoretical and application-related advantages. In the application parts, we show that the metric matrices associated with all Minkowski distance functions over four different points are linearly independent, and that the metric matrices associated with any four concyclic points are also linearly independent. [ABSTRACT FROM AUTHOR]

In this study, we investigate some new properties of the sequence of bi-periodic Fibonacci numbers with arbitrary initial conditions, through an approach that combines the matrix aspect and the fundamental Fibonacci system. Indeed, by considering the properties of the eigenvalues of their related 2 × 2 matrix, we provide a new approach to studying the analytic representations of these numbers. Moreover, the similarity of the associated 2 × 2 matrix with a companion matrix, allows us to formulate the bi-periodic Fibonacci numbers in terms of a homogeneous linear recursive sequence of the Fibonacci type. Therefore, the combinatorial aspect and other analytic representations formulas of the Binet type for the bi-periodic Fibonacci numbers are achieved. The case of bi-periodic Lucas numbers is outlined, and special cases are exposed. Finally, some illustrative examples are given. [ABSTRACT FROM AUTHOR]

We investigate the properties of matrices that emerge from the application of Fourier series to Jacobi matrices. Specifically, we focus on functions defined by the coefficients of a Fourier series expressed in orthogonal polynomials. In the operational formulation of integro-differential problems, these infinite matrices play a fundamental role. We have derived precise calculation formulas for their elements, enabling exact computation of these operational matrices. Numerical results illustrate the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

Non-Tarskian interpretations of many-valued logics have been widely explored in the logic literature. The development of non-tarskian conceptions of logical consequence set the theoretical foundations for rediscovering well-known (Tarskian) many-valued logics. One may find in distinct authors many novel interpretations of many-valued systems. They are produced through a type of procedure which consists in altering the semantic structure of Tarskian many-valued logics in order to output a non-Tarskian interpretation of these logics. Through this type of transformation the paper explores a uniform way of transforming finitely many-valued Tarskian logics into their non-Tarskian interpretation. Some general properties of carrying out this type of procedure are studied, namely the dualities between these logics and the conditions under which negation-explosive and negation-complete Tarskian logics become non-explosive. [ABSTRACT FROM AUTHOR]

This paper attempts to demystify the stability of CoCoSo ranking method via a comprehensive simulation experiment. In the experiment, matrices of different dimensions are generated via Python with fuzzy data. Stability is investigated via adequacy and partial adequacy tests. The test passes if the ranking order does not change even after changes are made to entities, and the partial pass signifies that the top ranked alternative remains intact. Results infer that CoCoSo method has better stability with respect to change of alternatives compared to criteria; and CoCoSo method shows better stability with respect to partial adequacy test for criteria. [ABSTRACT FROM AUTHOR]

Dynamic reduction algorithms have become an important part of attribute reduction research because of their ability to perform dynamic updates without the need to retrain the original model. To enhance the efficiency of variable precision reduction algorithms in processing dynamic data, research has been conducted from the perspective of the construction process of the discernibility matrix. By modifying the decision values of some samples through an absolute majority voting strategy, a connection between variable precision reduction and positive region reduction has been established. Considering the increase and decrease of samples, dynamic variable precision reduction algorithms have been proposed. For four cases of sample increase, four corresponding scenarios have been discussed, and judgment conditions for the construction of the discernibility matrix have been proposed, which has led to the development of a dynamic variable precision reduction algorithm for sample increasing (DVPRA-SI). Simultaneously, for the scenario of sample deletion, three corresponding scenarios have been proposed, and the judgment conditions for the construction of the discernibility matrix have been discussed, which has resulted in the development of a dynamic variable precision reduction algorithm for sample deletion (DVPRA-SD). Finally, the proposed two algorithms and existing dynamic variable precision reduction algorithms were compared in terms of the running time and classification precision, and the experiments demonstrated that both algorithms are feasible and effective. [ABSTRACT FROM AUTHOR]

This paper focuses on addressing the linear quadratic (LQ) optimal control problem on an infinite time horizon for stochastic systems controlled by impulses. No constraint regarding the sign of the quadratic functional is applied. That is why our first concern is to find conditions which guarantee that the considered optimal control problem is well posed. Then, when the optimal control problem is well posed, it is natural to look for conditions which guarantee the attainability of the optimal control problem that is being evaluated. The main tool involved in the solution of the problems stated before is a backward jump matrix linear differential equation (BJMLDE) with a Riccati-type jumping operator. This is formulated using the matrix coefficients of the controlled system and the weight matrices of the performance criterion. We show that the well posedness of the optimal control problem under investigation is guaranteed by the existence of the maximal and bounded solution of the associated BJMLDE with a Riccati-type jumping operator. Further, we show that when the associated BJMLDE with a Riccati-type jumping operator has a maximal solution which satisfies a suitable sign condition, then the optimal control problem is attainable if and only if it has an optimal control in a state feedback form, or if and only if the maximal solution of the BJMLDE with a Riccati-type jumping operator is a stabilizing solution. In order to make the paper more self-contained, we present a set of conditions that correspond to the existence of the maximal solution of the BJMLDE satisfying the desired sign condition. [ABSTRACT FROM AUTHOR]

Dual algebra plays an important role in kinematic synthesis and dynamic analysis, but there are still few studies on dual quaternion matrix theory. This paper provides an efficient method for solving the QLY least squares problem of the dual quaternion matrix equation A X B + C Y D ≈ E , where X, Y are unknown dual quaternion matrices with special structures. First, we define a semi-tensor product of dual quaternion matrices and study its properties, which can be used to achieve the equivalent form of the dual quaternion matrix equation. Then, by using the dual representation of dual quaternion and the GH -representation of special dual quaternion matrices, we study the expression of QLY least squares Hermitian solution of the dual quaternion matrix equation A X B + C Y D ≈ E . The algorithm is given and the numerical examples are provided to illustrate the efficiency of the method. [ABSTRACT FROM AUTHOR]

MAGMA (Matrix Algebra for GPU and Multicore Architectures) is a pivotal open-source library in the landscape of GPU-enabled dense and sparse linear algebra computations. With a repertoire of approximately 750 numerical routines across four precisions, MAGMA is deeply ingrained in the DOE software stack, playing a crucial role in high-performance computing. Notable projects such as ExaConstit, HiOP, MARBL, and STRUMPACK, among others, directly harness the capabilities of MAGMA. In addition, the MAGMA development team has been acknowledged multiple times for contributing to the vendors' numerical software stacks. Looking back over the time of the Exascale Computing Project (ECP), we highlight how MAGMA has adapted to recent changes in modern HPC systems, especially the growing gap between CPU and GPU compute capabilities, as well as the introduction of low precision arithmetic in modern GPUs. We also describe MAGMA's direct impact on several ECP projects. Maintaining portable performance across NVIDIA and AMD GPUs, and with current efforts toward supporting Intel GPUs, MAGMA ensures its adaptability and relevance in the ever-evolving landscape of GPU architectures. [ABSTRACT FROM AUTHOR]

In recent years, research on attribute-based encryption (ABE) has expanded into the quantum domain. Because a traditional single authority can cause the potential single point of failure, an improved lattice-based quantum-resistant identity authentication and policy attribute encryption scheme is proposed, in which the generation of random values is optimized by adjusting parameters in the Gaussian sampling algorithm to improve overall performance. Additionally, in the key generation phase, attributes are processed according to their shared nature, which reduces the computational overhead of the authorization authority. In the decryption phase, the basis transformation of the Lenstra–Lenstra–Lovász (LLL) lattice reduction algorithm is utilized to rapidly convert shared matrices into the shortest vector form, which can reduce the computational cost of linear space checks. The experimental results demonstrate that the proposed method not only improves efficiency but also enhances security compared with related schemes. [ABSTRACT FROM AUTHOR]

In this work, we present a new method for constructing self-dual codes over finite commutative rings R with characteristic 2. Our method involves searching for k × 2 k matrices M over R satisfying the conditions that its rows are linearly independent over R and M M ⊤ = α ⊤ α for an R-linearly independent vector α ∈ R k. Let C be a linear code generated by such a matrix M. We prove that the dual code C ⊥ of C is also a free linear code with dimension k, as well as C / H u l l (C) and C ⊥ / H u l l (C) are one-dimensional free R-modules, where H u l l (C) represents the hull of C. Based on these facts, an isometry from R x + R y onto R 2 is established, assuming that x + H u l l (C) and y + H u l l (C) are bases for C / H u l l (C) and C ⊥ / H u l l (C) over R, respectively. By utilizing this isometry, we introduce a new method for constructing self-dual codes from self-dual codes of length 2 over finite commutative rings with characteristic 2. To determine whether the matrix M M ⊤ takes the form of α ⊤ α with α being a linearly independent vector in R k , a necessary and sufficient condition is provided. Our method differs from the conventional approach, which requires the matrix M to satisfy M M ⊤ = 0 . The main advantage of our method is the ability to construct nonfree self-dual codes over finite commutative rings, a task that is typically unachievable using the conventional approach. Therefore, by combining our method with the conventional approach and selecting an appropriate matrix construction, it is possible to produce more self-dual codes, in contrast to using solely the conventional approach. [ABSTRACT FROM AUTHOR]

(p, q)-Pentadiagonal matrices have attracted considerable attention in the past few years, which are one of the generalizations of pentadiagonal matrices. In the current paper, we present an algorithm for tridiagonalization of (p, q)-pentadiagonal matrices using permutation matrices. Moreover, we give the explicit representations of the permutation matrices and describe the nonzero structure of the obtained tridiagonal matrix. As applications, the tridiagonalization provides us with some useful results such as a breakdown-free algorithm of the matrix determinants and permanents, efficient parallel matrix multiplications and integer powers, and a property of the (p, q)-pentadiagonal matrix over the finite field. Some other applications on numerical linear algebra are also discussed. [ABSTRACT FROM AUTHOR]

This paper introduces a nonconvex ℓ 1 − 2 -analysis model: min x (∥ D ∗ x ∥ 1 − ∥ D ∗ x ∥ 2) s.t.  A x = y , where A is a measurement matrix and D is a tight frame. Our main motivation is to generalize the sparse recovery via ℓ 1 − ℓ 2 minimization to this new model. As a nonconvex model, it is well known that its global minimizer and local minimizer are usually inconsistent. This paper provides a type of null space property (NSP) characterization which are necessary and sufficient conditions for the measurement matrix A such that a vector x can be recovered from A x with a tight frame D via ℓ 1 − 2 -analysis local minimization, or any vector x can be uniformly recovered from A x with a tight frame D via ℓ 1 − 2 -analysis minimization locally and globally. [ABSTRACT FROM AUTHOR]

In this paper, extesions of some inequalities involving matrix means and sector matrices are considered. Among other results, we prove that if two sector matrices A and B satisfy 0 < mIn ≥; ℜ(A),ℜ(B) ≥ MIn, then Φp (ℜ(Aσ B)) ≥ sec2p(θ)Kp(h)Φp(ℜ(Bσ ⊥A)), (0 ≥ p ≥ 2) for every unital positive linear map Φ and arbitrary mean σ, where K(h) := (M+m)² 4Mm is the Kantorovich constant with h := Mm. In addition, we present some norm, numerical radius and determinantal inequalities for sector matrices. [ABSTRACT FROM AUTHOR]

In this paper, we present some generalized inequalities for accretive-dissipative matrices involving convex and concave functions which extend some results of Jabbarzadeh and Kaleibary. Among other results, we show that if T1,T2, · · ·, Tn ∈Mn(C) are accretive-dissipative matrices, then for every non-negative increasing concave function f on [0,∞) and p ≥ 1, we have....Moreover, we also provide the generalized forms of Minkowski's determinant inequality and the Young type determinant inequality involving accretive-dissipative matrices. [ABSTRACT FROM AUTHOR]