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In this paper the construction of C k basis functions on meshes with hanging nodes for hp -finite element methods is proposed. The construction is limited to axis-parallel, rectangular meshes, which can, however, be of any dimension. The basis functions are composed by using hierarchical tensor product shape functions which are defined on patches of so-called constraining nodes and combine univariate Hermite interpolation polynomials and Gegenbauer polynomials. The paper discusses how to integrate the constructed C k basis functions into the assembling process of stiffness matrices and load vectors by using connectivity matrices. Numerical examples illustrate the applicability of the proposed concept. [ABSTRACT FROM AUTHOR]

The class of generalized S D D 1 (G S D D 1 ) matrices is a new subclass of H -matrices. In this paper, we focus on the subdirect sum of G S D D 1 matrices, and some sufficient conditions to ensure that the subdirect sum of G S D D 1 matrices with strictly diagonally dominant (S D D) matrices is in the class of G S D D 1 matrices are given. Moreover, corresponding examples are given to illustrate our results. The class of generalized matrices is a new subclass of -matrices. In this paper, we focus on the subdirect sum of matrices, and some sufficient conditions to ensure that the subdirect sum of matrices with strictly diagonally dominant matrices is in the class of matrices are given. Moreover, corresponding examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

In the field of radar target detection, the conventional approach is to employ the range profile energy accumulation method for detecting extended targets. However, this method becomes ineffective when dealing with non-stationary and non-uniform radar clutter scenarios, as well as long-distance targets with weak radar cross sections (RCSs). In such cases, the signal-to-noise ratio (SNR) of the target echo is severely degraded, rendering the energy accumulation detection algorithm unreliable. To address this issue, this paper presents a new extended target detection method based on the maximum eigenvalue of the Hermitian matrix. This method utilizes a detection model that incorporates observed data and employs the likelihood ratio test (LRT) theory to derive the maximum eigenvalue detector at low SNR. Specifically, the detector constructs a matrix using a sliding window block with the available data and then computes the maximum eigenvalue of the covariance matrix. Subsequently, the maximum eigenvalue matrix is transformed into a one-dimensional eigenvalue image, enabling extended target detection through analogy with the energy accumulation detection method. Furthermore, this paper analyzes the proposed extended target detection method from both theoretical and experimental perspectives, validating it through field-measured data. The results obtained from the measured data demonstrate that the method effectively enhances the SNR in low SNR conditions, thereby improving target detection performance. Additionally, the method exhibits robustness across different scattering center targets. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the necessary and sufficient conditions for solving a dual split quaternion matrix equation A X B   =   C , and present the general solution expression when the solvability conditions are met. As an application, we delve into the necessary and sufficient conditions for the existence of a Hermitian solution to this equation by using a newly defined real representation method. Furthermore, we obtain the solutions for the dual split quaternion matrix equations A X   =   C and X B   =   C . Finally, we provide a numerical example to demonstrate the findings of this paper. [ABSTRACT FROM AUTHOR]

This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, where we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. In addition, we also provide the least squares Hermitian solution in cases where the system of matrix equations is not consistent. To illustrate our main findings, in this paper we present two numerical algorithms and examples. [ABSTRACT FROM AUTHOR]

Dual quaternions have wide applications in automatic differentiation, computer graphics, mechanics, and others. Due to its application in control theory, matrix equation A X B = C has been extensively studied. However, there is currently limited information on matrix equation A X B = C regarding the dual quaternion algebra. In this paper, we provide the necessary and sufficient conditions for the solvability of dual quaternion matrix equation A X B = C , and present the expression for the general solution when it is solvable. As an application, we derive the ϕ -Hermitian solutions for dual quaternion matrix equation A X A ϕ = C , where the ϕ -Hermitian extends the concepts of Hermiticity and η -Hermiticity. Lastly, we present a numerical example to verify the main research results of this paper. [ABSTRACT FROM AUTHOR]

Plagued by heterogeneous clutter, it is a serious challenge for airborne radars to detect low-altitude, weak targets. To overcome this problem, a novel matrix information geometry detector for airborne multi-channel radar is proposed in this paper. The proposed detector applies the given steering vector and array structure information to the matrix information geometry detection method so that it can be used for space–time adaptive detection. While improving the detection performance, the matrix information geometry detector's original anti-clutter advantage is enhanced as well. The simulation experiment results indicate that the proposed detector has advantages in several of the properties related to space–time adaptive detection, while its computational complexity does not increase significantly. Moreover, experiment results based on the measured data verify the superior performance of the proposed method. Sea-detecting data-sharing-program data, mountaintop data, and phased-array radar data are employed to verify the performance advantage of the proposed method in heterogeneous clutter and the ability for weak target detection. [ABSTRACT FROM AUTHOR]

This study takes a detailed look at various inequalities related to the Euclidean operator radius. It examines groups of n-tuple operators, studying how they add up and multiply together. It also uncovers a unique power inequality specific to the Euclidean operator radius. The research broadens its scope to analyze how n-tuple operators, when used as parts of 2 × 2 operator matrices, illustrate inequalities connected to the Euclidean operator radius. By using the Euclidean numerical radius and Euclidean operator norm for n-tuple operators, the study introduces a range of new inequalities. These inequalities not only set limits for the addition, multiplication, and Euclidean numerical radius of n-tuple operators but also help in establishing inequalities for the Euclidean operator radius. This process involves carefully examining the Euclidean numerical radius inequalities of 2 × 2 operator matrices with n-tuple operators. Additionally, a new inequality is derived, focusing specifically on the Euclidean operator norm of 2 × 2 operator matrices. Throughout, the research keeps circling back to the idea of finding and understanding symmetries in linear operators and matrices. The paper highlights the significance of symmetry in mathematics and its impact on various mathematical areas. [ABSTRACT FROM AUTHOR]

Intuitionistic fuzzy (IF) β -minimal description operators can deal with noise data in the IF covering-based rough set theory. That is to say, they can be used to find data that we need in IF environments. For an IF β -covering approximation space (i.e., an IF environment) with a high cardinality, it would be tedious and complicated to use IF set representations to calculate them. Therefore, it is necessary to find a quick method to obtain them. In this paper, we present the notion of IF β -maximal description based on the definition of IF β -minimal description, along with the concepts of IF granular matrix and IF reduction. Moreover, we propose matrix calculation methods for IF covering-based rough sets, such as IF β -minimal descriptions, IF β -maximal descriptions, and IF reductions. Firstly, the notion of an IF granular matrix is presented, which is used to calculate IF β -minimal description. Secondly, inspired by IF β -minimal description, we give the notion of IF β -maximal description. Furthermore, the matrix representations of IF β -maximal descriptions are presented. Next, two types of reductions for IF β -covering approximation spaces via IF β -minimal and fuzzy β -minimal descriptions are presented, along with their matrix representations. Finally, the new calculation methods are compared with corresponding set representations by carrying out several experiments. [ABSTRACT FROM AUTHOR]

Transformer has emerged as one of the modern neural networks that has been applied in numerous applications. However, transformers' large and deep architecture makes them computationally and memory-intensive. In this paper, we propose the block g-circulant matrices to replace the dense weight matrices in the feedforward layers of the transformer and leverage the DCT-DST algorithm to multiply these matrices with the input vector. Our test using Portuguese-English datasets shows that the suggested method improves model memory efficiency compared to the dense transformer but at the cost of a slight drop in accuracy. We found that the model Dense-block 1-circulant DCT-DST of 128 dimensions achieved the highest model memory efficiency at 22.14%. We further show that the same model achieved a BLEU score of 26.47%. [ABSTRACT FROM AUTHOR]

Aiming at addressing the security and efficiency challenges during image transmission, an efficient image cryptosystem utilizing difference matrix and genetic algorithm is proposed in this paper. A difference matrix is a typical combinatorial structure that exhibits properties of discretization and approximate uniformity. It can serve as a pseudo-random sequence, offering various scrambling techniques while occupying a small storage space. The genetic algorithm generates multiple ciphertext images with strong randomness through local crossover and mutation operations, then obtains high-quality ciphertext images through multiple iterations using the optimal preservation strategy. The whole encryption process is divided into three stages: first, the difference matrix is generated; second, it is utilized for initial encryption to ensure that the resulting ciphertext image has relatively good initial randomness; finally, multiple rounds of local genetic operations are used to optimize the output. The proposed cryptosystem is demonstrated to be effective and robust through simulation experiments and statistical analyses, highlighting its superiority over other existing algorithms. [ABSTRACT FROM AUTHOR]

In this paper, the clique polynomial method (CPM) is proposed for the numerical solution of Singular Lane-Emden type equations. A new operational matrix of integration concerning clique polynomials of the complete graph has been generated in its generalized representation. This operational matrix is applied to differential equations and is transformed into a system of algebraic equations that can be solved efficiently with the help of Newton's iterative solver. The efficiency of the developed method is revealed by considering illustrative examples, and the obtained results are compared favorably with the corresponding exact solution and errors. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected graph of order n. In this paper, we introduce a new distance matrix called the universal distance matrix of G, denoted as UD (G) and it is defined as UD (G) = αTr (G) + βD(G) + γJ + δI, where Tr (G) is the diagonal matrix whose elements are the vertex transmissions, and D(G) is the distance matrix of G. Here J is the all-ones matrix, and I is the identity matrix and α, β, γ, δ ∈ R and β ̸= 0. This unified definition enables us to derive the spectra of different matrices associated with the distance matrix of graphs. The set of eigenvalues of the universal distance matrix namely, {ρ1, ρ2,.. ., ρn} is known as the universal distance spectrum of G. As a consequence, by taking appropriate values for α, β, γ, δ ∈ R and β ̸= 0, we obtain the eigenvalues of distance matrix, distance Laplacian matrix, distance signless Laplacian matrix, generalized distance matrix, distance Seidal matrix and distance matrices of graph complements. In this paper, we obtain the universal distance spectra of regular graph, join of two regular graphs, joined union of three regular graphs, generalized joined union of n disjoint graphs with one arbitrary graph H using the Schur complement of a block matrix. [ABSTRACT FROM AUTHOR]

This paper proposes a method to generate a synthetic rating matrix based on user's rational behavior, with the aim of generating a large-scale rating matrix at low cost. Collaborative filtering is one of the major techniques for recommender systems, which is widely used because it can recommend items using only a history of ratings given to the items by users. However, collaborative filtering has some problems such as the cold-start problem and the sparsity problem, both of which are caused by the shortage of ratings in a database (rating matrix). This problem is particularly serious for services that have just started operation or do not have a large number of users. The proposed method generates a rating matrix without missing values using users' rating probabilities, which are obtained from the distribution of their actual ratings. The final synthetic rating matrix is generated after adjusting its sparsity by introducing missing values. The validity of the proposed method is evaluated by comparing the synthetic rating matrix in terms of the similarity of the distribution of several statistics with that of the real data. The synthetic rating matrix is also evaluated by applying it to recommendation to actual users. The experimental results show that the proposed method can generate the synthetic rating matrix that has similar statistics to the real data, and recommendation models trained with the synthetic data achieve comparable recall to that trained with the real data when using the real data as test data. Based on the results of these experiments, this paper also tries to generate the synthetic rating matrix that contains richer information than the real data by increasing the number of users or reducing the sparsity of the rating matrix. The results of these experiments show the possibility that increasing the information contained in a rating matrix could improve recall. [ABSTRACT FROM AUTHOR]

This paper proposes explicit solutions for the algebraic Riccati matrix equation. For single‐input systems in controllable canonical form, the explicit Hermitian solutions of the non‐homogeneous Riccati equation are obtained using the entries of the system matrix, the closed‐loop system matrix, and the weighting matrix. The unknown entries of the closed‐loop system matrix are solved by scalar quadratic equations. For a homogeneous Riccati equation with a zero weighting matrix, the explicit solutions are proposed analytically in terms of the system eigenvalues. The advantages of the explicit solutions are threefold: first, if the system is controllable, the solution is directly given and the invariant subspaces of the Hamiltonian matrix are not required; second, if the system is near singularity, the explicit solution has higher numerical precision compared with the solution computed by numerical algorithms; third, for a real system in the controllable canonical form, the non‐negativity can be analysed for the explicit almost stabilizing solution. [ABSTRACT FROM AUTHOR]

The superposition of two independent Markovian arrival processes (MAPs) is also a Markovian arrival process of which the Markovian representation is given as the Kronecker sum of the transition rate matrices of the component processes. The moments of stationary intervals of the superposition can be obtained by differentiating the Laplace transform (LT) given in terms of the transition rate matrices. In this paper, we propose a streamlined procedure to determine the minimal LT of the merged process in terms of the minimal LT coefficients of the component processes. Combined with the closed-form transformation between moments and LT coefficients, our result enables us to determine the moments of the superposed process based on the moments of the component processes. The main contribution is that the whole procedure can be implemented without explicit Markovian representations. In order to transform the minimal LT coefficients of the component processes into the minimal LT representation of the merged process, we also introduce another minimal representation. A numerical example is provided to illustrate the procedure. [ABSTRACT FROM AUTHOR]

This paper deals with Fredholm properties of the one-sided coupled operator matrix M = A B 0 D I 0 L I by means of generalized Schur factorization and the associated space decompositions. For λ ∈ C , some sufficient conditions are given for λ - M to be Fredholm (resp. left or right Fredholm), and these conclusions are further used to determine the essential spectra of a delay equation and a wave equation with acoustic boundary conditions. [ABSTRACT FROM AUTHOR]

In recent years, the tropical polynomial factorization problem, the tropical matrix decomposition problem, and the tropical multivariate quadratic equation solving problem have been proved to be NP-hard. Some asymmetric cryptographic systems based on tropical semirings have been proposed, but most of them are insecure and have been successfully attacked. In this paper, a new key exchange protocol and a new encryption protocol are proposed based on the difficulty of finding the multiple exponentiation problem of the tropical Jones matrices. The analysis results indicate that our protocol can resist various existing attacks. The complexity of attacking an MEP by adversaries is raised due to the larger number of combinations in the tropical Jones matrices compared to regular matrix polynomials. Furthermore, the index semiring is the non-negative integer cyclic matrix semiring, leading to a higher efficiency in key generation. [ABSTRACT FROM AUTHOR]

This paper presents a numerical method based on an operational matrix of Legendre polynomials for resolving the class of time fractional diffusion (TFD) equations. The operational matrix of fractional order derivatives of the Legendre polynomials is derived as a product of matrices. The collocation method together with the operational matrix of Legendre polynomials are employed to transform the TFD equations into a set of algebraic equations. The perturbation method is applied to show the stability of the discussed method. The accuracy of the suggested method is validated using numerical experiments. The solution obtained by this method is in excellent agreement with the exact solution for the integer order of derivatives and is more precise than the solution obtained by the existing method in which Bernstein polynomials are taken as the basis polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the construction of symmetric matrix whose corresponding graph is connected and unicyclic using some pre-assigned spectral data. Spectral data for the problem consist of the smallest and the largest eigenvalues of each leading principal submatrices. Inverse eigenvalue problem (IEP) with this set of spectral data is generally known as the extremal IEP. We use a standard scheme of labeling the vertices of the graph, which helps in getting a simple relation between the characteristic polynomials of each leading principal submatrix. Sufficient condition for the existence of the solution is obtained. The proof is constructive, hence provides an algorithmic procedure for finding the required matrix. Furthermore, we provide the condition under which the same problem is solvable when two particular entries of the required matrix satisfy a linear relation. [ABSTRACT FROM AUTHOR]

The goods collection and delivery process often relates to distribution logistics problems. The task is to deliver goods from warehouses to customers under specific circumstances. Efforts to optimize the process are largely aimed at reducing overall costs of goods transportation. Among the prominent algorithms for solving the basic type of the delivery (or collection) problem, which includes a single depot and a homogeneous vehicle fleet, is the algorithm developed by Clarke and Wright in 1964. This algorithm minimizes transportation costs by maximizing the savings achieved through merging multiple routes into one. This paper primarily aims to solve the pickup and delivery problem where the goods must be delivered and empty packaging collected in a single process. The request of a customer can be routed from the depot or from another customer. Similarly, the destination of the request may be the depot or another customer. Unlike the original version of the Clarke and Wright algorithm, the initial routes are created to satisfy delivery orders, and therefore, the same customer can occur in multiple routes. Consequently, a situation may arise in which two routes containing one or more common vertices must be combined during the calculation. Furthermore, these vertices need not be the outermost vertices of the routes. This situation cannot be addressed by using the original version of the Clarke and Wright algorithm, and that is why we propose its modification. Merging routes through inner vertices means that the cost savings depend on the configurations of the routes, and therefore, they cannot be calculated a priori. Instead, the dynamic savings matrix must be used. [ABSTRACT FROM AUTHOR]

In this paper, by using the structure properties of algebras and the technique of matrix operation, we investigate the structure of local Lie triple derivations on triangular algebras. Under some mild conditions, we prove that each local Lie triple derivation of triangular algebras can be expressed as the sum of a derivation and a linear map from the triangular algebra to its center that vanishes at Lie triple products. Finally, we apply the main result to the problem of characterizing local Lie triple derivations on an upper triangular matrix algebra. [ABSTRACT FROM AUTHOR]

This paper introduces a method for determinant computation in square matrices. Our approach utilized recursion and the Schur formula to partition the matrix into submatrices. Determinant calculations were performed using the condensation method. To evaluate its computational ef- ficiency, we conducted a floating-point operation per second (FLOPS) analysis, using FLOPS to compare the efficiency of all reduction methods. Pseudocode was provided to demonstrate the computational efficiency of our method in terms of flops and execution time for square matrix determinant calculations. [ABSTRACT FROM AUTHOR]

Methods for interval priority weight estimation from a crisp pairwise comparison matrix were proposed in the interval analytic hierarchy process assuming the vagueness of human evaluation. The interval priority weights estimated by the conventional method do not reflect the intrinsic vagueness in the given pairwise comparison matrix (PCM). This paper proposes parameter-free methods based on minimal conceivable ranges for estimating interval priority weights from a crisp pairwise comparison matrix. The estimated interval priority weight vectors are required to satisfy (1) the potential reproducibility, (2) the normality, and (3) the preservation of the perfect consistent data. Estimation methods of interval priority weights are proposed based on the minimum possible range. We show those proposed methods satisfy the required three properties. The estimation problem of interval priority weights potentially has multiple solutions with which the associated interval PCMs are identical to one another. To make the further investigation simpler, we use an interval priority weight vector among multiple solutions such that the sum of the center values of interval priority weights is one. We compare the estimation methods of interval priority weights from the viewpoint of estimation accuracy by numerical experiments. Namely, by generating crisp pairwise comparison matrices randomly under true interval PCMs, we evaluate the accuracies of the estimated interval priority weight vectors by comparing the true interval priority weight vectors. [ABSTRACT FROM AUTHOR]

Community structure is a significant characteristic of complex networks, and community detection has valuable applications in network structure analysis. Non-negative matrix factorization (NMF) is a key set of algorithms used to solve the community detection issue. Nevertheless, the localization of feature vectors in the adjacency matrix, which represents the characteristics of complex network structures, frequently leads to the failure of NMF-based approaches when the data matrix has a low density. This paper presents a novel algorithm for detecting sparse network communities using non-negative matrix factorization (NMF). The algorithm utilizes local feature vectors to represent the original network topological features and learns regularization matrices. The resulting feature matrices effectively reveal the global structure of the data matrix, demonstrating enhanced feature expression capabilities. The regularized data matrix resolves the issue of localized feature vectors caused by sparsity or noise, in contrast to the adjacency matrix. The approach has superior accuracy in detecting community structures compared to standard NMF-based community detection algorithms, as evidenced by experimental findings on both simulated and real-world networks. [ABSTRACT FROM AUTHOR]

We study various perturbations and their holographic interpretation for non-Abelian T-dual of AdS5 × S5 where the T-duality is applied along the SU(2) of AdS5. This paper focuses on two types of perturbations, namely the scalar and the vector fields on NATD of AdS5 × S5. For scalar perturbations, the corresponding solutions could be categorised into two classes. For one of these classes of solutions, we build up the associated holographic dictionary where the asymptotic radial mode sources scalar operators for the (0 + 1)d matrix model. These scalar operators correspond to either a marginal or an irrelevant deformation of the dual matrix model at strong coupling. We calculate the two point correlation between these scalar operators and explore their high as well as low frequency behaviour. We also discuss the completion of these geometries by setting an upper cut-off along the holographic axis and discuss the corresponding corrections to the scalar correlators in the dual matrix model. Finally, we extend our results for vector perturbations where we obtain asymptotic solutions for a particular class of modes. These are further used to calculate the boundary charge density at finite chemical potential. [ABSTRACT FROM AUTHOR]

It is important to establish solid mathematical knowledge throughout the education period. Matrices are important mathematical objects commonly used in many diverse disciplines, including mathematics, engineering, and science. Most problems encountered in such disciplines are represented by mathematical models with various types of matrices, and solved through some applications of matrix algebra. Although simple or advanced operations of matrices can be performed by using modern programming languages, it usually results in a large fragment of code with a low level of readability due to a complicated sequence of control statements. On the other hand, special-purpose languages handle these operations via library functions, presenting poor integration with other programming environments, and less programming flexibility and practice. This paper addresses the design and development of a programming language, called MxPL, which supports matrix-related mathematics with the provision of some basic structures and functions. Firstly, a grammar that is compatible with the usual notations of matrices is constructed and the parser is produced. Then, the code verifier and interpreter for MxPL programs are implemented. Some code examples are presented to illustrate the performance of several sophisticated matrix operations. The comparative analysis of MxPL is conducted with modern programming languages based on language features. [ABSTRACT FROM AUTHOR]

This paper establishes the linear algebra of the (r, β)-Stirling matrix. Along the way, this paper derives various identities, such as its factorization and relationship to the Pascal matrix and the Stirling matrix of the second kind. Additionally, this paper develops a natural extension of the Vandermonde matrix, which can be used to study and evaluate successive power sums of arithmetic progressions. [ABSTRACT FROM AUTHOR]

This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer's cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a "cellular algebra Clifford theory" for the skew cellular algebras that arise as fixed point subalgebras of cellular algebras. As an application of this general theory, the main result of this paper proves that the Hecke algebras of type G(\ell,p,n) are graded skew cellular algebras. In the special case when p = 2 this implies that the Hecke algebras of type G(\ell,2,n) are graded cellular algebras. The proofs of all of these results rely, in a crucial way, on the diagrammatic Cherednik algebras of Webster and Bowman. Our main theorem extends Geck's result that the one parameter Iwahori-Hecke algebras are cellular algebras in two ways. First, our result applies to all cyclotomic Hecke algebras in the infinite series in the Shephard-Todd classification of complex reflection groups. Secondly, we lift cellularity to the graded setting. As applications of our main theorem, we show that the graded decomposition matrices of the Hecke algebras of type G(\ell,p,n) are unitriangular, we construct and classify their graded simple modules and we prove the existence of "adjustment matrices" in positive characteristic. [ABSTRACT FROM AUTHOR]

In this study, we introduce a system called Matrix Researcher2vec (MResearcher2vec) which generates researcher embedding vectors from their papers and research projects in researchmap and KAKENHI databases. The system includes data on 276,841 researchers, 6,161,592 papers, and research projects. Utilizing natural language processing techniques, the MResearcher2vec model extracts researcher vectors from the papers and research project summaries of KAKENHI grant recipients. The similarity between reseachers is then computed to visualize inter-researcher relationships. The machine learning results have been integrated into a web service, providing a novel approach for academic relationship mining. It can be applied in the matching of research contents and researchers in evaluation of industry-government-academia collaboration and joint research. It contributes in four aspects: (1) exchanges between researchers, (2) creation of opportunities for researchers and companies to connect, (3) further promotion of interdisciplinary research, and (4) reduction of lost opportunities for research institutions to acquire talents. [ABSTRACT FROM AUTHOR]

The class of H-matrices, also known as Generalized Diagonally Dominant (GDD) matrices, plays an important role in many areas of applied linear algebra, as well as in a wide range of applications, such as in dynamical analysis of complex networks that arise in ecology, epidemiology, infectology, neurology, engineering, economy, opinion dynamics, and many other fields. To conclude that the particular dynamical system is (locally) stable, it is sufficient to prove that the corresponding (Jacobian) matrix is an H-matrix with negative diagonal entries. In practice, however, it is very difficult to determine whether a matrix is a non-singular H-matrix or not, so it is valuable to investigate subclasses of H-matrices which are defined by relatively simple and practical criteria. Many subclasses of H-matrices have recently been discussed in detail demonstrating the many benefits they can provide, though one particular subclass has not been fully exploited until now. The aim of this paper is to attract attention to this class and discuss its relation with other more investigated classes, while showing its main advantage, based on its simplicity and elegance. This new approach, which we are presenting in this paper, will be compared with the existing ones, in three possible areas of applications, spectrum localization; maximum norm estimation of the inverse matrix in the point, as well as the block case; and error estimation for LCP problems. The main conclusion is that the importance of our approach grows with the matrix dimension. [ABSTRACT FROM AUTHOR]

If H ν = (ν n , k) n , k ≥ 0 is the matrix with entries ν n , k = ∫ [ 0 , ∞) t n + k n ! d ν (t) , where ν is a nonnegative Borel measure on the interval [ 0 , ∞) , the matrix H ν acts on the space of all entire functions f (z) = ∑ n = 0 ∞ a n z n and induces formally the operator in the following way: H ν (f) (z) = ∑ n = 0 ∞ (∑ k = 0 ∞ ν n , k a k) z n. In this paper, for 0 < p ≤ ∞ , we classify for which measures the operator H ν (f) is well defined on F p and also gets an integral representation, and among them we characterize those for which H ν is a bounded (resp., compact) operator between F p and F ∞ . [ABSTRACT FROM AUTHOR]

We identify the class of real square invertible matrices A for which the signs of the diagonal entries of A-1 match those of A, and begin their study. We say such matrices have the inverse diagonal property (IDP). This class includes many important classes: the positive definite matrices, the M-matrices, the totally positive matrices and some variants, the P-matrices, the diagonally dominant and H-matrices and their inverse classes, as well as triangular matrices. This class is closed under any real invertible diagonal multiplication on either the right or the left. So questions about this class can be reduced to the case of positive diagonal entries. Other basic properties are given. One theme is what conditions need be added to the IDP to insure membership in a familiar class. For example, the positive definite matrices are characterized as certain IDP matrices with special conditions on certain particular principal minors. The tridiagonal case is highlighted. Certain specially simple conditions on such matrices are mentioned that ensure them to be P-matrices, positive definite matrices or M-matrices. We also note that recent results about the invertibility of weakly diagonally dominant matrices are used. Examples are given throughout the paper. [ABSTRACT FROM AUTHOR]

The index of a signed graph is the largest eigenvalue of its adjacency matrix. Let Un,k,4 be the set of all signed complete graphs of order n whose negative edges induce a unicyclic graph of order k and girth at least 4. In this paper, we identify the signed graphs achieving the maximum index in the class Un,k,4. [ABSTRACT FROM AUTHOR]

Linear complementarity problems are fundamental in various mathematical fields, and it is widely recognized that these problems have a unique solution if and only if the matrix related to them is a P-matrix. However, determining whether a matrix is a P-matrix can be extremely challenging or even impossible, especially for large matrices. In this paper, we present a method for establishing the existence and uniqueness of the solution to linear complementarity problems without the need for P-matrix verification. [ABSTRACT FROM AUTHOR]

Arrowhead matrices have attracted attention due to their rich algebraic structures and numerous applications. In this paper, we focus on the enumeration of n × n arrowhead matrices with prescribed determinant over a finite field Fq and over a finite commutative chain ring R. The number of n×n arrowhead matrices over Fq of a fixed determinant a is determined for all positive integers n and for all elements a ∈ Fq. As applications, this result is used in the enumeration of n × n non-singular arrowhead matrices with prescribed determinant over R. Subsequently, some bounds on the number of n × n singular arrowhead matrices over R of a fixed determinant are given. Finally, some open problems are presented. [ABSTRACT FROM AUTHOR]

In this paper we introduce two new classes of inverses for square matrices, which are called 1Drazin (in short, 1D) inverse and Drazin1 (in short, D1) inverse. Next, we investigated the existence and uniqueness of 1D inverse and its dual D1 inverse. Some representation and characterizations of these inverses are derived. In addition to this, we obtain some properties of 1D and D1 inverses through idempotent and binary relations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of Harary Laplacian-energy-like for a simple undirected and connected graph G with order n. We also establish novel matrix results in this regard. Furthermore, by employing matrix order reduction techniques, we derive upper and lower bounds utilizing existing graph invariants and vertex connectivity. Finally, we characterize the graphs that achieve the aforementioned bounds by considering the generalized join operation of graphs. [ABSTRACT FROM AUTHOR]

In this paper, the one- and two-dimensional stochastic multi-order fractional diffusion-wave equations are introduced and a collocation procedure based on the shifted Jacobi polynomials is established to find their numerical solutions. Through this way, some operational matrices regarding classical and stochastic integrals as well as fractional and classical differentiations of these polynomials, are obtained. By representing the problem solution using an expansion of these polynomials (in which the coefficients of the expansion are unknown) and substituting it into the first problem, as well as by employing the obtained operational matrices, a system containing algebraic equations is obtained. Eventually, the coefficients of expansion and subsequently the solution of the original problem are found by solving this system. The correctness of the procedure is studied by solving four examples. [ABSTRACT FROM AUTHOR]

In recent years, long short-term memory (LSTM) networks have been extensively used in various domains such as machine interpretation, language displaying, and so forth. The paper proposes two architectures for accelerating matrix vector multiplication (MVM) in the LSTM network. The first proposal introduces an architecture for the MVM unit by incorporating the concept of Activity Span Reduction (ASR). Here, the partial products for inner product computations (IPC) are generated using ASR logic. This logic ensures that the computational units are active for an optimal duration, effectively utilizing time without compromising the functionality. This approach reduces power consumption as well as area requirements. The ASR technique is able to achieve a 65.9% reduction in power for MVM and occupies 43.77% fewer logic cell units when compared to the state of the art. The second proposal emphasizes the development of a two-stage recursive MVM architecture for the LSTM network. A partial product generator unit using recursive arithmetic architecture is proposed for the IPC in MVM, intended to reduce the power and area consumption. This MVM architecture provides a 6.8% improvement in the maximum clock frequency without an increase in power consumption. The utilization of the proposed designs in the development of the LSTM architecture provides additional evidence supporting the effectiveness of the proposed MVMs. [ABSTRACT FROM AUTHOR]

In this paper, we obtain necessary and sufficient conditions for the equivalence of two matrix summability methods. Some known results are also presented. [ABSTRACT FROM AUTHOR]

This article presents a definition of the pseudosimilar picture fuzzy matrix (PicFM) along with its associated properties. Detailed exploration is undertaken concerning both pseudo-similar and semi-similar picture fuzzy matrices. Furthermore, we analyze relationships pertaining to pseudosimilarity, the preservation of idempotency, and regularity within picture fuzzy matrices. The paper delves deeper into the characterizations of symmetric PicFMs and the group inverse of PicFMs. Finally, we introduce 2x2 Centrosymmetric and K-Centrosymmetric PicFMs, illustrating their attributes with pertinent examples. [ABSTRACT FROM AUTHOR]

This paper proposes a new method for solving a two-person zero-sum fuzzy matrix game with goals, payoffs, and decision variables represented as triangular fuzzy rough numbers. We created a pair of fully fuzzy rough linear programming problems for players. Triangular fuzzy rough numbers can be used to formulate two fuzzy linear programming problems for the first player in the form of upper approximation intervals and lower approximation intervals. Two problems for the second player can be created in the same way. These problems have been split into five sub-crisp problems for the player first and five sub-crisp problems for the player second. The solution to the game can be obtained by solving these ten fuzzy linear programming problems. To demonstrate the method, a numerical example is provided. Using Wolfram Cloud, optimal strategies and game values are calculated for various parameters. Sensitivity analysis is carried out by altering the values of parameters. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected graph with vertex set V (G) = fv1, v2, .... vng and edge set E(G). The Sombor matrix S(G) of a graph G is defined so that its (i; j)-entry is equal to q d2i+ d2j if the vertices vi and vj are adjacent, and zero otherwise, where di denotes the degree of vertex vi in G. In this paper, lower and upper bounds on the spectral radius, energy and Estrada index of the Sombor matrix of graphs are obtained, and the respective extremal graphs are characterized. [ABSTRACT FROM AUTHOR]

The symmetry of the spectrum and the completeness of the eigenfunction system of the Hamiltonian operator matrix have important applications in the symplectic Fourier expansion method in elasticity. However, the symplectic self-adjointness of Hamiltonian operator matrices is important to the characterization of the symmetry of the point spectrum. Therefore, in this paper, the symplectic self-adjointness of infinite dimensional Hamiltonian operators is studied by using the spectral method of unbounded block operator matrices, and some sufficient conditions of the symplectic self-adjointness of infinite dimensional Hamiltonian operators are obtained. In addition, the necessary and sufficient conditions are also investigated for some special infinite dimensional Hamiltonian operators. [ABSTRACT FROM AUTHOR]

In this paper, a maple code is presented to do many operations on interval valued neutrosophic matrices including entering the elements of the matrix, checking whether a given matrix is an interval valued neutrosophic matrix or not, finding complement of an interval valued neutrosophic matrix, finding score, accuracy, and certainty measures, union and intersection of two interval valued neutrosophic matrices, sum and product of two interval valued neutrosophic matrices and finding the transpose of a given interval valued of neutrosophic matrix. What distinguishes this code is its simplicity to understand and to call the functions. Many examples are presented and solved successfully. [ABSTRACT FROM AUTHOR]

Accurate evaluation of the critical nodes in the system is essential work for a multiplatform avionics system (MPAS) for resource allocation and other works. However, current evaluation methods are either limited to the aircraft level or the function module level. There is a lack of research on the evaluation using the information of these two levels. In view of this situation, this paper researches the two-level iterative method of evaluating the importance of aircraft function modules. The influence matrix was constructed by using the node access probability calculated by the PageRank algorithm and the function module weight calculated based on centrality. In addition, the importance of aircraft nodes was used to carry out two-level iteration, and finally, the importance of aircraft function modules was obtained. The experimental results show that this method can comprehensively utilize the information on aircraft cooperative network and function module cooperative network, solve the key problems of two-level iterative evaluation, and meet the requirement of evaluating critical nodes in a system. [ABSTRACT FROM AUTHOR]

This paper presents a new way to solve numerically multiterm variable-order fractional differential equations (MTVOFDEs) with initial conditions by using a class of modified shifted Jacobi polynomials (MSJPs). As their defining feature, MSJPs satisfy the given initial conditions. A key aspect of our methodology involves the construction of operational matrices (OMs) for ordinary derivatives (ODs) and variable-order fractional derivatives (VOFDs) of MSJPs and the application of the spectral collocation method (SCM). These constructions enable efficient and accurate numerical computation. We establish the error analysis and the convergence of the proposed algorithm, providing theoretical guarantees for its effectiveness. To demonstrate the applicability and accuracy of our method, we present five numerical examples. Through these examples, we compare the results obtained with other published results, confirming the superiority of our method in terms of accuracy and efficiency. The suggested algorithm yields very accurate agreement between the approximate and exact solutions, which are shown in tables and graphs. [ABSTRACT FROM AUTHOR]

The powerful parallel computing capability of GPU and the development of matrix processing unit in recent years provide more possibilities to improve the performance of convolutional neural network (CNN) on GPU. For the Winograd convolution algorithm, which is the most widely used in CNN and has the best performance, there are already some tuning results, but they all ignore the utilization of the matrix operation unit and fail to make full use of the computing resources of GPU. This paper introduces a single precision accelerated solution on GPU for CNN. According to the indicators of architecture, the optimal data layout, grid division and block division methods are derived. In order to adapt to a variety of padding in practical application, an efficient dynamic scheme for filling is designed, and by the use of matrix cores, a pipeline algorithm with operator fusion is implemented. The deep learning accelerated library MIOpen in AMD is used as the baseline. Taking several convolutional layers of ResNet50 as the experimental input, the evaluation shows that our approach outperforms MIOpen with the speedup of 1.41x on MI210, and reaches 74% of the peak value of single precision calculations. Applying this method to the training and inference of ResNet50, the speedup of 1.68x is obtained. [ABSTRACT FROM AUTHOR]

This paper focuses on presenting an accurate, stable, efficient, and fast pseudospectral method to solve tempered fractional differential equations (TFDEs) in both spatial and temporal dimensions. We employ the Chebyshev interpolating polynomial for g at Gauss–Lobatto (GL) points in the range [ − 1 , 1 ] and any identically shifted range. The proposed method carries with it a recast of the TFDE into integration formulas to take advantage of the adaptation of the integral operators, hence avoiding the ill-conditioning and reduction in the convergence rate of integer differential operators. Via various tempered fractional differential applications, the present technique shows many advantages; for instance, spectral accuracy, a much higher rate of running, fewer computational hurdles and programming, calculating the tempered-derivative/integral of fractional order, and its spectral accuracy in comparison with other competitive numerical schemes. The study includes stability and convergence analyses and the elapsed times taken to construct the collocation matrices and obtain the numerical solutions, as well as a numerical examination of the produced condition number κ (A) of the resulting linear systems. The accuracy and efficiency of the proposed method are studied from the standpoint of the L 2 and L ∞ -norms error and the fast rate of spectral convergence. [ABSTRACT FROM AUTHOR]