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Multiple algorithms exist for calculating Coulomb (J) or exchange (K) contributions to Fock-like matrices, and it is beneficial to develop a framework that allows the seamless integration and combination of different J and K construction algorithms. In Psi4, we have implemented the "CompositeJK" formalism for this purpose. CompositeJK allows for the combination of any J and K construction algorithms for any quantum chemistry method formulated in terms of J-like or K-like matrices (including, but not limited to, Hartree–Fock and density functional theory) in a highly modular and intuitive fashion, which is simple to utilize for both developers and users. Using the CompositeJK framework, Psi4 was interfaced to the sn-LinK implementation in the GauXC library, adding the first instance of noncommercial graphics processing unit (GPU) support for the construction of Fock matrix elements to Psi4. On systems with hundreds of atoms, the interface to the CPU sn-LinK implementation displays a higher performance than all the alternative JK construction methods available in Psi4, with up to x2.8 speedups compared to existing Psi4JK implementations. The GPU sn-LinK implementation, harnessing the power of GPUs, improves the observed performance gains to up to x7.0. [ABSTRACT FROM AUTHOR]

An outstanding property of any Hamiltonian system is the symplecticity of its flow, namely, the continuous trajectory preserves volume in phase space. Given a symplectic but discrete trajectory generated by a transition matrix applied at a fixed time-increment (τ > 0), it was generally believed that there exists a unique Hamiltonian producing a continuous trajectory that coincides at all discrete times (t = nτ with n integers) as long as τ is small enough. However, it is now exactly demonstrated that, for any given discrete symplectic dynamics of a harmonic oscillator, there exist an infinite number of real-valued Hamiltonians for any small value of τ and an infinite number of complex-valued Hamiltonians for any large value of τ. In addition, when the transition matrix is similar to a Jordan normal form with the supradiagonal element of 1 and the two identical diagonal elements of either 1 or −1, only one solution to the Hamiltonian is found for the case with the diagonal elements of 1, but no solution can be found for the other case. [ABSTRACT FROM AUTHOR]

Within the context of fewest-switch surface hopping (FSSH) dynamics, one often wishes to remove the angular component of the derivative coupling between states J and K . In a previous set of papers, Shu et al. [J. Phys. Chem. Lett. 11, 1135–1140 (2020)] posited one approach for such a removal based on direct projection, while we isolated a second approach by constructing and differentiating a rotationally invariant basis. Unfortunately, neither approach was able to demonstrate a one-electron operator O ̂ whose matrix element J O ̂ K was the angular component of the derivative coupling. Here, we show that a one-electron operator can, in fact, be constructed efficiently in a semi-local fashion. The present results yield physical insight into designing new surface hopping algorithms and are of immediate use for FSSH calculations. [ABSTRACT FROM AUTHOR]

Exact methods for the exponentiation of matrices of dimension N can be computationally expensive in terms of execution time (N3) and memory requirements (N2), not to mention numerical precision issues. A matrix often exponentiated in the natural sciences is the rate matrix. Here, we explore five methods to exponentiate rate matrices, some of which apply more broadly to other matrix types. Three of the methods leverage a mathematical analogy between computing matrix elements of a matrix exponential process and computing transition probabilities of a dynamical process (technically a Markov jump process, MJP, typically simulated using Gillespie). In doing so, we identify a novel MJP-based method relying on restricting the number of "trajectory" jumps that incurs improved computational scaling. We then discuss this method's downstream implications on mixing properties of Monte Carlo posterior samplers. We also benchmark two other methods of matrix exponentiation valid for any matrix (beyond rate matrices and, more generally, positive definite matrices) related to solving differential equations: Runge–Kutta integrators and Krylov subspace methods. Under conditions where both the largest matrix element and the number of non-vanishing elements scale linearly with N—reasonable conditions for rate matrices often exponentiated—computational time scaling with the most competitive methods (Krylov and one of the MJP-based methods) reduces to N2 with total memory requirements of N. [ABSTRACT FROM AUTHOR]

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]

The eccentricity matrix ɛ (G) of a strongly connected digraph G is defined as ɛ (G) i j = d (v i , v j) , if d (v i , v j) = min { e + (v i) , e − (v j) } , 0 , otherwise. , where e + (v i) = max { d (v i , v j) ∣ v j ∈ V (G) } is the out-eccentricity of the vertex v i of G , and e − (v j) = max { d (v i , v j) ∣ v i ∈ V (G) } is the in-eccentricity of the vertex v j of G. The eigenvalue of ɛ (G) with the largest modulus is called the eccentricity spectral radius of G. In this paper, we obtain lower bounds for the eccentricity spectral radius among all join digraphs with a fixed dichromatic number. We also give upper bounds for the eccentricity spectral radius of some special join digraphs with a fixed dichromatic number. [ABSTRACT FROM AUTHOR]

Summary: This paper develops a new image encryption algorithm based on a novel two‐dimensional chaotic map and bit‐level dynamic scrambling. First, multiple one‐dimensional chaotic maps are coupled to construct a novel two dimensions Logistic‐Sine‐coupling chaos map (2D‐LSCCM). The performance analysis shows that the 2D‐LSCCM has more complex chaotic characteristics and wider chaotic range than many extant 2D chaos maps. Second, original image matrix combines with hash algorithm SHA‐256 to generate a hash value. The initial values of 2D‐LSCCM are generated based on the hash value. Third, the original image matrix is divided into multiple sub‐matrices by wavelet transform, followed by scrambling by an improved Knuth shuffle algorithm. Fourth, the scrambled multiple sub‐matrices are stitched into an image matrix of M×N×3$$ M\times N\times 3 $$ and converted into a binary matrix. The chaotic sequence generated by 2D‐LSCCM is introduced as a control sequence to control the bit‐level scrambling of pixel points, which realizes the bit‐level dynamic scrambling. Finally, the diffusion operation is performed by parameter par and chaotic sequence to obtain the final encrypted image. The algorithm security analysis and simulation examples demonstrate the effectiveness of the proposed encryption scheme. [ABSTRACT FROM AUTHOR]

For a finite simple undirected graph G , the universal adjacency matrix U (G) is a linear combination of the adjacency matrix A (G) , the degree diagonal matrix D (G) , the identity matrix I and the all-ones matrix J , that is U (G) = α A (G) + β D (G) + γ I + η J , where α , β , γ , η ∈ ℝ and α ≠ 0. The cozero-divisor graph Γ ′ (R) of a finite commutative ring R with unity is a simple undirected graph with the set of all nonzero nonunits of R as vertices and two vertices x and y are adjacent if and only if x ∉ y R and y ∉ x R. In this paper, we study structural properties of Γ ′ (R) by defining an equivalence relation on its vertex set in terms of principal ideals of the ring R. Then we obtain the universal adjacency eigenpairs of Γ ′ (R) and its complement, and as a consequence one may obtain several spectra like the adjacency, Seidel, Laplacian, signless Laplacian, normalized Laplacian, generalized adjacency and convex linear combination of the adjacency and degree diagonal matrix of Γ ′ (R) and Γ ′ (R) ¯ in an unified way. Moreover, we get the universal adjacency eigenpairs of the cozero-divisor graph and its complement for a reduced ring and the ring of integers modulo m in a simpler form. [ABSTRACT FROM AUTHOR]

Motivated by the inverse formula of the distance matrix of a tree and the Moore–Penrose inverse of a circum-Euclidean distance matrix (CEDM), in this paper, we study a general real square matrix M whose Moore–Penrose inverse can be expressed as the sum of a Laplacian-like matrix L and a rank one matrix. In particular, for a symmetric hollow matrix M, under an assumption, we show that M is a Euclidean distance matrix if and only if L is positive semidefinite. Based on this, we obtain a new characterization for CEDMs involving their Moore–Penrose inverses. As an application, we show that the distance matrices of block graphs and odd-cycle-clique graphs are CEDMs. Finally, we establish an interlacing property between the eigenvalues of a Euclidean distance matrix M (including the singular case) and its associated Laplacian-like matrix L, which generalizes the interlacing property proved for the distance matrices of trees. [ABSTRACT FROM AUTHOR]

We define the Wasserstein mean of $ n \times n $ n × n unipotent matrices by solving its corresponding matrix equation, and study its fundamental properties. Under a binomial expansion for the weighted geometric mean of unipotent matrices, we provide the explicit formula of two-variable Wasserstein means. Furthermore, we show that the explicit formula of multi-variable Wasserstein mean for $ n= 2,3. $ n = 2 , 3. [ABSTRACT FROM AUTHOR]

In this paper, closed formulas for the eigenvectors of a particular class of matrices generated by generalized permutation matrices, named generalized circulant matrices, are presented. [ABSTRACT FROM AUTHOR]

This research study's primary goal is to create an efficient wavelet collocation technique to resolve a kind of nonlinear fractional order systems of ordinary differential equations that arise in the modeling of autocatalytic chemical reaction problems. Here, we created the functional matrix of integration for the Fibonacci wavelets. The Fibonacci wavelet collocation method is employed to find the numerical solution of the system of nonlinear coupled ordinary differential equations of both integer and fractional order. The nonlinear Brusselator system is transformed into an algebraic equation system using the operational matrices of fractional derivative and collocation technique. These algebraic equations are treated by the Newton–Raphson method, and obtained unknown coefficient values are substituted in the approximation. We demonstrate our method's computational effectiveness and accuracy using different model constraints in the numerical examples. The effectiveness and consistency of the developed strategy's performance are shown in graphs and tables. Comparisons with existing methods available in the literature demonstrate the high accuracy and robustness of the developed Fibonacci wavelet collocation method. Mathematical software Mathematica has been used to perform all calculations. [ABSTRACT FROM AUTHOR]

A general review of quantum molecular similarity structure and applications is presented. The backbone of the discussion corresponds to the general problem of the data structure associated with the mathematical representation of a molecular set. How to standardize, and how to compare it to any other problem. This computational track describes the exact isometric vectors of the similarity matrix in a Minkowskian space. The further aim is to construct a set of origin-shifted vectors forming the vertices of a molecular polyhedron. From here, one can calculate a set of statistical-like momenta, providing a set of scalars that describe in a compact form the attached molecular set. Finally, the definition of a quantum QSPR operator permits building up a system of equations that can be further employed to determine the unknown properties of molecules in the original set. This last achievement leads to a quantum QSPR algorithm comparable with the classical QSPR counterpart but described in molecular space, not parameter space. [ABSTRACT FROM AUTHOR]

In this paper, we consider how to accurately solve the singular value problem and the linear system for a class of Polynomial-Vandermonde-type (PVT) matrices, which belongs to the class of negative matrices introduced by Huang and Xue (Adv Comput Math 47:73, 2021), and these negative matrices arise in some applications such as interpolation problems. In order to parameterize PVT matrices, we present the explicit expressions of initial minors of such matrices. An algorithm is designed to accurately compute the parametrization matrix for PVT matrices. Based on the accurate parametrization algorithm, all the singular values, both large and small, of PVT matrices are computed, and the linear systems associated with PVT matrices are solved to high relative accuracy. Numerical experiments are performed to confirm the claimed high relative accuracy. [ABSTRACT FROM AUTHOR]

We present a method for computing actions of the exponential-like φ -functions for a Kronecker sum K of d arbitrary matrices A μ . It is based on the approximation of the integral representation of the φ -functions by Gaussian quadrature formulas combined with a scaling and squaring technique. The resulting algorithm, which we call phiks, evaluates the required actions by means of μ -mode products involving exponentials of the small sized matrices A μ , without forming the large sized matrix K itself. phiks, which profits from the highly efficient level 3 BLAS, is designed to compute different φ -functions applied on the same vector or a linear combination of actions of φ -functions applied on different vectors. In addition, thanks to the underlying scaling and squaring techniques, the desired quantities are available simultaneously at suitable time scales. All these features allow the effective usage of phiks in the exponential integration context. In fact, our newly designed method has been tested in popular exponential Runge–Kutta integrators of stiff order from one to four, in comparison with state-of-the-art algorithms for computing actions of φ -functions. The numerical experiments with discretized semilinear evolutionary 2D or 3D advection–diffusion–reaction, Allen–Cahn, and Brusselator equations show the superiority of the proposed μ -mode approach. [ABSTRACT FROM AUTHOR]

Catalan–Stieltjes matrices and their corresponding Catalan-like numbers are very important in combinatorics. In this paper, we give a sufficient condition for the Catalan–Stieltjes matrix to be triangular strictly totally positive (Δ STP), that is, all its minors are positive except those which are zero due to the triangular structure. On this basis, we show that it is possible to compute the bidiagonal factorization of a class of Catalan–Stieltjes matrices with high relative accuracy (HRA). We also demonstrate that this class of matrices includes many well-known combinatorial triangles, such as the Catalan triangle of Aigner and the Bell triangle. Moreover, we construct an accurate algorithm to compute the bidiagonal factorization of some Catalan–Stieltjes matrices. The algorithm guarantees HRA for several linear algebraic computations associated with these matrices, like computing their singular values, their inverses, as well as the solutions of some linear systems. We show the accuracy and effectiveness of the proposed algorithm through numerical experiments. [ABSTRACT FROM AUTHOR]

As a significant part of modern decision science, multi-attribute decision-making (MADM) has attracted the interests of more and more researchers. In the process of decision analysis, decision-makers or experts usually give decision information in the forms of preference order, utility values or preference relation. Utility values, i.e. attribute values, generally include real-values, interval-values and linguistic-values. This article studies ternary hybrid decision matrices (THDMs), which contain the above three types of attributes. Firstly, we give a calculation method for attribute weight vectors in THDMs, and transform THDMs into fuzzy preference relations (FPRs). Secondly, we consider additive consistency of induced FPRs and introduce an inconsistency repairing method. Thirdly, we propose an MADM method for THDMs in view of (T, S)-fuzzy rough sets (FRSs) with FPRs. Finally, we give an example to explain the raised method in detail and conduct comparative analysis to illustrate the rationality and practicability of our raised method. [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]

We revisit the seminal Brill–Noether algorithm for plane curves with ordinary singularities. Our new approach takes advantage of fast algorithms for polynomials and structured matrices. We design a new probabilistic algorithm of Las Vegas type that computes a Riemann–Roch space in expected sub-quadratic time. [ABSTRACT FROM AUTHOR]

An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to a row-equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse row-equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the ℓ 1 -norm of the Graver basis is bounded by a function of the maximum ℓ 1 -norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix row-equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such a row-equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the ℓ 1 -norm of the Graver basis of the constraint matrix, when parameterized by the ℓ 1 -norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix. [ABSTRACT FROM AUTHOR]

Due to the increasing applications of dual quaternion and their matrices in recent years, as well as the significance of LU decomposition as a matrix decomposition technique, in this paper, we propose dual quaternion Gaussian transformation and obtain dual quaternion LU decomposition by using Gaussian transformation. We also use the total order of dual numbers to obtain the partial pivoting dual quaternion LU decomposition. Based on the real structure-preserving algorithm of quaternion matrix, we propose the real structure-preserving algorithms of LU decomposition and partial pivoting LU decomposition for dual quaternion matrix. Numerical experiments have verified the effectiveness of the new real structure-preserving approaches. [ABSTRACT FROM AUTHOR]

We propose a novel method for the estimation of the number of factors in approximate factor models. The model is based on a penalized maximum likelihood approach incorporating an adaptive hierarchical lasso penalty function that enables setting entire columns of the factor loadings matrix to zero, which corresponds to removing uninformative factors. Additionally, the model is capable of estimating weak factors by allowing for sparsity in the non zero columns. We prove that the proposed estimator consistently estimates the true number of factors. Simulation experiments reveal superior selection accuracies for our method in finite samples over existing approaches, especially for data with cross-sectional and serial correlations. In an empirical application on a large macroeconomic dataset, we show that the mean squared forecast errors of an approximate factor model are lower if the number of included factors is selected by our method. [ABSTRACT FROM AUTHOR]

For a finite, positive Borel measure \mu on (0,1) we consider an infinite matrix \Gamma _\mu, related to the classical Hausdorff matrix defined by the same measure \mu, in the same algebraic way that the Hilbert matrix is related to the Cesáro matrix. When \mu is the Lebesgue measure, \Gamma _\mu reduces to the classical Hilbert matrix. We prove that the matrices \Gamma _\mu are not Hankel, unless \mu is a constant multiple of the Lebesgue measure, we give necessary and sufficient conditions for their boundedness on the scale of Hardy spaces H^p, \, 1 \leq p < \infty, and we study their compactness and complete continuity properties. In the case 2\leq p<\infty, we are able to compute the exact value of the norm of the operator. [ABSTRACT FROM AUTHOR]

This article, using generalized inverses of matrices, studies the spaces whose elements are matrices. Based on the results obtained, for linear models for random matrices, the article explores the role of ordinary least square estimators in identifying linear estimable functions, and the role of minimum norm ordinary least square estimators in creating linear unbiased estimators. With added conditions on the covariance matrix for vectorized response, it is shown that a linear unbiased estimator constructed from the minimum norm ordinary least square estimator is a best linear unbiased estimator with respect to the risk induced from squared distance loss. [ABSTRACT FROM AUTHOR]

Often, reliability systems suffer shocks from external stress factors, stressing the system at random. These random shocks may have non ignorable effects on the reliability of the system. In this article, we provide sufficient (and necessary) conditions on components' lifetimes and their survival probabilities from random shocks for comparing the lifetimes of two parallel systems with independent components by means of usual stochastic and reversed hazard rate orders, when the matrix of parameters changes to another matrix in a mathematical sense. We also discuss stochastic comparisons of series systems with independent components under random shocks in the sense of usual stochastic and hazard rate orders, by using the concept of vector majorization and related orders. The results developed in this article generalize some known results in the literature. [ABSTRACT FROM AUTHOR]

This study introduces delay-aware cloudlet placement with a modified k-means (DACP-mk) algorithm to address the complexities of mobility and optimize cloudlet placement in a mobility-enhanced environment. DACP-mk enhances the traditional k-means algorithm by incorporating modified regional and global delay matrices to enable the simulation of a delay-aware cloudlet movement and placement strategy using a cluster-based approach within a smart healthcare platform. An optimized cloudlet position is generated to achieve a uniform distribution of mobile devices among placed cloudlets, maximize device coverage, and minimize network access delay within the cloudlet coverage distance. The proposed algorithm determines the optimal positions for cloudlets at coordinates (161.90, 103.24), (080.22, 139.52), and (080.29, 051.21) with an increase of 02.42%, 00.54%, 38.71%, and 49.00%, respectively, in terms of global mobile device coverage compared to modified k-means Manhattan (mkMAN), k-means, energy-efficient cloudlet placement method (ECPM) and enhanced adaptive cloudlets placement with covering algorithm (EACP-CA), and a decrease of 00.71% and 58.27% in global network access delay compared to mkMAN and k-means algorithms. The proposed algorithm offers enhanced healthcare support and cost-effective services, improving the Quality of Service (QoS) and ultimately contributing to saving lives. [ABSTRACT FROM AUTHOR]

The central objective of this paper is to investigate the lacunary statistical convergence of order γ and strong lacunary summability of order γ for sequences of fuzzy functions by using the modulus function and a regular matrix. We examine the relevant inclusion relations between these spaces under specific conditions. Additionally, we study properties related to these spaces and establish several interesting results. [ABSTRACT FROM AUTHOR]

A subalgebra of the full matrix algebra M n (K) , K a field, satisfying the identity [ x 1 , y 1 ] [ x 2 , y 2 ] ⋯ [ x q , y q ] = 0 is called a D q subalgebra of M n (K). In the paper we deal with the structure, conjugation and isomorphism problems of maximal D q subalgebras of M n (K). We show that a maximal D q subalgebra A of M n (K) is conjugated with a block triangular subalgebra of M n (K) with maximal commutative diagonal blocks. By analysis of conjugations, the sizes of the obtained diagonal blocks are uniquely determined. It reduces the problem of conjugation of maximal D q subalgebras of M n (K) to the analogous problem in the class of commutative subalgebras of M n (K). Further examining conjugations, in case A is contained in the upper triangular matrix algebra U n (K) , we prove that A is already in a block triangular form. We consider the isomorphism problem in a certain class of maximal D q subalgebras of M n (K) which contain all D q subalgebras of M n (K) with maximum dimension. In case K is algebraically closed, we invoke Jacobson's characterization of maximal commutative subalgebras of M n (K) with maximum (K -)dimension to show that isomorphic subalgebras in this class are already conjugated. To illustrate it, we invoke results from [19] and find all isomorphism (equivalently conjugation) classes of D q subalgebras of M n (K) with maximum possible dimension, in case K is algebraically closed. [ABSTRACT FROM AUTHOR]

A linear Lie algebra is splittable if it contains the semisimple and nilpotent parts of each element. It is early known that a solvable linear Lie algebra g is splittable if and only if g = a + n , where a is an abelian subalgebra of g composed of semisimple elements and n is the ideal of all nilpotent matrices of g. In this paper, using elementary linear algebra we give a direct proof of the theorem and related results. Besides, we determine the structure of linear Lie algebras composed of semisimple or nilpotent elements. [ABSTRACT FROM AUTHOR]

The Gau-Wu number is an important matrix invariant describing the geometry of the numerical range. In this work, the group of non-singular linear preservers of the Gau-Wu number is completely characterized. [ABSTRACT FROM AUTHOR]

We give a characterization when a nonzero square matrix with integer entries is periodic using cyclotomic polynomials. We also determine the set of all possible periods of periodic matrices with integer entries via Euler Phi functions. [ABSTRACT FROM AUTHOR]

Let U be an associative unital algebra containing a non-trivial idempotent e. We consider U as a superalgebra whose Z 2 -grading is induced by e. This paper aims to describe Lie superderivations of U . In particular, we characterize the general form of Lie superderivations of U and apply it to present the necessary and sufficient conditions for a Lie superderivation on U to be proper. Similar results have been presented for triangular algebras as superalgebras, wherein their Z 2 -grading is also obtained concerning standard idempotent. The main result is subsequently applied to full matrix algebras and upper triangular matrix algebras. [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 paper presents a novel statistical analysis-based approach to non-parametric damage detection in truss bridges. The method utilizes the normalized acceleration response time histories (NARTHs) of a bridge under random excitation. The coefficients of variation matrices are calculated using NARTHs for the truss bridge in both its baseline and damaged states. The results are shown as the difference in covariance matrices (DCMs) between the two conditions (pristine and damaged). The row or column-wise summation of the DCM matrix yields a summation of the difference in covariance matrix (SDCM) bar plot having one distinct peak for the damaged member. The location and relative severity of the damage can be determined by examining the DCM matrix and the SDCM bar plot. The variation in magnitude of the coefficients demonstrates the relative severity of damage. Experimental investigation of the proposed method for detecting damage on the truss bridge shows promising results. The approach is then numerically validated using a finite element model of a truss bridge. The proposed method effectively identified, located, and evaluated damage, even when the acceleration time histories are contaminated with noise. The case of multiple-member damage detection in the truss structure is further investigated in the study. The early detection of damage and monitoring of its progression through this method can assist in creating efficient maintenance strategies for truss bridge infrastructure. [ABSTRACT FROM AUTHOR]

Image encryption is a vital area in our current digital age, playing an important role in protecting information and improving data quality. Encryption protects privacy and enhances security in various applications, such as communications, cloud storage, and digital transmission. As the size and complexity of images increase, the importance of using parallel methods in image processing and encryption becomes more prominent. These methods allow exploiting the multiple processing capabilities available in modern devices, such as multi-core processors, which enhance efficiency and speed in processing large data sets.in this paper, we present a modified Blowfish algorithm and a hybrid parallel approach to overcome the known weaknesses of the traditional Blowfish algorithm. First, we use a Pascal matrix to permuting the image pixels, and the results of this operation are used as input to the modified version of the Blowfish algorithm. In this version, the P matrix is modified using a hybrid chaos approach, which improves the encryption process. Furthermore, this encryption is implemented using a hybrid parallel processing approach, which enhances the performance and efficiency of data processing. Tests and Result are shown using test images (256*256) from the USC-SIPI and CVG-UGR databases. were quicker outcomes and more secure encryption. In addition, the average execution time of encryption and decryption reached (0.00618ms, 0.003292 ms) the information entropy screening rate reached 7.99735, which is close to the optimal ratio of 8. and NPCR and UACI reached (99.639, 33.42825). The algorithm has achieved a high level of security. [ABSTRACT FROM AUTHOR]

In this paper we analyze and solve eigenvalue programs, which consist of the task of minimizing a function subject to constraints on the "eigenvalues" of the decision variable. Here, by making use of the FTvN systems framework introduced by Gowda, we interpret "eigenvalues" in a broad fashion going beyond the usual eigenvalues of matrices. This allows us to shed new light on classical problems such as inverse eigenvalue problems and also leads to new applications. In particular, after analyzing and developing a simple projected gradient algorithm for general eigenvalue programs, we show that eigenvalue programs can be used to express what we call vanishing quadratic constraints. A vanishing quadratic constraint requires that a given system of convex quadratic inequalities be satisfied and at least a certain number of those inequalities must be tight. As a particular case, this includes the problem of finding a point x in the intersection of m ellipsoids in such a way that x is also in the boundary of at least ℓ of the ellipsoids, for some fixed ℓ > 0 . At the end, we also present some numerical experiments. [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]

Let f (x , y) > 0 be a real symmetric function. For a connected graph G , the weight of edge v i v j is equal to the value f (d i , d j) , where d i is the degree of vertex v i. The degree-based weighted adjacency matrix is defined as A f (G) , in which the (i , j) -entry is equal to f (d i , d j) if v i v j is an edge of G and 0 otherwise. In this paper, we first give some bounds of the weighted adjacency eigenvalue λ 1 (A f (G)) in terms of λ 1 (A f (H)) , where H is obtained from G by some kinds of graph operations, including deleting vertices, deleting an edge and subdividing an edge, and examples are given to show that bounds are tight. Second, we obtain some bounds for the largest weighted adjacency eigenvalue λ 1 (A f (G)) of irregular weighted graphs. [ABSTRACT FROM AUTHOR]

A randomly generated complex symmetric matrix using the adaptive Monte Carlo method is taken as a general form of Majorana neutrino mass matrix, which is diagonalized by the use of eigenvectors. We extract all the neutrino oscillation parameters, i.e. two mass-squared differences (Δm212 and Δm322), three mixing angles (θ12, θ13, θ23) and three phases, i.e. one Dirac CP violating phase (δCP) and two Majorana phases (α1 and α2). The Charge-Parity (CP) violating phases are extracted from the mixing matrix constructed with the eigenvectors of the Hermitian matrix formed by the complex symmetric mass matrix. All the latest neutrino oscillation parameters within 3σ bound are only allowed in Normal Hierarchy (NH) consistent with the latest Planck cosmological upper bound, ∑|mi|<0.12eV. Three cases of one-zero texture i.e. m11=0, m12=0, m13=0 and two cases of two-zero texture, i.e. m11,m12=0 and m11,m13=0 are allowed in normal hierarchy whereas none of zero texture is allowed in inverted hierarchy. We also study effective neutrino masses mβ in tritium beta decay and mββ in neutrinoless double beta decay and calculate numerically the allowed range of mass elements of Majorana mass matrix. [ABSTRACT FROM AUTHOR]

ABSTRACT In this article, we propose an adaptive Kalman filtering with adaptive window length based on Q‐learning for dynamic systems with unknown model information. The iteration step length of the Q‐function is quantitatively adjusted through the influence function. The adaptive Kalman filtering algorithm is used to set an appropriate weight matrix for the Q‐function to estimate unknown model parameters. One numerical example and a practice‐oriented case are given to illustrate the effectiveness of the proposed method. It is shown that this filtering can provide state estimates of best accuracy among all the compared methods when the model mismatch and noise statistical characteristics change. [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]

The inverse eigenvalue problem studies the possible spectra among matrices whose off-diagonal entries have their zero-nonzero patterns described by the adjacency of a graph G. In this paper, we refer to the i -nullity pair of a matrix A as (null (A) , null (A (i)) , where A (i) is the matrix obtained from A by removing the i -th row and column. The inverse i -nullity pair problem is considered for complete graphs, cycles, and trees. The strong nullity interlacing property is introduced, and the corresponding supergraph lemma and decontraction lemma are developed as new tools for constructing matrices with a given nullity pair. [ABSTRACT FROM AUTHOR]

The paper considers the convergence of the complex block Jacobi diagonalization methods under the large set of the generalized serial pivot strategies. The global convergence of the block methods for Hermitian, normal and J -Hermitian matrices is proven. In order to obtain the convergence results for the block methods that solve other eigenvalue problems, such as the generalized eigenvalue problem, we consider the convergence of a general block iterative process which uses the complex block Jacobi annihilators and operators. [ABSTRACT FROM AUTHOR]

Let F be a field and q (x) a quadratic polynomial in F [ x ] with q (0) ≠ 0. We denote by T ∞ (F) the algebra of all infinite upper triangular matrices over the field F. A matrix A ∈ T ∞ (F) is called a quadratic matrix with respect to q (x) if q (A) = 0. In this paper, we first investigate the subgroup in T ∞ (F) generated by all quadratic matrices with respect to q (x) and then present some applications. [ABSTRACT FROM AUTHOR]