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

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

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

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

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

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

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

Let un = un(k) denote the generalized Leonardo number defined recursively by un = un−1 + un−2 + k for n ≥ 2, where u0 = u1 = 1. Terms of the sequence un(1) are referred to simply as Leonardo numbers. In this paper, we find expressions for the determinants of several Toeplitz–Hessenberg matrices having generalized Leonardo number entries. These results are obtained as special cases of more general formulas for the generating function of the corresponding sequence of determinants. Special attention is paid to the cases 1 ≤ k ≤ 7, where several connections are made to entries in the On-Line Encyclopedia of Integer Sequences. By Trudi's formula, one obtains equivalent multi-sum identities involving sums of products of generalized Leonardo numbers. Finally, in the case k = 1, we also provide combinatorial proofs of the determinant formulas, where we make extensive use of sign-changing involutions on the related structures. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the subset-strong product of graphs and give a method for calculating the adjacency spectrum of this product. In addition, exact expressions for the first and second Zagreb indices of the subset-strong products of two graphs are reported. Examples are provided to illustrate the applications of this product in some growing graphs and complex networks. [ABSTRACT FROM AUTHOR]

In this paper, a novel measurement matrix construction method based on adaptive cross-approximation (ACA) is proposed to improve the performance of the compressive sensing-based method of moments (CS-MoM) for analyzing electromagnetic scattering problems. ACA is based on a weight scheme and is able to recognize the rows and columns that contribute significantly to the matrix. Thus, the object is divided into multiple blocks, and the impedance matrix is partitioned into near-field and far-field groups to establish the condition for applying ACA. Then, the row indexes are extracted from the group with the highest number of ACA recognized rows in the far-field groups of each block. Finally, by combining all row indexes to extract the impedance matrix, a lower-dimensional and deterministic measurement matrix is constructed, thereby improving computational efficiency. Numerical simulation results validate the accuracy and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The role of an expert in the decision-making process is crucial. If we ask an expert to help us to make a decision we assume their honesty. But what if the expert is dishonest? Then, the answer on how difficult it is for an expert to provide manipulated data in a given case of decision-making process becomes essential. In the presented work, we consider manipulation of a ranking obtained by the Geometric Mean Method applied to a pairwise comparisons matrix. More specifically, we propose an algorithm for finding an almost optimal way to swap the positions of two selected alternatives in a ranking. We also define a new index which measures how difficult such manipulation is in a given case. [ABSTRACT FROM AUTHOR]

In this paper, a new micromechanical hysteresis loop constitutive model of C/SiC composites with different interphases was developed considering the probabilistic‐statistical matrix fragmentation process. The lengths of matrix fragmentation were divided into three types, that is, long matrix fragments (LMFs), medium matrix fragments (MMFs), and short matrix fragments (SMFs). The distributions of the LMFs, MMFs, and SMFs with increasing tensile stress were determined using the probabilistic‐stochastic model by assuming the two‐parameter matrix strength distribution. The micro stress field of the LMFs, MMFs, and SMFs upon unloading and reloading was obtained and adopted to determine the corresponding stress‐strain relations. The interaction of matrix fragmentation lengths, especially for the LMFs with large debonding energy (LDE) and SMFs, was considered in the closed‐form constitutive model and hysteresis‐based inverse tangent modulus (ITMs) damage parameter. Synergistic effects of the fiber volumes, peak stresses, and interface debonding energy on the interface damage state, mechanical hysteresis loops, and related ITMs with small debonding energy and LDE were also analyzed. Comparisons of the mechanical hysteresis loops using the new hysteresis models considering matrix stochastic fragmentation and hysteresis models considering constant matrix fragmentation were also discussed. Experimental cyclic tensile hysteresis loops and unloading/reloading ITMs of C/(PyC)/SiC and C/(PyC+SiC)/SiC composites with different interphase thickness (i.e., t = 300, 600, 1000, and 2000 nm) were predicted using the developed constitutive model. Evolution of the unloading/reloading interface slip ratio was analyzed for different tensile peak stresses. [ABSTRACT FROM AUTHOR]

AbstractThis paper introduces a method for efficiently approximating the inverse of the Fisher information matrix, a crucial step in achieving effective variational Bayes inference. A notable aspect of our approach is the avoidance of analytically computing the Fisher information matrix and its explicit inversion. Instead, we introduce an iterative procedure for generating a sequence of matrices that converge to the inverse of Fisher information. The natural gradient variational Bayes algorithm without analytic expression of the Fisher matrix and its inversion is provably convergent and achieves a convergence rate of order O( log s/s) , with s the number of iterations. We also obtain a central limit theorem for the iterates. Implementation of our method does not require storage of large matrices, and achieves a linear complexity in the number of variational parameters. Our algorithm exhibits versatility, making it applicable across a diverse array of variational Bayes domains, including Gaussian approximation and normalizing flow Variational Bayes. We offer a range of numerical examples to demonstrate the efficiency and reliability of the proposed variational Bayes method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an inexact Noda iteration method featuring inner and outer iterations for computing the smallest eigenvalue and corresponding eigenvector of an irreducible monotone matrix. The proposed method includes two primary relaxation steps designed to compute the smallest eigenvalue and its associated eigenvector. These steps are influenced by specific relaxation factors, and we examine how these factors impact the convergence of the outer iterations. By applying two distinct relaxation factors to solve the inner linear systems, we demonstrate that the convergence can be globally linear or superlinear, contingent upon the relaxation factor used. Additionally, the relaxation factor affects the rate of convergence. The inexact Noda iterations we propose are structure-preserving and ensure the positivity of the approximate eigenvectors. Numerical examples are provided to demonstrate the practicality of the proposed method, consistently preserving the positivity of approximate eigenvectors. [ABSTRACT FROM AUTHOR]

Starting from the very first principles we derive explicit parameterizations of the ortho-symplectic matrices in the real four-dimensional Euclidean space. These matrices depend on a set of four real parameters which splits naturally as a union of the real line and the three-dimensional space. It turns out that each of these sets is associated with a separate Lie algebra which after exponentiations generates Lie groups that commute between themselves. Besides, by making use of the Cayley and Fedorov maps, we have arrived at alternative realizations of the ortho-symplectic matrices in four dimensions. Finally, relying on the fundamental structure results in Lie group theory we have derived one more explicit parameterization of these matrices which suggests that the obtained earlier results can be viewed as a universal method for building the representations of the unitary groups in arbitrary dimension. [ABSTRACT FROM AUTHOR]

We study a class of almost diagonal matrices compatible with the mixed-norm α-modulation spaces Mp,qs,α(ℝn), α ∈ [0, 1], introduced recently by Cleanthous and Georgiadis. The class of almost diagonal matrices is shown to be closed under matrix multiplication and we connect the theory to the continuous case by identifying a suitable notion of molecules for the mixed-norm α-modulation spaces. It is shown that the “change of frame” matrix for a pair of time-frequency frames for the mixed-norm α-modulation consisting of suitable molecules is almost diagonal. As examples of applications, we use the almost diagonal matrices to construct compactly supported frames for the mixed-norm α-modulation spaces, and to obtain a straightforward boundedness result for Fourier multipliers on the mixed-norm α-modulation spaces. [ABSTRACT FROM AUTHOR]

Based on double-compressed sampling, a hyperspectral spectral unmixing algorithm (SU_DCS) is proposed, which could directly complete the endmember extraction and abundance estimation. On the basis of the linear mixed model (LMM), we designed spatial and spectral sampling matrices, obtained spatial and spectral measurement data, and constructed a joint unmixing model containing endmember and abundance information. By using operator separation and Lagrangian multiplier algorithm, the endmember matrix, abundance matrix and remixing image can be quickly obtained by matrix operation. The parameters of the unmixing algorithm, including regularization parameter, convergence threshold and spatial sampling rate, are determined using synthetic simulated hyperspectral data. The proposed algorithm is applied to two kinds of real hyperspectral data, with or without ground truth, in order to verify the effectiveness and reliability of the algorithm. Firstly, we provide the performance of the algorithm on real datasets without ground truth. Compared with algorithm VCA_FCLS and algorithm CPPCA_VCA_FCLS, the endmember spectral curve extracted by the proposed SU_DCS is almost consistent with that obtained by VCA_FCLS, and is more smooth than that of obtained by CPPCA_VCA_FCLS. Additionally, the abundance estimation map estimated by the SU_DCS has consistency with the results obtained by VCA_FCLS. Moreover, the proposed SU_DCS has higher peak signal-to-noise ratio (PSNR) for remixing images with higher computational efficiency. Secondly, we provide the performance of the proposed algorithm on four real datasets with ground truth, including dataset Cuprite, dataset Samson, dataset Jasper and dataset Urban. We provide the results of endmember extraction and abundance estimation from the compressed data under different sampling rate conditions. The extracted endmember maintains good consistency with the true spectral curves, and the estimated abundance map can also maintain good spatial consistency with the ground truth. The comparison results with other four comparative algorithms also indicate that the proposed algorithm can obtain relatively accurate endmembers and abundance information from compressed data, the reliability and validity of the proposed algorithm have been proved. In summary, the main innovation of the proposed algorithm is that it can extract endmembers and estimate abundance with high accuracy from a small amount of measurement data. [ABSTRACT FROM AUTHOR]

A signed graph Σ = (G , σ) consists of an underlying graph G = (V , E) with a sign function σ : E → { − 1 , 1 } . Let A (Σ) be the adjacency matrix of Σ and λ 1 (Σ) denote the largest eigenvalue (index) of Σ. Define (K n , H −) as a signed complete graph whose negative edges induce a subgraph H. In this paper, we focus on the following question: which spanning tree T with a given number of pendant vertices makes the λ 1 (A (Σ)) of the unbalanced (K n , T −) as large as possible? To answer the question, we characterize the extremal signed graph with maximum λ 1 (A (Σ)) among graphs of type (K n , T −) . [ABSTRACT FROM AUTHOR]

This study introduces a 4 × 4 matrix eigenvalue problem and develops an integrable hierarchy with a bi-Hamiltonian structure. Integrability is ensured by the zero-curvature condition, while the Hamiltonian structure is supported by the trace identity. Explicit derivations yield second-order and third-order integrable equations, illustrating the integrable hierarchy. [ABSTRACT FROM AUTHOR]

The measure of sphericity for positive semi-definite matrices plays a crucial role in understanding their geometric properties, especially in high-dimensional settings. This paper introduces a robust measure of sphericity, which remains invariant under orthogonal transformations and scaling. We explore its behavior in finite-dimensional cases. Additionally, we investigate the stochastic case by considering a normal distribution, analyzing the asymptotic normality of random matrices and its implications on the convergence properties of the proposed measure. [ABSTRACT FROM AUTHOR]

This paper proposes an implicit family of sub-step integration algorithms grounded in the explicit singly diagonally implicit Runge–Kutta (ESDIRK) method. The proposed methods achieve third-order consistency per sub-step, and thus, the trapezoidal rule is always employed in the first sub-step. This paper demonstrates for the first time that the proposed s-sub-step implicit method with s ≤ 6 can reach sth-order accuracy when achieving dissipation control and unconditional stability simultaneously. Hence, this paper develops, analyzes, and compares four cost-optimal high-order implicit algorithms within the present s-sub-step method using three, four, five, and six sub-steps. Each high-order implicit algorithm shares identical effective stiffness matrices to achieve optimal spectral properties. Unlike the published algorithms, the proposed high-order methods do not suffer from the order reduction for solving forced vibrations. Moreover, the novel methods overcome the defect that the authors' previous algorithms require an additional solution to obtain accurate accelerations. Linear and nonlinear examples are solved to confirm the numerical performance and superiority of four novel high-order algorithms. [ABSTRACT FROM AUTHOR]

Let A be an m × n matrix with nonnegative real entries. The psd rank of A is the smallest k for which there exist two families (P 1 , ... , P m) and (Q 1 , ... , Q n) of positive semidefinite Hermitian k × k matrices such that A (i | j) = tr (P i Q j) for all i, j. Several questions on the algorithmic complexity of related matrix invariants were posed in recent literature: (i) by Stark (for the psd rank as defined above), (ii) by Goucha, Gouveia (for phaseless rank, which appears if the matrices P i and Q j are required to be of rank one in the above definition), (iii) by Gribling, de Laat, Laurent (for cpsd rank, which corresponds to the situation when A is symmetric and P i = Q i for all i). We solve these questions by proving that the decision versions of the corresponding invariants are ∃ R -complete. In addition, we give a polynomial time recognition algorithm for matrices of bounded cpsd rank. [ABSTRACT FROM AUTHOR]

This article mainly focuses on presenting a new accurate technique (Fourier Transform Method) for solving linear of mth order Difference Equations with constant coefficients. Also, a new lower triangular matrix was introduced to overcome problems related to finding the Fourier Transform of polynomials by rewriting standard-based polynomials through the fallen power polynomial base. Besides, five examples have been presented to illustrate the validity and accuracy of this method. The results reveal that the Fourier transform method is very effective and attractive in solving the difference equations. [ABSTRACT FROM AUTHOR]

In the four-dimensional Minkowski space, hypersurfaces classified as right conoids with a time-like axis are introduced and studied. The computation of matrices associated with the fundamental form, the Gauss map, and the shape operator specific to these hypersurfaces is included in our analysis. The intrinsic curvatures of these hypersurfaces are determined to provide a deeper understanding of their geometric properties. Additionally, the conditions required for these hypersurfaces to be minimal are established, and detailed calculations of the Laplace–Beltrami operator are performed. Illustrative examples are provided to enhance our comprehension of these concepts. Finally, the umbilical condition is examined to determine when these hypersurfaces become umbilic, and also the Willmore functional is explored. [ABSTRACT FROM AUTHOR]

In the paper, we introduce the signless Laplacian A B C -matrix Q ̃ (G) = D ¯ (G) + A ̃ (G) , where D ¯ (G) is the diagonal matrix of A B C -degrees and A ̃ (G) is the A B C -matrix of G. The eigenvalues of the matrix Q ̃ (G) are the signless Laplacian A B C -eigenvalues of G. We give some basic properties of the matrix Q ̃ (G) , which includes relating independence number and clique number with signless Laplacian A B C -eigenvalues. For bipartite graphs, we show that the signless Laplacian A B C -spectrum and the Laplacian A B C -spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian A B C -eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian A B C -eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix Q ̃ (G) − t r (Q ̃ (G)) n I , called the signless Laplacian A B C -energy of G. We obtain some upper and lower bounds for signless Laplacian A B C -energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the A B C -energy with the signless Laplacian A B C -energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian A B C -energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same A B C -energy. [ABSTRACT FROM AUTHOR]

We prove sharp spectral transition in the arithmetics of phase between localization and singular continuous spectrum for Diophantine almost Mathieu operators. We also determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices throughout the localization region. This uncovers a universal structure in their behavior governed by the exponential phase resonances. The structure features a new type of hierarchy, where self-similarity holds upon alternating reflections [ABSTRACT FROM AUTHOR]

Spectral graph theory explores the relationship between the eigenvalues and the structural properties of graphs. This study focuses on the fundamental matrices associated with simple graphs, specifically the adjacency matrix. This work contributes to the spectral graph theory literature by introducing a new family of simple connected graphs and investigating their eigenvalue properties. We examine conditions under which these graphs satisfy specific spectral properties, such as the strong antireciprocal eigenvalue property. Additionally, we construct another family of graphs from this new family, proving that each graph within this derived family satisfies the strong antireciprocal eigenvalue property. Through these contributions, we aim to deepen the understanding of the spectral characteristics of graph families and their applications in theoretical and applied contexts. [ABSTRACT FROM AUTHOR]

Time series similarity matrices (informally, recurrence plots or dot-plots), are useful tools for time series data mining. They can be used to guide data exploration, and various useful features can be derived from them and then fed into downstream analytics. However, time series similarity matrices suffer from very poor scalability, taxing both time and memory requirements. In this work, we introduce novel ideas that allow us to scale the largest time series similarity matrices that can be examined by several orders of magnitude. The first idea is a novel algorithm to compute the matrices in a way that removes dependency on the subsequence length. This algorithm is so fast that it allows us to now address datasets where the memory limitations begin to dominate. Our second novel contribution is a multiscale algorithm that computes an approximation of the matrix appropriate for the limitations of the user's memory/screen-resolution, then performs a local, just-in-time recomputation of any region that the user wishes to zoom-in on. Given that this largely removes time and space barriers, human visual attention then becomes the bottleneck. We further introduce algorithms that search massive matrices with quadrillions of cells and then prioritize regions for later examination by either humans or algorithms. We will demonstrate the utility of our ideas for data exploration, segmentation, and classification in domains as diverse as astronomy, bioinformatics, entomology, and wildlife monitoring. [ABSTRACT FROM AUTHOR]

This work tackles the performance and energy consumption analysis of a legacy scientific application, the VASP (Vienna Ab-initio Simulation Package), an application commonly used by physicists and chemists for modeling materials at the atomic scale. Many of these scientific applications have been implemented in Fortran, where energy metrics instrumentation is not straightforward. We obtained performance figures (execution time and energy consumption) by instrumenting the source code using EML. This energy measurement library has been modified to introduce Fortran interfaces for these metrics. The analysis was carried out using different matrix algebra libraries, parallelization techniques, and hardware platforms, emphasizing on the MPI, OpenMP, and CUDA parallel implementations of the algorithms used in VASP. We employ various material specifications (atomic structures) and molecular sizes of a silicon-based crystal to create a set of benchmarks for these specifications, leading to some recommendations for final users regarding performance improvements. The proposed benchmarking technique assists the user in selecting the right combination of problem size, compilers, and parallelization options available in VASP. For a given system platform, the user will be able to determine not only the architecture to use (GPU or multicore processors), but also the appropriate library and parallelization according to the atomic structure and molecular size. [ABSTRACT FROM AUTHOR]

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]

In this paper, we address the extremal inverse eigenvalue problem for pentadiagonal matrices. We provide sufficient conditions for their existence and realizability through new constructions that consider spectral data of its leading principal submatrices. Finally, we present some examples generated from the algorithmic procedures derived from our results. [ABSTRACT FROM AUTHOR]

In this paper, an improved zeroing neural network (ZNN) model is proposed to obtain the positive definite solutions of the continuous coupled Lyapunov matrix equations (CLMEs) associated with continuous‐time Markovian jump (CMJ) systems. To achieve this, a general ZNN model is established by constructing a matrix‐valued error function. Then, to accelerate the convergence rate of the proposed ZNN model, the latest estimation is introduced to obtain an improved ZNN model. Some convergence conditions have been derived for the presented improved ZNN model through Lyapunov theory. Comparisons among the improved ZNN model and the existing results are conducted to illustrate the advantages of the proposed improved ZNN model in numerical examples. [ABSTRACT FROM AUTHOR]

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

In the generalized linear model (GLM) setup, the information matrix depends on the unknown parameters of the model. In such a situation, an experimenter has to adopt the approach of finding local optimal designs i.e. first guess the best value for the parameters and then calculate the optimal designs [see Chernoff (1953)]. In this paper, locally R-optimal designs for the Poisson regression model with two parameters including intercept are discussed. The R-optimality criterion has been proposed in the literature as an alternative to the most frequently used D-optimality criterion when the experimenter wishes to minimize the volume of the confidence region for unknown parameters based on Bonferroni t-intervals [see Dette (1997)]. The necessary and sufficient conditions of this optimality criterion are confirmed through the equivalence theorem [ABSTRACT FROM AUTHOR]

A Pythagorean triple is a triple of positive integers (a, b, c) such that a² + b² = c². If a, b are coprime, then it is called a primitive Pythagorean triple. It is known that every primitive Pythagorean triple can be generated from the triple (3, 4, 5) using multiplication by unique number and order of three specific 3 × 3 matrices, which yields a ternary tree of triplets. Two such trees were described by Berggren and Price, respectively. A different approach is to view the primitive Pythagorean triples as points in the three-dimensional Euclidean space. In this paper, we prove that the triple of descendants of any primitive Pythagorean triple in Berggren's or Price's tree forms a triangle (and therefore defines a plane), and we present our results related to these triangles (and these planes). [ABSTRACT FROM AUTHOR]

Let H be a complex Hilbert space and let T represent a bounded linear operator on H. In this paper we introduce, a new class of non normal operators, the (M, k)-quasi paranormal operator. An operator T is said to be a (M,k)-quasi paranormal operator, for a non negative integer k and a real positive number M if it satisfies ||Tk+1x||2 < M||Tk+2x|| ||Tkx||, for all x ∈ H. This new class of operators is generalization of some of the non normal operators, such as, the k-quasi paranormal and M-paranormal operators. We prove the basic properties, the structural and spectral properties and also the matrix representation of this new class of operators. [ABSTRACT FROM AUTHOR]

Motivation Imaging Mueller polarimetry has already proved its potential for biomedicine, remote sensing, and metrology. The real-time applications of this modality require both video rate image acquisition and fast data post-processing algorithms. First, one must check the physical realizability of the experimental Mueller matrices in order to filter out non-physical data, i.e. to test the positive semi-definiteness of the 4 × 4 Hermitian coherency matrix calculated from the elements of corresponding Mueller matrix pixel-wise. For this purpose, we compared the execution time for the calculations of (i) eigenvalues, (ii) Cholesky decomposition, (iii) Sylvester's criterion, and (iv) coefficients of the characteristic polynomial (two different approaches) of the Hermitian coherency matrix, all calculated for the experimental Mueller matrix images (600 pixels × 700 pixels) of mouse uterine cervix. The calculations were performed using C++ and Julia programming languages. Results Our results showed the superiority of the algorithm (iv) based on the simplification via Pauli matrices over other algorithms for our dataset. The sequential implementation of latter algorithm on a single core already satisfies the requirements of real-time polarimetric imaging. This can be further amplified by the proposed parallelization (e.g. we achieve a 5-fold speed up on six cores). Availability and implementation The source codes of the algorithms and experimental data are available at https://github.com/pogudingleb/mueller_matrices. [ABSTRACT FROM AUTHOR]

We show that the rank decomposition of a real matrix r, which is a Spin(3,1) tensor, leads to 2k Majorana bispinors, where k = rank r. The Majorana bispinors are determined up to local GL(k,R) transformations. The bispinors are combined in pairs to form k complex Dirac fields. We analyze in detail the case k = 1, in which there is just one Dirac field with the well-known standard Lagrangian. The GL(1,R) symmetry gives rise to a new conserved current, different from the well-known U(1) current. The U(1) symmetry is present too. All global continuous internal symmetries in the k = 1 case form the SO(2,1) group. As a side result, we clarify the discussed in literature issue whether there exist algebraic constraints for the matrix r which would be equivalent to the condition rank r = 1. [ABSTRACT FROM AUTHOR]

In this study, we introduced a generalization of distance Fibonacci sequences and investigate some of its basic properties. We then proposed a generalization of distance Fibonacci sequences for negative integers and investigated some basic properties. Additionally, we explored the construction of matrix generators for these sequences and offered a graphical representation to clarify their structure. Furthermore, we demonstrated how these generalizations can be applied to obtain the Padovan and Narayana sequences for specific parameter values. [ABSTRACT FROM AUTHOR]

In this paper, the impulsive synchronization of a class of nonlinear multiple neural networks (MNNs) with multi-delays was considered under a dynamic event-based mechanism. To achieve a more comprehensive synchronization outcome and mitigate the conservativeness of impulsive control due to predetermined time sequences, we integrated a dynamic event-triggered strategy. This approach formed a novel control framework for generalized MNNs, where impulsive inputs were applied only under specific conditions governed by event-triggering rules. Towards the above objectives, the impulsive jumping system, resulting from dynamic component, and matrix measure method were invoked to directly increase the computational simplicity and extensibility of the study. As the outcome, the synchronization criteria for the MNNs could be achieved, and the exponential convergence rate is resolved by considering both the generalized comparison principle regarding impulsive systems and the variable parameter formula. Moreover, Zeno-freeness of the achieved triggering regulation is ensured. Finally, two numerical examples confirmed the validity of the designed approach. [ABSTRACT FROM AUTHOR]

Let a 0 , a 1 , ... , a n − 1 be real numbers and let A = C i r c (a 0 , a 1 , ... , a n − 1) be a circulant matrix with f (x) = Σ j = 0 n − 1 a j x j . First, we prove that C i r c (a 0 , a 1 , ... , a n − 1) must be invertible if the sequence a 0 , a 1 , ... , a n − 1 is a strictly monotonic sequence and a 0 + a 1 + ⋯ + a n − 1 ≠ 0. Next, we reduce the calculation of f (ε 0) f (ε) ... f (ε n − 1) for a prime n by using the techniques on finite fields, where ε is a primitive n -th root of unity. Finally, we provide two examples to explain how to use the obtained results to calculate the determinant of a circulant matrix. [ABSTRACT FROM AUTHOR]

The correlation filter object tracking algorithm has gained extensive attention from scholars in the field of tracking because of its excellent tracking performance and efficiency. However, the mathematical modeling relationships of correlation filter tracking frameworks are unclear. Therefore, many forms of correlation filters are susceptible to confusion and misuse. To solve these problems, we attempted to review various forms of the correlation filter and discussed their intrinsic connections. First, we reviewed the basic definitions of the circulant matrix, convolution, and correlation operations. Then, the relationship among the three operations was discussed. Considering this, four mathematical modeling forms of correlation filter object tracking from the literature were listed, and the equivalence of the four modeling forms was theoretically proven. Then, the fast solution of the correlation filter was discussed from the perspective of the diagonalization property of the circulant matrix and the convolution theorem. In addition, we delved into the difference between the one-dimensional and two-dimensional correlation filter responses as well as the reasons for their generation. Numerical experiments were conducted to verify the proposed perspectives. The results showed that the filters calculated based on the diagonalization property and the convolution property of the cyclic matrix were completely equivalent. The experimental code of this paper is available at https://github.com/110500617/Correlation-filter/tree/main. [ABSTRACT FROM AUTHOR]

The recently developed data-driven eigenmatrix method shows very promising reconstruction accuracy in sparse recovery for a wide range of kernel functions and random sample locations. However, its current implementation can lead to numerical instability if the threshold tolerance is not appropriately chosen. To incorporate regularization techniques, we have proposed to regularize the eigenmatrix method by replacing the computation of an ill-conditioned pseudo-inverse by the solution of an ill-conditioned least squares system, which can be efficiently treated by Tikhonov regularization. Extensive numerical examples confirmed the improved effectiveness of our proposed method, especially when the noise levels were relatively high. [ABSTRACT FROM AUTHOR]

For any parameter vector (r , t , ω) in ℍ : = ℝ × ℝ n × ℝ n , the representation matrix S (r , t , ω) of the Heisenberg group (ℍ , ∗) is extended to a new form S B , W , A (r , t , ω) with nonsingular matrices B , W , A which are related to the directional translation B t : f → f (⋅ − B t) , directional modulation ℳ W ω : f (x) → e i x ′ W ω and dilation A : f → | det (A) | − 1 2 f (A − 1 ⋅) if B ′ W = I n , the identity matrix of order n. In the case B ′ W ≠ I n , an intrinsic group structure under the matrix S B , W , A (r , t , ω) is investigated and a unitary operator representation is offered. [ABSTRACT FROM AUTHOR]

In this paper, we compute paradeterminants and parapermanents of some triangular matrices that give bihyperbolic numbers of the Fibonacci type. Using connections between the paradeterminant of triangular matrix and the lower Hessenberg determinant, we also obtain the general term of these sequences. [ABSTRACT FROM AUTHOR]

Decoupling control is a commonly employed technique for achieving high precision in multiple-input multiple-output (MIMO) motion control systems. A static decoupling matrix, which can be determined using geometric construction, is widely used due to its practicality and simplicity. However, inaccurate geometric parameters will lead to a coarse decoupling matrix, and result in interactions among the system axes and performance deterioration. To tackle this challenge, various attempts have been undertaken to calibrate the decoupling matrix. Data-driven on-line approaches have gained considerable attention for their ability to calibrate the decoupling matrix without interrupting the normal operation of the system. This paper presents a data-driven approach to calibrate the decoupling matrix for MIMO and linear time invariant (LTI) systems. Through some reasonable assumptions, a calibrated static decoupling matrix can be derived to improve the performance of the system. Moreover, considering the inevitable presence of measurement noise, the consistency of the proposed method has been analyzed. As a result, the instrument variable is introduced in the improved method to eliminate the impact of the measurement noise. Finally, the effectiveness and practicality of the proposed method are demonstrated through both numerical simulations and experiments carried out on an ultraprecision wafer stage. • The performance of MIMO systems declines if mechanical parameters are inaccurate. • Traditional calibration of these parameters is complicated. • Optimizing the decoupling matrix directly with measurement data. • A calibrated static decoupling matrix with consistency is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider large-scale low-rank Sylvester differential matrix equations. We present two iterative methods for the approximate solution of such difierential matrix equations. The first approach is based on the integral expression of the exact solution which exploits an extended block Krylov subspace method to compute the exponential of a matrix times a block of vectors. In the second method, we first project the initial value problem onto an extended block Krylov subspace and acquire a low-dimensional Sylvester difierential matrix equation with a low-rank constant term. Then the reduced Sylvester difierential matrix equation is solved by the backward difierentiation formula method (BDF) and the derived solution is used to construct the low-rank approximate solutions of the original initial value problem. The iterative approaches are followed until some certain accuracy is obtained. We give some theoretical results and some numerical examples to show the eficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, a class of matrices, namely, Drazin-dagger ma- trices, in which the Drazin inverse and the Moore-Penrose inverse com- mute, is introduced. Also, some properties of the generalized inverses of these matrices, are investigated. Moreover, some results about the Moore- Penrose inverse, the Drazin inverse and the numerical range of some re- ciprocal matrices are obtained. In particular, the relations between re- ciprocal matrices, Drazin-Dagger matrices and star order are established. Also, some properties of the generalized inverses of the conjugate EP ma- trices are studied. To illustrate the results, some numerical examples are also given. [ABSTRACT FROM AUTHOR]

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

The aim of this paper is to show that if H is the real quaternion division ring and n is an integer greater than 1, then every matrix in the special linear group SLn(H) can be expressed as a product of at most three commutators of unipotent matrices of index 2. [ABSTRACT FROM AUTHOR]