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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 this paper, we first develop the modified gradient‐based iterative (MGI) method for the complex conjugate and transpose matrix equations A1XB1+A2X‾B2+A3XTB3+A4XHB4=E$$ {A}_1X{B}_1+{A}_2\overline{X}{B}_2+{A}_3{X}^T{B}_3+{A}_4{X}^H{B}_4=E $$. By adopting the updated technique, we can make full use of the latest information to compute the next result, which leads to a faster convergence rate. In theory, we apply the real representation of a complex matrix and the vec‐operator to prove the convergence properties. Furthermore, we extend the MGI algorithm to solve the generalized complex conjugate and transpose matrix equations. Then, the necessary and sufficient conditions for convergence of the MGI algorithm are presented. Lastly, three numerical examples are introduced to testify the efficiency of our methods. [ABSTRACT FROM AUTHOR]

In the past few decades, there has been a significant focus on computational methods for periodic systems. In some of these works, various considerations were outlined to establish and categorize algorithms for designing periodic discrete‐time state‐space systems. In a recent work, a direct algorithm was proposed to obtain low‐dimensional realizations from a proper lifted system in state‐space form. This algorithm, which lies in a category often called "fast" algorithms, takes advantage on the structure of lifted representations being highly efficient. In a 1993 work by Ching‐An Lin and Chwan‐Wen King, another direct algorithm was proposed that appears to be quite similar to the algorithm outlined recently. These two algorithms generate state‐space periodic realizations that are both uniform, meaning that the dimension of the state remains constant at each time instant. The algorithms proposed in these two works require both a rank test applied to two different types of matrices, Kℓ$$ {K}_{\ell } $$ and ℓM$$ {}^{\ell }M $$, respectively (in chronologic order). Although the former algorithm establishes an upper bound for the state dimension à priori, by determining the maximum rank of the matrices Kℓ$$ {K}_{\ell } $$, the latter obtains an upper bound à posteriori, based on the rank of matrices ℓM$$ {}^{\ell }M $$. In this work, we proved that these matrices Kℓ$$ {K}_{\ell } $$ and ℓM$$ {}^{\ell }M $$ have the same rank for all ℓ$$ \ell $$, and we use this information to revise the latter algorithm and assess whether it represents an improvement. The algorithm presented as well its modified versions that were coded using the Python programming language. Surprisingly, the revised algorithm did not demonstrate any significant improvement in terms of computational efficiency. [ABSTRACT FROM AUTHOR]

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

In this paper, approximate solutions for stochastic Fitzhugh–Nagumo partial differential equations are obtained using two‐dimensional shifted Legendre polynomial (2DSLP) approximation. The problem's suitability and solvability are confirmed. The convergence analysis for the proposed methodology and the error analysis in the L2$$ {L}^2 $$ norm are carried out. Using Maple software, an algorithm is created and implemented to arrive at the numerical solution. The solution thus obtained is compared with the exact solution and the solution obtained using the explicit order RK1.5 method. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate similarity and consimilarity of hyper‐dual generalized quaternions and their matrices. For this purpose, we give different conjugates according to the generalized quaternionic units i,j,k$$ i,j,k $$. We present i$$ i $$‐consimilarity of hyper‐dual generalized quaternions and their matrices except hyper‐dual 14$$ \frac{1}{4} $$‐quaternions. For the generalization consisting of hyper‐dual coefficients quaternion and split quaternion, we search j$$ j $$‐consimilarity and k$$ k $$‐consimilarity with the help of j$$ j $$‐conjugate and k$$ k $$‐conjugate. We also give i,j,k$$ i,j,k $$‐coneigenvalues and i,j,k$$ i,j,k $$‐coneigenvectors of the matrices of these generalizations. In addition, we examine right coneigenvalue problem in generalized quaternion matrices for real and split quaternions. The complex matrix representation obtained through the complex adjoint matrix representation of this generalization is introduced, and its properties are presented. Besides, we give algebraic methods for the concept of right coneigenvalues and coneigenvectors for matrices, which are the generalization of real quaternion and split quaternion. [ABSTRACT FROM AUTHOR]

A map Φ between matrices is said to be zero product preserving if $$\begin{align*} \Phi(A)\Phi(B) = 0 \quad {\rm whenever}\ AB = 0. \end{align*}$$ Φ (A) Φ (B) = 0 whenever AB = 0. In this paper, we give concrete descriptions of an additive/linear zero product preserver $ \Phi : \mathbf{M}_n(\mathbb {F}) \rightarrow \mathbf{M}_r(\mathbb {F}) $ Φ : M n (F) → M r (F) between matrix algebras of different dimensions over an arbitrary field $ \mathbb {F} $ F , and $ n\geq 2 $ n ≥ 2. In particular, we show that if Φ is linear and preserves zero products then $$\begin{align*} \Phi(A)= S\begin{pmatrix} R_1 \otimes A & 0 \\ 0 & \Phi_0(A)\end{pmatrix} S^{-1}, \end{align*}$$ Φ (A) = S ( R 1 ⊗ A 0 0 Φ 0 (A) ) S − 1 , for some invertible matrices $ R_1 $ R 1 in $ \mathbf{M}_k(\mathbb {F}) $ M k (F) , S in $ \mathbf{M}_r(\mathbb {F}) $ M r (F) and a zero product preserving linear map $ \Phi _0: \mathbf{M}_n(\mathbb {F}) \rightarrow \mathbf{M}_{r-nk}(\mathbb {F}) $ Φ 0 : M n (F) → M r − nk (F) into nilpotent matrices. If $ \Phi (I_n) $ Φ (I n) is invertible, then $ \Phi _0 $ Φ 0 is vacuous. In general, the structure of $ \Phi _0 $ Φ 0 could be quite arbitrary, especially when $ \Phi _0(\mathbf{M}_n({\mathbb F})) $ Φ 0 (M n (F)) has trivial multiplication, i.e. $ \Phi _0(X)\Phi _0(Y) = 0 $ Φ 0 (X) Φ 0 (Y) = 0 for all X, Y in $ \mathbf{M}_n(\mathbb {F}) $ M n (F). We show that if $ \Phi _0(I_n) = 0 $ Φ 0 (I n) = 0 or $ r-nk \le n+1 $ r − nk ≤ n + 1 , then $ \Phi _0(\mathbf{M}_n({\mathbb F})) $ Φ 0 (M n (F)) indeed has trivial multiplication. More generally, we characterize subspaces $ \mathbf{V} $ V of square matrices satisfying XY = 0 for any $ X, Y \in \mathbf{V} $ X , Y ∈ V . Similar results for double zero product preserving maps are obtained. [ABSTRACT FROM AUTHOR]

Recently, W. Berndt and S. Sra (Hlawka-Popoviciu inequalities on positive definite tensors. Linear Algebra Appl. 2015;486:317–327) proved a Popoviciu type inequality for any arbitrary finite number of positive definite matrices. In this paper, we proved a quantitative Popoviciu type inequality for four positive semi-definite matrices, which is stronger than Berndt-Sra's corresponding result and also a generalization of Hong-Qi's (Refinements of two determinantal inequalities for positive semidefinite matrices. Math Inequal Appl. 2022;25(3):673–678) result. As applications, we partially recovered our early result in F. Wang (A quantitative Popoviciu type inequality, submitted.] and obtained a quantitative Hartfiel inequality. [ABSTRACT FROM AUTHOR]

We study periodic Lyapunov matrix equations for a general discrete-time linear periodic system $ B_px_p-A_px_{p-1}=f_p $ B p x p − A p x p − 1 = f p , where the matrix coefficients $ B_p $ B p and $ A_p $ A p can be singular. The block coefficients of the inverse operator of the system are referred to as the Green matrices. We derive new decay estimates of the Green matrices in terms of the spectral norms of special solutions to the periodic Lyapunov matrix equations. The study is based on the periodic Schur decomposition of matrices. [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]

In this work, we consider the hyponormality of Toeplitz operators on the Bergman space of the annulus with a general radial weight. We give necessary conditions when the symbol is of the form g1+g2‾, where g1 and g2 are analytic on the annulus z∈ℂ,1/2<|z|<1. [ABSTRACT FROM AUTHOR]

In this work, we consider the hyponormality of Toeplitz operators on the Bergman space of the annulus with a logarithmic weight. We give necessary conditions when the symbol is of the form ψ+φ‾, where ψ and φ are analytic on the annulus z∈ℂ,1/2<|z|<1. [ABSTRACT FROM AUTHOR]

For a matrix A which satisfies Crouzeix's conjecture, we construct several classes of matrices from A for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which shows that Crouzeix's Conjecture holds in full generality if and only if it holds for the differentiation operator on a class of analytic functions. We pose several open questions, which if proved, will prove Crouzeix's conjecture. We also begin an investigation into Crouzeix's conjecture for symmetric matrices and in the case of 3 × 3 matrices, we show Crouzeix's conjecture holds for symmetric matrices if and only if it holds for analytic truncated Toeplitz operators. [ABSTRACT FROM AUTHOR]

The main aim of the present paper is to establish inversion formulas of Gohberg-Semencul type for Toeplitz-plus-Hankel matrices. In particular, it is shown how the inverse of such a structured matrix A n of order n is computed by means of their first two and last two columns or rows under the additional assumption that a certain 2 × 2 matrix is nonsingular. Moreover, a formula for the inverse of the Toeplitz-plus-Hankel matrix A n − 2 of order n − 2 in the center of A n is established and sufficient invertibility criteria for both A n and A n − 2 are obtained. Hereby a main tool is to use known inversion formulas involving not only rows or columns of the inverse but also solutions of equations the right hand sides of which depend on entries of the Toeplitz-plus-Hankel matrix itself. [ABSTRACT FROM AUTHOR]

We investigate the convex polytope Ω n (123 ‾) of doubly stochastic matrices spanned by the n × n permutation matrices that avoid an increasing pattern of length 3, the 123 ‾ -permutation matrices. We determine some of its facets and other faces, including faces of small dimension, and their connection to facets and faces of the polytope Ω n of all n × n doubly stochastic matrices. The paper concludes with some relations concerning the frequencies of the positions of the 1's in n × n 123 ‾ -permutation matrices, and the vertex-edge graph of Ω n (123 ‾). [ABSTRACT FROM AUTHOR]

In this paper, we introduce multi-layer quivers and show how to construct an (n + 1) -slice algebra of infinite type from an n -slice algebra of infinite type using the bound quivers. This leads to constructing (n + 1) -slice algebras of infinite type as matrix algebra and as tensor algebra of an n -slice algebra and equivalences of their module categories as the module categories of diagram of some quiver of type A ̃ n + 1 . [ABSTRACT FROM AUTHOR]

Let 풰 be an algebra over the field 픽. We say that 풰 is two-sided zero product determined if every bilinear functional ϕ : 풰×풰→ 픽 the following holds: if ϕ(x,y) = 0 whenever xy = yx = 0, then there exist linear functionals F1 and F2 on 풰 such that ϕ(x,y) = F1(xy)+F2(yx) for all x,y ∈풰. We show that the unital triangular algebra 풯 = 풜ℳ0 ℬ is a two-sided zero product determined algebra if and only if 풜 and ℬ are two-sided zero product determined algebras, and then we get various results about this property for generalized triangular algebras and block upper triangular matrix algebras. We also provide an application of the main result to determine the structure of commutativity preserving maps at commutative zero products on triangular algebras. We note that some of the previous results about the two-sided zero product determined property have been generalized. [ABSTRACT FROM AUTHOR]

This article presents an asynchronous event‐triggered scheme for switched Takagi–Sugeno (T–S) fuzzy systems against dual‐channel hybrid cyber attacks. Different from existing results, both sensor and controller channels are subjected to aperiodic denial‐of‐service attacks and random false data injection attacks. To efficiently utilize dual‐channel network communication resources while resisting hybrid attacks, two resilient event‐triggered mechanisms (ETMs) are constructed. Considering asynchronous ETMs and hybrid cyber attacks, the time‐delay switched T–S fuzzy system is derived by utilizing model transformation methods. Thereby, the stability conditions are derived by utilizing multiple Lyapunov functions technique, and slack matrices are introduced to further relax the conditions. Finally, two examples are given to demonstrate the effectiveness of the developed event‐based security control strategy. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a frame diagonalization of matrices in $ M_n({\mathbb {C}}) $ Mn(C), called QR-frame diagonalization, by using QR-factorization. Furthermore, we show that every matrix A in $ M_n({\mathbb {C}}) $ Mn(C) is QR-frame diagonalizable. Also we introduce another frame diagonalization via Hadamard matrices in $ M_n({\mathbb {R}}) $ Mn(R), called Hadamard frame diagonalization, by using Hadamard matrices for n = 2, 4 or multiple of 4. We show that every matrix A in $ M_n({\mathbb {R}}) $ Mn(R) which n = 2, 4 or multiple of 4 is Hadamard frame diagonalizable. In this frame diagonalization, we can find entries on main diagonal Δ by using inner product. Moreover we introduce frame diagonalization of matrices on left quaternionic Hilbert spaces $ M_n({\mathbb {H}}) $ Mn(H). [ABSTRACT FROM AUTHOR]

Let Mq(n) be the standard quantized matrix algebra (introduced by Faddeev, Reshetikhin and Takhtajan). It is shown that Mq(n) is Auslander regular, Cohen–Macaulay, Artin–Schelter regular, and a maximal order in its quotient division algebra. [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]

The continuous-time algebraic Riccati equation (ARE) is often utilized in control, estimation, and optimization. For a linear system with a second-order structure of size n, the ARE required to be solved to get the control values in standard control problems results in complex subequations in terms of the second-order system matrices. The computational costs of solving the algebraic Riccati equation through standard methods such as the Hamiltonian matrix pencil approach increase substantially as matrix sizes increase for a second-order system, due to the eigendecomposition of the 2n × 2n system matrices involved. This work introduces a new solution that does not require the eigendecomposition of the 2n × 2n system matrices, while satisfying all of the requirements of the solution to the Riccati equation (e.g., detectability, stabilizability, and positive semidefinite solution matrix). [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]

The quaternionic Schrödinger equation ∂ ∂ t | f ⟩ = - A | f ⟩ is the crucial part of the study of quaternionic quantum mechanics and plays indispensable roles in related fields. One of the practical and special cases that has received more attention from mathematicians and physicists is that A is a Hermitian quaternion matrix. The problem can be equivalent to a Hermitian quaternion right eigenvalue problem A α = α λ by discretization. This paper, by means of a complex representation method, studies the Hermitian quaternion Schrödinger equation problem, and proposes a novel algebraic method (complex structure-preserving method) for right eigenvalue problems of Hermitian quaternion matrices. Moreover, the complex structure-preserving method is superior and formally simple compared to previous methods, and numerical experiments also demonstrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

Nonsymmetric algebraic Riccati equations are a class of nonlinear matrix equations with wide applications. In this paper, numerical solution of nonsymmetric algebraic Riccati equations is studied. A generalized ALI iteration method is proposed to solve the equation, which is an extension of the ALI iteration method and the MALI iteration method. Theoretical analysis and numerical experiments show that the proposed method is feasible and is more effective than the ALI iteration method and the MALI iteration method. [ABSTRACT FROM AUTHOR]

In this paper, we derive and analyze an algorithm for inverting quaternion matrices. The algorithm is an analogue of the Frobenius algorithm for the complex matrix inversion. On the theory side, we prove that our algorithm is more efficient than other existing methods. Moreover, our algorithm is optimal in the sense of the least number of complex inversions. On the practice side, our algorithm outperforms existing algorithms on randomly generated matrices. We argue that this algorithm can be used to improve the practical utility of recursive Strassen-type algorithms by providing the fastest possible base case for the recursive decomposition process when applied to quaternion matrices. [ABSTRACT FROM AUTHOR]

In Meurant et al. (Numer. Algorithms 88(3), 1337–1359, 2021), we presented an adaptive estimate for the energy norm of the error in the conjugate gradient (CG) method. Here we consider algorithms for solving linear approximation problems with a general, possibly rectangular matrix that are based on applying CG to a system with a positive (semi-)definite matrix built from the original matrix. We discuss algorithms based on Hestenes–Stiefel-like implementation (often called CGLS and CGNE in the literature) as well as on bidiagonalization (LSQR and CRAIG), and both unpreconditioned and preconditioned variants. Each algorithm minimizes a certain quantity at each iteration (within the current Krylov subspace), that is related to some error norm. We call this "the quantity of interest". We show that the adaptive estimate used in CG can be extended for these algorithms to estimate the quantity of interest. Throughout, we emphasize the applicability of the estimate during computations in finite-precision arithmetic. We carefully derive the relations from which the estimate is constructed, without exploiting the global orthogonality that is not preserved during computation. We show that the resulting estimate preserves its key properties: it can be cheaply evaluated, and it is numerically reliable in finite-precision arithmetic under some mild assumptions. These properties make the estimate suitable for use in stopping the iterations. The numerical experiments confirm the robustness and satisfactory behaviour of the estimate. [ABSTRACT FROM AUTHOR]

This paper identifies necessary and sufficient conditions for the existence of iterative algorithms that achieve approximate Byzantine consensus in arbitrary directed graphs, where each directed link represents a communication channel between a pair of nodes. The class of iterative algorithms considered in this paper ensures that, after each iteration of the algorithm, the state of each fault-free node remains in the convex hull of the states of the fault-free nodes at the end of the previous iteration. We present the necessary and sufficient condition for the existence of such iterative consensus algorithms in synchronous arbitrary point-to-point networks in presence of Byzantine faults in two different equivalent forms. We prove the necessity using an indistinguishability argument. For sufficiency, we develop a proof framework, which first uses a series of "transition matrices" to model the state evolution of the fault-free nodes using our algorithm, and then proves the correctness by identifying important properties of the matrices. The proof framework is useful for other iterative fault-tolerant algorithms. We discuss the extensions to asynchronous systems and the Byzantine links fault model. [ABSTRACT FROM AUTHOR]

Modern data are often collected in a matrix form. In this paper, we consider modelling the varying coefficient regression with matrix valued covariate X and scalar index variable U. The proposed model simultaneously makes principal component analysis for both the row and column dimensions of the matrix objects, maintaining the matrix structure while achieving substantial dimension reduction. We develop an iterative estimation method for the involved principal parameters and nonparametric functions. Under regularity conditions, the asymptotic distributions of the estimators are derived. In addition, by incorporating the estimation with the adaptive group Lasso and the group SCAD penalties, variables of X in entire rows or columns are selected. The proximal gradient algorithm is further utilised to solve the regularised optimisation problems. The asymptotic properties of the penalised estimators are also studied. Our model and estimation methods are demonstrated by simulated experiments. Real applications to the primary biliary cirrhosis (PBC) data reveal that the effects of the blood measurements to the survival time vary with levels of age. [ABSTRACT FROM AUTHOR]

This paper proves new results on spectral and scattering theory for matrix-valued Schrödinger operators on the discrete line with non-compactly supported perturbations whose first moments are assumed to exist. In particular, a Levinson theorem is proved, in which a relation between scattering data and spectral properties (bound and half-bound states) of the corresponding Hamiltonians is derived. The proof is based on stationary scattering theory with prominent use of Jost solutions at complex energies that are controlled by Volterra-type integral equations. [ABSTRACT FROM AUTHOR]

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

In this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph K m , n , extending a method from de Klerk et al. (SIAM J Discrete Math 20:189–202, 2006) and the subsequent reduction by De Klerk, Pasechnik and Schrijver (Math Prog Ser A and B 109:613–624, 2007). We exploit the full symmetry of the problem using a novel decomposition technique. This results in a full block-diagonalization of the underlying matrix algebra, which we use to improve bounds on several concrete instances. Our results imply that cr (K 10 , n) ≥ 4.87057 n 2 - 10 n , cr (K 11 , n) ≥ 5.99939 n 2 - 12.5 n , cr (K 12 , n) ≥ 7.25579 n 2 - 15 n , cr (K 13 , n) ≥ 8.65675 n 2 - 18 n for all n. The latter three bounds are computed using a new and well-performing relaxation of the original semidefinite programming bound. This new relaxation is obtained by only requiring one small matrix block to be positive semidefinite. [ABSTRACT FROM AUTHOR]

Let G be a graph and d i be the degree of a vertex v i in G. For a symmetric real function f (x , y) , one can get an edge-weighted graph in such a way that for each edge v i v j of G , the weight of v i v j is assigned by f (d i , d j). Hence, we have a weighted adjacency matrix A f (G) of G , in which the ij -entry is equal to f (d i , d j) if v i v j ∈ E (G) and 0 otherwise. In this paper, we use a unified approach to deal with the spectral properties of A f (G) for f (x , y) to be the functions of graphical or topological function-indices. Firstly, we obtain uniform interlacing inequalities for the weighted adjacency eigenvalues. For the edge-weight functions defined by almost a half of popularly used topological indices, it can be shown that our inequalities cannot be improved. Secondly, we establish a uniform equivalent condition for a connected graph G to have m distinct weighted adjacency eigenvalues. As an application, a combinatorial characterization for a graph to have two and three distinct weighted adjacency eigenvalues is presented, respectively. Moreover, bipartite graphs and unicyclic graphs with three distinct weighted adjacency eigenvalues are characterized. This paper attempts to unify the spectral study for weighted adjacency matrices of graphs with degree-based edge-weights. [ABSTRACT FROM AUTHOR]

Denote by $ \omega _n $ ω n the convex polytope of $ n\times n $ n × n doubly substochastic matrices and by $ \Omega _n $ Ω n the convex polytope of $ n\times n $ n × n doubly stochastic matrices. We generalize the Bruhat order for permutation matrices and doubly stochastic matrices to the subpermutation matrices and doubly substochastic matrices, respectively. Moreover we study the Bruhat shadow of a subpermutation matrix and the Bruhat faces of $ \omega _n $ ω n . The relations between the Bruhat faces in $ \omega _n $ ω n and those in $ \Omega _n $ Ω n are also given. [ABSTRACT FROM AUTHOR]

In this work, we characterize the fixed points of the doubly stochastic operators defined on discrete spaces $ \ell ^p(I) , $ ℓ p (I) , where I is a nonempty set and $ p\in [1,+\infty). $ p ∈ [ 1 , + ∞). We use our results to characterize all Markov chains with infinite state space and doubly stochastic transition probability matrix that have at least one stationary distribution vector. [ABSTRACT FROM AUTHOR]

We define quantum automorphisms and isomorphisms of Hadamard matrices. We show that every Hadamard matrix of size N\ge 4 has quantum symmetries and that all Hadamard matrices of a fixed size are mutually quantum isomorphic. These results pass also to the corresponding Hadamard graphs. We also define quantum Hadamard matrices acting on quantum spaces and bring an example thereof over matrix algebras. [ABSTRACT FROM AUTHOR]

We extend the stability and spectral invariance of convolution-dominated matrices to the case of quasi-Banach algebras p < 1 . As an application, we construct a spectrally invariant quasi-Banach algebra of pseudodifferential operators with non-smooth symbols that generalize Sjöstrand's results. [ABSTRACT FROM AUTHOR]

This paper aims to quantitatively analyze the relationship between forces acting on the tool tip and tool movement during drilling operations. The study encompasses axial and lateral vibrations superimposed on the nominal tool movement, arising from rigid body motion (rotational and axial velocities). Specifically, only forces attributed to the cutting process are considered, excluding considerations of indentation forces around the chisel edge. The research adopts a generalized approach, spanning from tool measurements to establishing the force model. The investigation involves measuring cutting forces and correlating them with the varying rake and inclination angles of the drill's cutting edges. An analytical model is proposed to describe the distribution of all local force components along drill edges, considering the evolution of forces and geometry. The dynamic coefficient matrix is evaluated by using the identified cutting coefficient and tool geometry. Validation of the proposed methodology is demonstrated through drilling experiments on Ti6Al4V alloy, utilizing three solid carbide drills with distinct geometries. The proposed procedure allows complete identification of the dynamic characteristics from the measurements taken at the entrance stage of hole drilling operation. Moreover, the influence of tool geometry on cutting coefficients and dynamic coefficient matrices are discussed. • A novel method is developed to achieve minimized experimental effort to identify cutting coefficients in drilling process. • Experiments with industrially relevant tools validated the accurate predictions of transient drilling axial force and torque. • The method is extended to inversely obtain the dynamic coefficient matrix, for the first time, from experimentally measured forces. • Effects of geometry and dynamic interactions were demonstrated using three solid carbide drills with distinct geometries. [ABSTRACT FROM AUTHOR]

This article investigates the H∞$$ {H}_{\infty } $$ control problem of switched systems with time‐varying delays under asynchronous switching. By reconstructing the switching rule of local controllers and empolying multiple Lyapunov functionals and integral inequalities, sufficient conditions without the constraint between the switching delay and the dwell time of submodes to solve the H∞$$ {H}_{\infty } $$ control problem are derived. The control gain matrices can be conveniently obtained by the proposed linear search algorithm. [ABSTRACT FROM AUTHOR]

This paper deals with a time-space fractional Schrödinger equation with homogeneous Dirichlet boundary conditions. A common strategy for discretizing time-fractional operators is finite difference schemes. In these methods, the time-step size should usually be chosen sufficiently small, and subsequently, too many iterations are required which may be time-consuming. To avoid this issue, we utilize the Laplace transform method in the present work to discretize time-fractional operators. By using the Laplace transform, the equation is converted to some time-independent problems. To solve these problems, matrix transformation and improved matrix transformation techniques are used to approximate the spatial derivative terms which are defined by the spectral fractional Laplacian operator. After solving these stationary equations, the numerical inversion of the Laplace transform is used to obtain the solution of the original equation. The combination of finite difference schemes and the Laplace transform creates an efficient and easy-to-implement method for time-space fractional Schrödinger equations. Finally, some numerical experiments are presented and show the applicability and accuracy of this approach. [ABSTRACT FROM AUTHOR]

Let G be a connected graph and S be a k element subset of the vertex set V (G). The Steiner-k distance dG(S) between vertices of S is the minimum size among all connected subgraphs whose vertex set contains S. In this paper, we have defined the Steiner k-eccentric connectivity index and derived a closed formula for the same in case of some standard graphs. Also, we have used Steiner 3-eccentric connectivity index to predict values of boiling point of some primary and secondary amines, cross sectional area and molar refraction of alcohols. For each, regression model is developed and statistical analysis is conducted and these have ensured at least 97% accuracy. [ABSTRACT FROM AUTHOR]

Mathematical chemistry is a field of mathematics where chemical compounds are studied by associating a graph to it. A topological index serves as a mathematical invariant that elucidates the underlying topological arrangement of molecules or networks. This paper explores the neighborhood M-Polynomial concerning various graph operations of regular graphs. Additionally, it addresses and rectifies several erroneous results pertaining to cycle-related graphs that were previously reported. Furthermore, we examine the applications of the neighborhood M-Polynomial to the VPHX[m, n] nanotubes and VPHY[m, n] nanotori, presenting their potential in real-world. Through this comprehensive investigation, we aim to advance the understanding of topological indices and their practical implications. [ABSTRACT FROM AUTHOR]

The soft SVD is a robust matrix decomposition algorithm and a key component of matrix completion methods. However, computing the soft SVD for large sparse matrices is often impractical using conventional numerical methods for the SVD due to large memory requirements. The Rank-Restricted Soft SVD (RRSS) algorithm introduced by Hastie et al. addressed this issue by sequentially computing low-rank SVDs that easily fit in memory. We analyze the convergence of the standard RRSS algorithm and we give examples where the standard algorithm does not converge. We show that convergence requires a modification of the standard algorithm, and is related to non-uniqueness of the SVD. Our modification specifies a consistent choice of sign for the left singular vectors of the low-rank SVDs in the iteration. Under these conditions, we prove linear convergence of the singular vectors using a technique motivated by alternating subspace iteration. We then derive a fixed point iteration for the evolution of the singular values and show linear convergence to the soft thresholded singular values of the original matrix. This last step requires a perturbation result for fixed point iterations which may be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper, exponential stabilization criteria for neutral hybrid nonlinear systems are established via aperiodically intermittent stochastic noise on the basis of average skills. By the It$ \hat{o} $ formula and M-matrix method, we prove that the neutral hybrid nonlinear systems via aperiodically intermittent stochastic noise is $ p $th moment exponentially stable and almost surely exponentially stable. Under the average skills, the delay $ \tau $ is no longer subject to the minimum control upper bound, but it is controlled by the average noise control rate (ANCR) $ \theta $ and the average noise control period (ANCP) $ T $. Lastly, two examples are presented to illustrate our theories. [ABSTRACT FROM AUTHOR]