Diagonal Elements Research Articles

A TWO-STEP MODULUS-BASED MULTISPLITTING ITERATION METHOD FOR THE NONLINEAR COMPLEMENTARITY PROBLEM

In this paper, we construct a two-step modulus-based multisplitting iteration method based on multiple splittings of the system matrix for the nonlinear complementarity problem. And we prove its convergence when the system matrix is an $ H $-matrix with positive diagonal elements. Numerical experiments show that the proposed method is efficient.

Hardware-Friendly Laplacian-Based Multi-Focus Image Fusion in DCT Domain for Visual Sensor Network

Visual sensor network (VSN) requires a multi-focus image or video frame fusion technique involving focus measure computation in the DCT-domain to generate an all-in-focus image. Such techniques are implemented on resource-constrained on-board systems requiring hardware-friendly implementations. In this article, we first show that components of the Laplacian matrix are related to the discrete cosine transform (DCT) basis. The relation is that the eigenvalues of the Laplacian with proper boundary condition form the diagonal elements of the diagonal matrix generated by the DCT operation on the Laplacian. Exploiting this relation, we propose a focus measure which works on the DCT coefficients reflecting the spatial-domain Laplacian operation. Certain simplifications allow our focus measure computation through hardware-friendly integer multiplication and summation, where matrix multiplication involves just N scalar multiplications for an N × N 2D signal. Finally, we propose an approach which suitably fuses multi-focus images or video frames in DCT based image or video coding framework through detection of properly focused area and neighborhood consistency analysis. We show that our proposed approach is hardwarefriendly, computationally simple, and is fast enough for VSN. Through experimental results, we show that our approach outperforms the relevant state-of-the-art in multi-focus image fusion for VSN both quantitatively and subjectively. We also show that our approach is effective in comparison to the state-of-the-art and a few latest generic multi-focus image fusion techniques in terms of quantitative and subjective evaluations.

A High-Resolution and Low-Complexity DOA Estimation Method with Unfolded Coprime Linear Arrays

The direction-of-arrivals (DOA) estimation with an unfolded coprime linear array (UCLA) has been investigated because of its large aperture and full degrees of freedom (DOFs). The existing method suffers from low resolution and high computational complexity due to the loss of the uniform property and the step of exhaustive peak searching. In this paper, an improved DOA estimation method for a UCLA is proposed. To exploit the uniform property of the subarrays, the diagonal elements of the two self-covariance matrices are averaged to enhance the accuracy of the estimated covariance matrices and therefore the estimation performance. Besides, instead of the exhaustive peak searching, the polynomial roots finding method is used to reduce the complexity. Compared with the existing method, the proposed method can achieve higher resolution and better estimation performance with lower computational complexity.

Generalized P and CP transformations in the three-Higgs-doublet model

We study generalised P and CP transformations in the three-Higgs-doublet model (3HDM) with Higgs and gauge fields only. We find that there are two equivalence classes, with respect to flavour transformations, of generalised P transformations and there is only one class of CP transformations. We discuss the conditions the potential has to satisfy in order to be invariant under these transformations. We apply the method of bilinears which we briefly review. We discuss the relation to the conventional basis, where the potential is written in terms of scalar products of the doublet fields. In particular we reproduce the known result that a potential is invariant under CP transformations if and only if there is a conventional basis where all parameters are real. Eventually we study standard P and CP transformations in the $n$-Higgs-doublet model (nHDM). We show that for the bilinears of the nHDM the standard CP transformation corresponds to a diagonal linear transformation with only $\pm 1$ as diagonal elements. We give this matrix explicitly for arbitrary $n$.

Entanglement and matrix elements of observables in interacting integrable systems.

We study the bipartite von Neumann entanglement entropy and matrix elements of local operators in the eigenstates of an interacting integrable Hamiltonian (the paradigmatic spin-1/2 XXZ chain), and we contrast their behavior with that of quantum chaotic systems. We find that the leading term of the average (over all eigenstates in the zero magnetization sector) eigenstate entanglement entropy has a volume-law coefficient that is smaller than the universal (maximal entanglement) one in quantum chaotic systems. This establishes the entanglement entropy as a powerful measure to distinguish integrable models from generic ones. Remarkably, our numerical results suggest that the volume-law coefficient of the average entanglement entropy of eigenstates of the spin-1/2 XXZ Hamiltonian is very close to, or the same as, the one for translationally invariant quadratic fermionic models. We also study matrix elements of local operators in the eigenstates of the spin-1/2 XXZ Hamiltonian at the center of the spectrum. For the diagonal matrix elements, we show evidence that the support does not vanish with increasing system size, while the average eigenstate-to-eigenstate fluctuations vanish in a power-law fashion. For the off-diagonal matrix elements, we show that they follow a distribution that is close to (but not quite) log-normal, and that their variance is a well-defined function of ω=E_{α}-E_{β} ({E_{α}} are the eigenenergies) proportional to 1/D, where D is the Hilbert space dimension.

Eigenstate Thermalization and Rotational Invariance in Ergodic Quantum Systems.

Generic rotationally invariant matrix models satisfy a simple relation: the probability distribution of half the difference between any two diagonal elements and the one of off-diagonal elements are the same. In the spirit of the eigenstate thermalization hypothesis, we test the hypothesis that the same relation holds in quantum systems that are nonlocalized, when one considers small energy differences. The relation provides a stringent test of the eigenstate thermalization hypothesis beyond the Gaussian ensemble. We apply it to a disordered spin chain, the Sachdev-Ye-Kitaev model, and a Floquet system.

Ensemble Average of Three-dimensional Minkowski Tensors of a Gaussian Random Field in Redshift Space

Abstract We present the ensemble expectation values for the translation-invariant, rank-2 Minkowski tensors in three dimensions, for a linearly redshift-space-distorted Gaussian random field. The Minkowski tensors , are sensitive to global anisotropic signals present within a field, and by extracting these statistics from the low-redshift matter density one can place constraints on the redshift-space distortion parameter β = f/b. We begin by reviewing the calculation of the ensemble expectation values , for isotropic, Gaussian random fields, then consider how these results are modified by the presence of a linearly anisotropic signal. Under the assumption that all fields remain Gaussian, we calculate the anisotropic correction due to redshift-space distortion in a coordinate system aligned with the line of sight, finding inequality between the diagonal elements of , . The ratio of diagonal elements of these matrices provides a set of statistics that are sensitive only to the redshift-space distortion parameter β. We estimate the Fisher information that can be extracted from the Minkowski tensors, and find is more sensitive to β than , and a measurement of accurate to ∼1% can yield a constraint on β. Finally, we discuss the difference between using the matrix elements of the Minkowski tensors directly against measuring the eigenvalues. For the purposes of cosmological parameter estimation we advocate the use of the matrix elements, to avoid spurious anisotropic signals that can be generated by the eigenvalue decomposition.

The Infimum on Laplacian Eigenvalues of a Connected Extended Graph: An Edge-Grafting Perspective

Consider a connected extended graph G of size N+1 generated by a connected graph G of size N together with one additional node and some directed edges, in which the extra node is named the virtual leader and the directed edges are originated from the virtual leader to some nodes in G. Define an N × N matrix L + H as the extended Laplacian matrix of G, in which L is the Laplacian matrix of G and H is an nonnegative diagonal matrix with at least one diagonal element being 1. In this brief, we focus on the eigenvalues of the extended Laplacian matrix L + H (i.e., the Laplacian eigenvalues) associated with a connected G and acquire the infimum for them by using the edge-grafting approach. It is found that, the infimum on the Laplacian eigenvalues of a connected G obtained here corresponds to a chain together with a virtual leader and a directed edge, in which either one of the two ending nodes with degree 1 in the chain is informed by the virtual leader. Also with the help of the edge-grafting approach, the infimum on the algebraic connectivity associated with a connected G is gotten.

Riemannian Optimal Identification Method for Linear Systems With Symmetric Positive-Definite Matrix

This article develops identification methods for linear continuous-time symmetric systems, such as electrical network systems, multiagent network systems, and temperature dynamics in buildings. To this end, we formulate three system identification problems for the corresponding discrete-time systems. The first is a least-squares problem in which we wish to minimize the sum of squared errors between the true and model outputs on the product manifold of the manifold of symmetric positive-definite matrices and two Euclidean spaces. In the second problem, to reduce the search dimensions, the product manifold is replaced with the quotient set under a specified group action by the orthogonal group. In the third problem, the manifold of symmetric positive-definite matrices in the first problem is replaced by the manifold of matrices with only positive diagonal elements. In particular, we examine the quotient geometry in the second problem. We propose Riemannian conjugate gradient methods for the three problems, and select initial points using a popular subspace method. The effectiveness of our proposed methods is demonstrated through numerical simulations and comparisons with the Gauss–Newton method, which is one of the most popular approach for solving least-squares problems.

A Divergent Sequence of Romberg Integrals

Romberg integrals are built in order to accelerate the convergence of sequences of trapezoidal rules for approximating the definite integral of a continuous function f. While every sequence of trapezoidal rules with decreasing step length converges whenever f is continuous, this does not always hold for Romberg integrals. In this note we present a concrete example for which the sequence formed by the diagonal elements of the Romberg table diverges when the number of points used to compute the trapezoidal rules grows too slowly.

Partial distinguishability as a coherence resource in boson sampling

Quantum coherence is a useful resource that is consumed to accomplish several tasks that classical devices are hard to fulfill. Especially, it is considered to be the origin of quantum speedup for many computational algorithms. In this work, we interpret the computational time cost of boson sampling with partially distinguishable photons from the perspective of coherence resource theory. With incoherent operations that preserve the diagonal elements of quantum states up to permutation, which we name \emph{permuted genuinely incoherent operation} (pGIO), we present some evidence that the decrease of coherence corresponds to a computationally less complex system of partially distinguishable boson sampling. Our result shows that coherence is one of crucial resources for the computational time cost of boson sampling. We expect our work presents an insight to understand the quantum complexity of the linear optical network system.

Computation of Impulse-Response Gramian for Interval Systems

This paper presents an impulse response gramian for the interval systems. To reduce the higher-order interval systems, two methods used to evaluate the impulse response gramian are computed in the frequency domain. The first method is based on Routh technique with () table to evaluate energy integrals for finding the diagonal impulse response gramian. The second method is based on the Inners technique to compute the diagonal elements of the Gramian matrix, while the off-diagonal elements are obtained by the Markov parameters. The method is computationally simple but the reduced-order stable interval systems are difficult to obtain. To overcome this problem, the reduced-order model is developed based on Kharitonov's theorem. The advantage of the proposed technique is useful for system identification and reduction technique. Illustrative examples are included and results are compared and stability of the reduced-order model is guaranteed.

RC Beam–Column Connections Retrofitted by Steel Prop: Experimental and Analytical Studies

This paper studies the efficiency of a proposed retrofit technique to boost the seismic behavior of beam–column connections in existing deficient reinforced concrete (RC) moment frame systems with the consideration of constraint conditions such as the height of beams, using analytical, experimental, and numerical methods. The proposed retrofit method, called the “single steel prop and curbs”, consists of a diagonal steel prop element and two steel curbs that to the beam and column are laterally mounted. The internal force diagrams of the retrofitted exterior beam–column connections by analytical formulations are investigated and design strategies for promoting the efficiency of the single steel prop to achieve the expected efficiency proposed. To assay the validity and reliability of the proposed analytical procedure, experimental and numerical assessments were also conducted independently. Therefore, four deficient RC beam–column connections by single steel prop were retrofitted using three different cross-sectional areas and revival sheets and then accompanied by a control specimen subjected to cyclic loading. Next, numerical models were calibrated in ABAQUS software. Finally, by derivation of the props’ average axial force from experimental and numerical results, the beam shear coefficient, β, is calculated based on the analytical relations. These results confirmed good conformity between the experimental and numerical outputs as well as reliability of the analytical formulations. Also, the output results indicated that the retrofitted specimens had 53–78% bearing capacity and 146–217% dissipated energy more than the control specimen.

Probability of Deriving a Yearly Transition Probability Matrix for Land-Use Dynamics

Takada’s group developed a method for estimating the yearly transition matrix by calculating the mth power roots of a transition matrix with an interval of m years. However, the probability of obtaining a yearly transition matrix with real and positive elements is unknown. In this study, empirical verification based on transition matrices from previous land-use studies and Monte-Carlo simulations were conducted to estimate the probability of obtaining an appropriate yearly transition probability matrix. In 62 transition probability matrices of previous land-use studies, 54 (87%) could provide a positive or small-negative solution. For randomly generated matrices with differing sizes or power roots, the probability of obtaining a positive or small-negative solution is low. However, the probability is relatively large for matrices with large diagonal elements, exceeding 90% in most cases. These results indicate that Takada et al.’s method is a powerful tool for analyzing land-use dynamics.

A Perspective on the Mathematical and Psychometric Aspects of Factor Indeterminacy

It is generally known that the factors in the common factor model (CFM) cannot be solved for uniquely even if the factor model fits perfectly and the factor loadings are known. This indeterminacy of the factor variables (often referred to as ‘factor scores’) stems from the fact that the CFM is a set of underdetermined linear equations that has more unknowns than equations—a situation which produces an infinity of alternative solutions. In the first section of this inquiry, factor indeterminacy for all factors, common and unique, is examined in the context of The Theory of Generalized Inverses (TGI). Casting factor indeterminacy in the context of TGI provides some simpler results than older approaches. It is shown, for example, that essentially all statistical properties of the infinity of solutions are summarized in the elements of a single, idempotent matrix—symbolized here as H. In the final section of this article, a psychometric view of indeterminacy is presented in which it is argued that the two causes of indeterminacy are low reliabilities and low communalities of the observed variables. It is shown that the diagonal elements of the matrix H are reliability coefficients; therefore, the Spearman-Brown equation can be brought to bear on how much indeterminacy can be reduced by repeated sampling from an Infinite Behavior Domain (IBD). A brief discussion is presented on the effects of factor indeterminacy in IRT models.

Developmental transcriptomes of the sea star, Patiria miniata, illuminate how gene expression changes with evolutionary distance

Understanding how changes in developmental gene expression alter morphogenesis is a fundamental problem in development and evolution. A promising approach to address this problem is to compare the developmental transcriptomes between related species. The echinoderm phylum consists of several model species that have significantly contributed to the understanding of gene regulation and evolution. Particularly, the regulatory networks of the sea star, Patiria miniata (P. miniata), have been extensively studied, however developmental transcriptomes for this species were lacking. Here we generated developmental transcriptomes of P. miniata and compared these with those of two sea urchins species. We demonstrate that the conservation of gene expression depends on gene function, cell type and evolutionary distance. With increasing evolutionary distance the interspecies correlations in gene expression decreases. The reduction is more severe in the correlations between morphologically equivalent stages (diagonal elements) than in the correlation between morphologically distinct stages (off-diagonal elements). This could reflect a decrease in the morphological constraints compared to other constraints that shape gene expression at large evolutionary divergence. Within this trend, the interspecies correlations of developmental control genes maintain their diagonality at large evolutionary distance, and peak at the onset of gastrulation, supporting the hourglass model of phylotypic stage conservation.

Pole representation of the dislocation segment generalized susceptibility diagonal matrix elements

Diagonal elements of the generalized susceptibility matrix in the form of expansion in a series of poles are presented. Screw dislocation segment considered. The calculation of the series coefficients for different values of the dislocation segment normalized lengths has been carried out. The real and imaginary parts of the diagonal matrix elements are calculated. The results of the calculations are presented in the figures. The figures show a good agreement of the diagonal matrix elements direct calculation results with the calculation carried out in this work.

Location of eigenmodes of Euler-Bernoulli beam model under fully non-dissipative boundary conditions.

The Euler-Bernoulli beam model with non-conservative feedback-type boundary conditions is investigated. Components of the two-dimensional input vector are shear and moment at the right end, and components of the observation vector are time derivative of displacement and slope at the right end. The boundary matrix containing four control parameters relates input and observation. The following results are presented: (i) if one and only one of the control parameters is positive and the rest of them are equal to zero, then the set of the eigenmodes is located in the open left half-plane of the complex plane, which means that all eigenmodes are stable; (ii) if the diagonal elements of the boundary matrix are positive and off-diagonal elements are zeros, then the set of the eigenmodes is located in the open left half-plane, which implies stability of all eigenmodes; (iii) specific combinations of the diagonal and off-diagonal elements have been found to ensure the stability results. To prove the results, two special relations between the eigenmodes and mode shapes of the non-self-adjoint problem and clamped-free self-adjoint problem have been established.

Direction of Arrival Estimation by a Modified Orthogonal Propagator Method With Spline Interpolation

This paper presents a modified orthogonal propagator method (OPM) with spline interpolation for direction of arrival (DoA) estimation. The conventional OPM is based on the noise-free signal model. When noise exists, the main contribution of noise (additive temporally and spatially white Gaussian noise) is in the diagonal elements of the data covariance matrix. Therefore, the influence of noise can be minimized by restoring the noise free diagonal elements through an interpolation procedure and the propagator can then be directly extracted from the denoised data covariance matrix, which is the main idea of the proposed method in this paper. The performance of the proposed modified OPM is tested through numerical simulations. Simulation results show that the proposed modified OPM achieves low root-mean square errors (RMSEs) with small computational load in DoA estimation.

Analysis of acoustic wave propagation in composite laminates via a spectral element method

AbstractThe analysis of the acoustic or elastic wave propagation phenomena in the time‐domain is computationally expensive. An often‐cited rule of thumb is to use at least ten nodes per wavelength if using quadratic shape functions, and even more nodes are necessary in order to achieve highly accurate numerical results. For large‐scale problems, this results in a large system of linear algebraic equations, which must be solved for several thousands of time‐steps. This requires a vast amount of the computing time. In this paper, a spectral element method (SEM), which is a variant of the finite element method (FEM) by using high‐order shape functions with non‐equidistant node distributions, is presented in order to investigate the acoustic wave propagation problems taking the fluid‐structure interaction into account with a minimum of required degrees of freedoms (DOFs). Hence, the computing time for solving the system of linear algebraic equations is reduced while maintaining a high accuracy. The element matrices are computed using the Lobatto quadrature method, which results in a diagonal element mass matrix. For fluid‐structure interaction problems in acoustics, a non‐symmetric sparse mass matrix is obtained. By using an explicit time‐integration scheme only the mass matrix becomes the coefficient matrix of the linear system and, due to its high sparsity, the solution process is very efficient. In this work, the SEM is implemented and applied for the analysis of the transient behavior of the laminated composite structures excited by acoustic waves. Numerical examples are presented and discussed to show the efficiency of the SEM in the time‐domain for applications in acoustic wave propagation problems considering the fluid‐structure interaction.