Dominant Matrix Research Articles

The Schur complements of [formula omitted]-diagonally and product [formula omitted]-diagonally dominant matrix and their disc separation

In this paper, we obtain some estimates for the γ - diagonally and product γ - diagonally dominant degree of the Schur complement of matrices. As application we present some bounds for the eigenvalues of Schur complement by the entries of the original matrix. Based on these results, we give a kind of iteration called the Schur-based iteration, which can solve large scale linear systems though reducing the order by the Schur complement and can compute out the results faster.

Metamagnetism Seeded by Nanostructural Features of Single‐Crystalline Gd5Si2Ge2

Metamagnets are materials with first-order phase transition from one magnetically ordered phase to another that could provide environmentally friendly, highly efficient refrigeration near ambient temperatures using a phenomenon known as the magnetocaloric effect. Profoundly important to fundamental research and the application of room-temperature magnetic refrigeration (a solid-state cooling method that takes advantage of the entropy change associated with changes in magnetic order), is the need to understand how the metamagnetic transition evolves, and whether it can be manipulated by nanoengineering, or chemical substitutions to be sharper, occur at a lower magnetic driving fields, or be less hysteretic. Here, using Hall probe imaging, we show that inclusions of mesocopic platelets of Gd5Si1.5Ge1.5, that are inherent to all single crystals and polycrystals of this compound, seed the global metamagnetic transition to the ferromagnetic state across the Gd5Si2Ge2 crystal. We show that these inclusions, as well as well-defined mosaic boundaries, play a dramatic role on the transition properties. Strain and the associated field-driven volume changes underpin the phenomena that we have observed, and show that nanostructuring is the likely route to a more complete control of the critical properties in these materials. Themagnetic cooling field has a historic past, first conceived in the 1920’s by P. Debye and W. Giauque; the concepts have been used to reach extremely low temperatures in a process known as adiabatic demagnetization. Over the past ten years, the field has experienced a resurgence of interest after the discovery of giant magnetocaloric effects in a family of materials known as metamagnets, which are compounds that undergo a first-order transition to a magnetically ordered state, often accompanied by a coincident transformation of the crystal structure. Gd5Si2Ge2 has attracted a great deal of interest in this important area because it could play a pivotal role for energy-efficient magnetic cooling near room temperatures. It is a paramagnet (PM) above 274K with a monoclinic crystal structure; by applying a magnetic field, the material can be converted to a ferromagnetic (FM) state with a different, yet closely related, orthorhombic structure via a first-order transition. It has been shown that alloying Gd5Si2Ge2 with different elements substituting for either Si and Ge, or Gd can tune the critical temperature,Tc, but this leads to a significant shift in the magnetic character from first-order towards second-order behavior, which has an adverse effect on the magnitude of the magnetocaloric effect. Although these are interesting facts, it is true to say that a proper fundamental understanding of how to engineer the metamagnetic transition to optimize the magnetocaloric effect has yet to be established. Conventional structural characterization techniques, such as scanning and transmission electron microscopies (SEM, TEM), and X-ray diffraction have proved to be vitally important for understanding of the Gd5Si2Ge2 compound. High-resolution TEM has shown that all the Gd5Si2Ge2 crystals have a complex microstructure consisting of a dominant matrix of the singlephase parent material that is interspersed with extremely thin-plates (200 nm thick) of a second phase Gd5(SixGe1–x)3 material. In this sense the crystals are not pure, but single-crystal-like diffraction patterns are observed by backscatter Laue technique, and so for all intents and purposes these samples can indeed be considered single crystals. The platelets arrange themselves in a criss-cross pattern when viewed perpendicular or nearly perpendicular to the [010] face (see Fig. 1), with platelets extending for several hundred micrometers. When viewed perpendicular to both the [100] and [001] faces, the platelets

H∞ control for linear systems with state saturation nonlinearities

AbstractThe problem of H∞ control for a class of linear systems with state saturation nonlinearities is considered in this paper. By introducing a row diagonally dominant matrix with negative diagonal elements and a diagonal matrix with positive elements, the H∞ control problem is reduced to a matrix inequality feasibility problem that can be solved by the proposed iterative linear matrix inequality algorithm. The effectiveness of the presented method is demonstrated by a numerical example. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Linear system based approach for solving some related problems of [formula omitted]-matrices

This paper develops a novel linear system based approach for computing ‖A-1‖∞, the Skeel condition number of an M-matrix A, and the positive diagonal matrices D guaranteeing that AD be a strictly diagonally dominant matrix. Theoretic analysis and simulation results justify the validity of the proposed approach. Moreover, the proposed linear system model can be implemented by application-specific integrated circuits, and then it has asynchronous parallel processing ability and can achieve high computing performance.

On the parallel GSAOR method for block diagonally dominant matrices

In this paper we consider the parallel generalized SAOR iterative method based on the generalized AOR iterative method presented by James for solving large nonsingular system. We obtain some convergence theorems for the case when coefficient matrix is a block diagonally dominant matrix or a generalized block diagonal dominant matrix. A numerical example is given to illustrate to our results.

An acceleration method for the BEM-based cooling simulation of injection molding

Mold cooling process in injection molding is of great importance and most cooling analyses are based on the boundary element (BEM) approach. The BEM leads to dense influence matrices, which greatly affect the computational efficiency and scale. In this paper, an acceleration method is presented to speed up the solution procedure. In this method, the dense BEM matrix is split into a sparse dominant matrix and a dense residual matrix. The residual item is transformed from the inner iteration to the outer iteration, and the dominant item can be stored in the RAM memory so that the resulting system of equations will be solved much more quickly. Numerical experiments show that this developed approach is a cost-effective method for BEM-based simulation of injection mold cooling.

Effect of landscape structure on the spatial distribution of Mediterranean dung beetle diversity

ABSTRACTAim Landscape structure influences the distribution of animals, altering their movements and their ability to reach habitat patches. We analysed the spatial patterns of dung beetle species diversity in three differently structured natural landscapes in a Mediterranean protected area in the centre of the Iberian Peninsula.Location Cabañeros National Park, Central Spain.Methods Diversity components within (α) and among (β) the three main vegetation types in the reserve were compared by using a hierarchical nested design. These were forests, scrublands and grasslands embedded in three different landscapes, where each was the most dominant and structurally connected habitat.Results Species richness of grassland habitat did not vary across landscapes, but forest habitat showed lower species richness in the grassland‐dominated landscape. Scrubland was the least species‐rich habitat, but here again there was no significant variation across landscapes. However, in all cases, there was a significant influence of habitat context (configuration of habitat patches within landscape matrix) on similarity of species composition. These tended to be more similar to the dominant landscape matrix where they were embedded, rather than to the same habitat type in other landscapes. Additive partitioning of diversity showed higher than expected values of β in all landscapes, which indicated a structured response. Highest values of β in the grassland‐dominated landscape suggest that this was the least connected landscape for dung beetles.Main conclusions Our results suggest that in homogeneous conditions of climate and trophic resources, landscape structure may well be more important than habitat type as a determinant of dung beetle distribution in the Mediterranean.

Strict diagonal dominance and a Geršgorin type theorem in Euclidean Jordan algebras

For complex square matrices, the Levy–Desplanques theorem asserts that a strictly diagonally dominant matrix is invertible. The well-known Geršgorin theorem on the location of eigenvalues is equivalent to this. In this article, we extend the Levy–Desplanques theorem to an object in a Euclidean Jordan algebra when its Peirce decomposition with respect to a Jordan frame is given. As a consequence, we prove a Geršgorin type theorem for the spectral eigenvalues of an object in a Euclidean Jordan algebra.

Water losses through paddy bunds: Methods, experimental data, and simulation studies

Summary Irrigated rice landscapes increasingly experience water scarcity, and the bunds surrounding paddy rice fields (also referred to as dikes, banks or levees) have been pointed out as possible causes for low water productivities. Yet, detailed experimental data to confirm this assumption is missing, as well as an identification of the most decisive factors for bund infiltration. The objective of this study was therefore to quantify water fluxes through bunds in a terraced paddy landscape both experimentally and by numerical analysis. Two bunds with a respective age of 20 (young bund: BY) and 100 years (old bund: BO) were investigated at the end of the rice-growing season. Soil core samples revealed a compacted hard pan at 30 cm depth in both bunds. Hydraulic conductivity was very high and variable especially in the upper part of the bunds. For direct measurements of infiltration into the bund, a double-frame infiltrometer was introduced analogous to a double ring infiltrometer. Measured infiltration rates in bund BO at ponding water depths (PWDs) of 3, 5 and 7 cm amounted to 480, 270 and 550 cm2 d−1, respectively, and no cross-flow was observed. At bund BY, the infiltrating water resulted in an increasing water level on the lower field, and the cross-flow was 807, 4205 and 5506 cm2 d−1 at corresponding PWDs. However, no reliable infiltration measurements could be recorded. Tensiometers installed in the bunds during the infiltration experiments revealed a nearly instantaneous rise in pressure head after flooding throughout the profile in bund BY, indicating preferential water fluxes. In bund BO, contrarily, pressure heads underneath the hard pan rose slowly and with a time lag and thus show dominant matrix fluxes. Different infiltration rates in the investigated bunds could be mainly attributed to a lower bulk density and enhanced macroporosity in the young bund BY. The existence of a hard pan in a bund does not necessarily reduce percolation losses, as it might become perforated by macropores originating from bio-activity. A sensitivity analysis realized with HYDRUS 1.05 revealed that under-bund percolation is most strongly influenced by the hydraulic conductivity in the upper and middle bund. Ponding water depth and groundwater table are less important, while soybean cultivation, bund width and elevation difference only play a minor role. For each conductivity profile, there is a critical bund width underneath which cross-flow occurred. A detailed description of the bunds’ water fluxes needs to be included in hydrologic modelling of irrigated rice landscapes.

Stability Analysis of Discrete Hopfield Neural Networks with the Nonnegative Definite Monotone Increasing Weight Function Matrix

The original Hopfield neural networks model is adapted so that the weights of the resulting network are time varying. In this paper, the Discrete Hopfield neural networks with weight function matrix (DHNNWFM) the weight changes with time, are considered, and the stability of DHNNWFM is analyzed. Combined with the Lyapunov function, we obtain some important results that if weight function matrix (WFM) is weakly (or strongly) nonnegative definite function matrix, the DHNNWFM will converge to a stable state in serial (or parallel) model, and if WFM consisted of strongly nonnegative definite function matrix and column (or row) diagonally dominant function matrix, DHNNWFM will converge to a stable state in parallel model.

The eigenvalue distribution on Schur complement of nonstrictly diagonally dominant matrices and general H-matrices

The paper studies the eigenvalue distribution of Schur complements of some specialmatrices, including nonstrictly diagonally dominant matrices and general H−matrices. Zhang, Xu, and Li [Theorem 4.1, The eigenvalue distribution on Schur complements of H-matrices. Linear Algebra Appl., 422:250–264, 2007] gave a condition for an n×n diagonally dominant matrix A to have |JR+(A)| eigenvalues with positive real part and |JR_ (A)| eigenvalues with negative real part, where |JR+ (A)| (|JR_ (A)|) denotes the number of diagonal entries of A with positive (negative) real part. This condition is applied to establish some results about the eigenvalue distribution for the Schur complements of nonstrictly diagonally dominant matrices and general H−matrices with complexdiagonal entries. Several conditions on the n × n matrix A and the subset α ⊆ N = {1, 2, · · · , n} arepresented so that the Schur complement A/α of A has |JR+ (A)|−|JαR+ (A)| eigenvalues with positivereal part and |JR_ (A)| − |JαR_ (A)| eigenvalues with negative real part, where |JαR+ (A)| (|JαR_ (A)|) denotes the number of diagonal entries of the principal submatrix A(α) of A with positive (negative) real part.

Relative Perturbation Bounds for Eigenvalues of Symmetric Positive Definite Diagonally Dominant Matrices

For a symmetric positive semidefinite diagonally dominant matrix, if its off-diagonal entries and its diagonally dominant parts for all rows (which are defined for a row as the diagonal entry subtracted by the sum of absolute values of off-diagonal entries in that row) are known to a certain relative accuracy, we show that its eigenvalues are known to the same relative accuracy. Specifically, we prove that if such a matrix is perturbed in a way that each off-diagonal entry and each diagonally dominant part have relative errors bounded by some $\epsilon$, then all its eigenvalues have relative errors bounded by $\epsilon$. The result is extended to the generalized eigenvalue problem.

Combinatorial Preconditioners for Scalar Elliptic Finite-Element Problems

We present a new preconditioner for linear systems arising from finite-element discretizations of scalar elliptic partial differential equations (PDE's). The solver splits the collection $\{K_{e}\}$ of element matrices into a subset of matrices that are approximable by diagonally dominant matrices and a subset of matrices that are not approximable. The approximable $K_{e}$'s are approximated by diagonally dominant matrices $L_{e}$'s that are assembled to form a global diagonally dominant matrix L. A combinatorial graph algorithm then approximates L by another diagonally dominant matrix M that is easier to factor. Finally, M is added to the inapproximable elements to form the preconditioner, which is then factored. When all the element matrices are approximable, which is often the case, the preconditioner is provably efficient. Approximating element matrices by diagonally dominant ones is not a new idea, but we present a new approximation method which is both efficient and provably good. The splitting idea is simple and natural in the context of combinatorial preconditioners, but hard to exploit in other preconditioning paradigms. Experimental results show that on problems in which some of the $K_{e}$'s are ill conditioned, our new preconditioner is more effective than an algebraic multigrid solver, than an incomplete-factorization preconditioner, and than a direct solver.

Schur complements of generally diagonally dominant matrices and a criterion for irreducibility of matrices

As is well known, the Schur complements of strictly or irreducibly diagonally dominant matrices are H−matrices; however, the same is not true of generally diagonally dominant matrices. This paper proposes some conditions on the generally diagonally dominant matrix A and the subset α ⊂{ 1,2,...,n} so that the Schur complement matrix A/α is an H−matrix. These conditions are then applied to decide whether a matrix is irreducible or not.

Special H-matrices and their Schur and diagonal-Schur complements

It is well known, see [D. Carlson, T. Markham, Schur complements of diagonally dominant matrices, Czech. Math. J. 29 (104) (1979) 246–251 [2]; J. Liu, J. Li, Z. Huang, X. Kong, Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices, Linear Alg. Appl. 428 (2008) 1009–1030] [14], that the Schur complement of a strictly diagonally dominant matrix is strictly diagonally dominant, as well as its diagonal-Schur complement. Also, if a matrix is an H-matrix, then its Schur complement and diagonal-Schur complement are H-matrices, too, see [J. Liu, Y. Huang, Some properties on Schur complements of H-matrices and diagonally dominant matrices, Linear Alg. Appl. 389 (2004) 365–380] [13]. Recent research, see [J. Liu, Y. Huang, F. Zhang, The Schur complements of generalized doubly diagonally dominant matrices, Linear Alg. Appl. 378 (2004) 231–244 [12]; J. Liu, J. Li, Z. Huang, X. Kong, Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices, Linear Alg. Appl. 428 (2008) 1009–1030] [14], showed that the similar statements hold for some special subclasses of H-matrices. The aim of this paper is to give more invariance results of this type, and simplified proofs for some already known results, by using scaling approach.

Theoretical studies of active power/angle sub-matrix in power flow Jacobian for power system analysis

Properties of the active power/angle sub-matrix in the power flow Jacobian for power system analysis are studied. The sub-matrix is a dominant and irreducible matrix under very general conditions of power systems, so that it is invertible. Also the necessary conditions for its singularity are given. These theoretical results can be used to clarify the ambiguous understanding of the sub-matrix in current literature, and also provide the theoretical foundations for the applications based on reduced power flow Jacobian. Numerical simulation on the IEEE 118-bus power system is used to illustrate our results.

A coupled finite volume solver for the solution of incompressible flows on unstructured grids

This paper reports on a newly developed fully coupled pressure-based algorithm for the solution of laminar incompressible flow problems on collocated unstructured grids. The implicit pressure-velocity coupling is accomplished by deriving a pressure equation in a procedure similar to a segregated SIMPLE algorithm using the Rhie–Chow interpolation technique and assembling the coefficients of the momentum and continuity equations into one diagonally dominant matrix. The extended systems of continuity and momentum equations are solved simultaneously and their convergence is accelerated by using an algebraic multigrid solver. The performance of the coupled approach as compared to the segregated approach, exemplified by SIMPLE, is tested by solving five laminar flow problems using both methodologies and comparing their computational costs. Results indicate that the number of iterations needed by the coupled solver for the solution to converge to a desired level on both structured and unstructured meshes is grid independent. For relatively coarse meshes, the CPU time required by the coupled solver on structured grid is lower than the CPU time required on unstructured grid. On dense meshes however, this is no longer true. For low and moderate values of the grid aspect ratio, the number of iterations required by the coupled solver remains unchanged, while the computational cost slightly increases. For structured and unstructured grid systems, the required number of iterations is almost independent of the grid size at any value of the grid expansion ratio. Recorded CPU time values show that the coupled approach substantially reduces the computational cost as compared to the segregated approach with the reduction rate increasing as the grid size increases.

Restoring the rare forb Erodium macrophyllum to exotic grassland in southern California

Erodium macrophyllum is a rare annual forb that was once widespread in California, but has become largely restricted to clay soil outcrops. An effort was made to restore it to a soil where it was once likely abundant. This soil is now dominated by exotic annual grasses and forbs from the Mediterranean that grow in patches with native forbs. To determine whether there is a competitive hierarchy of E. macrophyllum with the dominant matrix species, we examined establishment and seed production of E. macrophyllum in patches of forbs and grasses in a southern California exotic grassland over 3 yr. We sowed seeds of E. macrophyllum into patches where the dominant species was either Amsinckia menziesii (native forb), Erodium brachycarpum (exotic forb) or Bromus spp. (exotic grasses) and into adjacent manually weeded control plots. All 3 competitors negatively impacted E. macrophyllum seed production, but seldom affected seedling establishment. Although we expected that a native forb would be most compatible with E. macrophyllum, A. menziesii in fact had the most negative effect or was equivalent to E. brachycarpum, and Bromus spp. had the least negative effect. Bromus plants had the lowest density of the 3 matrix species, the lowest rate of recol- onization following removal, and grew in soil with the highest levels of extractable N and P. In this case, the competitive hierarchy was confounded by soil nutrients, so that establishment of E. macro- phyllum was greatest in patches of Bromus. However, both native and exotic competitors reduced E. macrophyllum establishment, so weeding all known species of competitors will be needed to increase chances of seed production and promote restoration of this rare annual forb.

Expression of Sox and fibrillar collagen genes in lamprey larval chondrogenesis with implications for the evolution of vertebrate cartilage

Lampreys possess unique types of cartilage in which elastin-like proteins are the dominant matrix component, whereas gnathostome cartilage is mainly composed of fibrillar collagen. Despite the differences in protein composition, the Sox-col2a1 genetic cascade was suggested to be conserved between lamprey pharyngeal cartilage and gnathostome cartilage. We examined whether the cascade is conserved in another type of lamprey cartilage, the trabecular cartilage. We found that SoxD and SoxE are expressed in both trabecular and pharyngeal cartilages. However, trabecular cartilage shows no clade A fibrillar collagen gene expression, including genes expressed in pharyngeal cartilage of this animal. On the basis of these observations, we propose that lampreys possess an ancestral type of cartilage that is similar to amphioxus gill cartilage, and in this respect, gnathostome cartilage can be regarded as derived for the loss of elastin-like protein as a cartilage component and recruitment of fibrillar collagen, which is included as a minor component in the ancestral cartilage, as the main component.

Quantum dot-labeled aptamer nanoprobes specifically targeting glioma cells

Two new techniques, aptamer-based specific recognition and quantum dot (QD)-basedfluorescence labeling, are becoming increasingly important in biosensing. In this study,these two techniques have been coupled together to construct a new kind of fluorescentQD-labeled aptamer (QD–Apt) nanoprobe by conjugating GBI-10 aptamer to the QDsurface. GBI-10 is a single-stranded DNA (ssDNA) aptamer for tenascin-C, whichdistributes on the surface of glioma cells as a dominant extracellular matrix protein. TheQD–Apt nanoprobe can recognize the tenascin-C on the human glioma cell surface, whichwill be helpful for the development of new convenient and sensitive in vitro diagnosticassays for glioma. The QD–Apt nanoprobe has particular features such as strongfluorescence, stability, monodispersity and uniformity. In addition, this probe preparationmethod is universal, so it is expected to provide a new type of stable nanoprobe forhigh-throughput and fast biosensing detection and bioimaging. New methodsfor real-time and dynamic tracking and imaging can be accordingly developed.