Constrained Matrices Research Articles

Non-linear buckling analysis of delaminated hat-stringer panels using variational asymptotic method

This research proposes a computationally efficient methodology using a Constrained Variational Asymptotic Method (C-VAM) for non-linear buckling analysis on a hat-stringer panel with delamination defects. Starting with the geometrically non-linear kinematics, the VAM procedure reduces the three-dimensional (3-D) strain energy functional to an analogous 2-D plate model and evaluates the closed form warping solutions. Utilising the resulting warping solutions and recovery relations for the skin and the stringer, displacement continuity at the three-dimensional level is enforced between the stringer and the skin based on the pristine and delaminated interface regions. Consequently, the constrained matrices obtained from C-VAM is incorporated into an in-house developed non-linear finite element framework. Using the developed formulation, a stiffened panel with delamination of 40 mm between the stringer and the skin is analysed under compression. The results have been validated locally and globally, employing experimental data and 3-D finite element analysis (FEA). Experiments are carried out on the co-cured panel by applying quasi-static loading with displacement-controlled conditions, and 3-D FEA is carried out in Abaqus. Load-response plots have been obtained to validate the results at the global level, and they are in excellent agreement with experiments and 3-D FEA. Subsequently, out-of-plane displacement contour plots are obtained; the number of half waves and wave intensity in 3-D FEA and C-VAM are comparable, although there are minor differences compared to the experimental findings. The proposed framework is shown to be computationally efficient by over 55% as compared to 3-D FEA for performing non-linear buckling analysis on the stiffened composite structure considered in the current work.

Three special kinds of least squares solutions for the quaternion generalized Sylvester matrix equation

<abstract><p>In this paper, we propose an efficient method for some special solutions of the quaternion matrix equation $ AXB+CYD = E $. By integrating real representation of a quaternion matrix with $ \mathcal{H} $-representation, we investigate the minimal norm least squares solution of the previous quaternion matrix equation over different constrained matrices and obtain their expressions. In this way, we first apply $ \mathcal{H} $-representation to solve quaternion matrix equation with special structure, which not only broadens the application scope of $ \mathcal{H} $-representation, but further expands the research idea of solving quaternion matrix equation. The algorithms only include real operations. Consequently, it is very simple and convenient, and it can be applied to all kinds of quaternion matrix equation with similar problems. The numerical example is provided to illustrate the feasibility of our algorithms.</p></abstract>

Law of large numbers for permanents of random constrained matrices

Permanents of random matrices with independent and identically distributed (i.i.d.) entries have extensively studied in literature and convergence and concentration properties are known under varying assumptions on the distributions. In this paper we consider constrained \(n \times n\) random matrices where each row has a deterministic number \(r = r(n)\) zero entries and the rest of the entries are independent random variables that are positive almost surely. The positions of the zeros within each row are also random and we establish sufficient conditions for the existence of a weak law of large numbers for the permanent, appropriately scaled and centred. As a special case, we see that if \(r\) grows faster than \(\sqrt{n},\) then the permanent of a randomly chosen constrained \(0-1\) matrix is concentrated around the van der Waerden bound with high probability, i.e. with probability converging to one as \(n \rightarrow \infty .\)

Reconfigurable Intelligent Surface Empowered Downlink Non-Orthogonal Multiple Access

Power-domain non-orthogonal multiple access (NOMA) has become a promising technology to exploit the new dimension of the power domain to enhance the spectral efficiency of wireless networks. However, most existing NOMA schemes rely on the strong assumption that users’ channel gains are quite different, which may be invalid in practice. To unleash the potential of power-domain NOMA, we propose a reconfigurable intelligent surface (RIS)-empowered NOMA scheme to introduce desirable channel gain differences among the users by adjusting the phase shifts at the RIS. Our goal is to minimize the total transmit power by jointly optimizing the beamforming vectors at the base station, the phase-shift matrix at the RIS, and user ordering. To address challenge due to the highly coupled optimization variables, we present an alternating optimization framework to decompose the non-convex bi-quadratically constrained quadratic problem under a specific user ordering into two rank-one constrained matrices optimization problems via matrix lifting. To accurately detect the feasibility of the non-convex rank-one constraints and improve performance by avoiding early stopping in the alternating optimization procedure, we equivalently represent the rank-one constraint as the difference between nuclear norm and spectral norm. A difference-of-convex (DC) algorithm is further developed to solve the resulting DC programs via successive convex relaxation, followed by establishing the convergence of the proposed DC-based alternating optimization method. We further propose an efficient user ordering scheme with closed-form expressions, considering both the channel conditions and users’ target data rates. Simulation results validate the ability of an RIS in enlarging the channel-gain difference when the users’ original channel conditions are similar and the superiority of the proposed DC-based alternating optimization method in reducing the total transmit power.

Long-Time Asymptotics for the Focusing Hirota Equation with Non-Zero Boundary Conditions at Infinity Via the Deift-Zhou Approach

Support constrained generator matrix for a linear code has been an active topic in recent years. The necessary and sufficient condition for the existence of MDS codes over small fields with support constrained generator matrices were conjectured by Dau, Song, Yuen in 2014. This GM-MDS conjecture was proved independently by Lovett and Yildiz-Hassibi in 2018. In this paper we propose the necessary and sufficient conditions for support constrained generator matrices of general linear codes based on the generalized Hamming weights. It is proved that the direct generalization of the GM-MDS conjecture for $2$-MDS codes and algebraic geometry codes is not true over arbitrary fields. We propose and prove a GHW-based sufficient condition for support constrained matrices of general linear codes. This is the first sufficient condition for the existence of support constrained generator matrices for general linear codes over arbitrary finite fields. Moreover a weaker GHW-based sufficient condition for support constrained generator matrices are given for the binary simplex code and the first order $q$-ary Reed-Muller codes.

An efficient real representation method for least squares problem of the quaternion constrained matrix equation AXB + CY D = E

Let and represent the sets of all η-Hermitian quaternion matrices and η-anti-Hermitian quaternion matrices, respectively. On the basis of the real representation matrix of a quaternion matrix and its particular structure, we convert the least squares problem of the quaternion matrix equation AXB + CY D = E over into the corresponding problem of the real matrix equation over free variables, and then we establish its unique minimal norm least squares solution. Our resulting expressions are expressed only by real matrices, and the algorithm only includes real operations. Consequently, they are very simple and convenient. Compared with the existing method [S.F. Yuan, Q.W. Wang, and X. Zhang, Least-squares problem for the quaternion matrix equation AXB + CYD = E over different constrained matrices, Int. J. Comput. Math. 90 (2013), pp. 565–576], the final two examples show that our method is more efficient and superior.

Direct synthesis of fixed‐order multi‐objective controllers

SummaryThis paper introduces a new methodology for the design of fixed‐order multi‐objective output feedback controllers. The problem comprises a set of linear matrix inequalities and an additional rank constraint. The primary idea is to classify convex subsets of the set of rank constrained matrices in such formulations, based on which two noniterative and relatively fast methods are developed. The proposed methods require solving a convex optimization problem at each step and can be applied with any weighted summation of design objectives such as performance, performance, passivity, and regional pole assignment. Several benchmark systems with performance criteria including H∞, mixed H2/H∞, and pole placement are used to demonstrate a marked improvement in comparison to the existing methods.

Iterative solutions of generalized inverse eigenvalue problem for partially bisymmetric matrices

In this paper, for given , , an iterative algorithm is constructed to find the solutions of generalized inverse eigenvalue problem , where A and B should be partially bisymmetric under a prescribed submatrix constraint. For any initial constrained matrices, a solution pair can be obtained in finite iteration steps by this iterative algorithm in the absence of roundoff errors. The least norm solution can be obtained by choosing a special kind of initial matrix pencil. In addition, the unique optimal approximation solution to a given matrix pencil in the solution set of the above problem can also be obtained by this iterative algorithm. Numerical examples are given to illustrate efficiency of the proposed algorithm.

Least-squares problem for the quaternion matrix equation AXB+CYD=E over different constrained matrices

For any A=A 1+A 2 j∈Q n×n and η∈<texlscub>i, j, k</texlscub>, denote A η H =−η A H η. If A η H =A, A is called an $\\eta$-Hermitian matrix. If A η H =−A, A is called an η-anti-Hermitian matrix. Denote η-Hermitian matrices and η-anti-Hermitian matrices by η HQ n×n and η AQ n×n , respectively. By using the complex representation of quaternion matrices, the Moore–Penrose generalized inverse and the Kronecker product of matrices, we derive the expressions of the least-squares solution with the least norm for the quaternion matrix equation AXB+CYD=E over X∈η HQ n×n and Y∈η AQ n×n .

Modeling the influence of nucleus elasticity on cell invasion in fiber networks and microchannels

Cell migration in highly constrained extracellular matrices is exploited in scaffold-based tissue engineering and is fundamental in a wide variety of physiological and pathological phenomena, among others in cancer invasion and development. Research into the critical processes involved in cell migration has mainly focused on cell adhesion and proteolytic degradation of the external environment. However, rising evidence has recently shown that a number of cell-derived biophysical and mechanical parameters, among others nucleus stiffness and cell deformability, plays a major role in cell motility, especially in the ameboid-like migration mode in 3D confined tissue structures. We here present an extended cellular Potts model (CPM) first used to simulate a micro-fabricated migration chip, which tests the active invasive behavior of cancer cells into narrow channels. As distinct features of our approach, cells are modeled as compartmentalized discrete objects, differentiated in the nucleus and in the cytosolic region, while the migration chamber is composed of channels of different widths. We find that cell motile phenotype and velocity in open spaces (i.e., 2D flat surfaces or large channels) are not significantly influenced by cell elastic properties. On the contrary, the migratory behavior of cells within subcellular and subnuclear structures strongly relies on the deformability of the cytosol and of the nuclear cluster, respectively. Further, we characterize two migration dynamics: a stepwise way, characterized by fluctuations in cell length, within channels smaller than nucleus dimensions and a smooth sliding (i.e., maintaining constant cell length) behavior within channels larger than the nuclear cluster. These resulting observations are then extended looking at cell migration in an artificial fiber network, which mimics cell invasion in a 3D extracellular matrix. In particular, in this case, we analyze the effect of variations in elasticity of the nucleus on cell movement. In order to summarize, with our simulated migration assays, we demonstrate that the dimensionality of the environment strongly affects the migration phenotype and we suggest that the cytoskeletal and nuclear elastic characteristics correlate with the tumor cell's invasive potential.

LSQR iterative method for generalized coupled Sylvester matrix equations

An iterative method is proposed to solve generalized coupled Sylvester matrix equations, based on a matrix form of the least-squares QR-factorization (LSQR) algorithm. By this iterative method on the selection of special initial matrices, we can obtain the minimum Frobenius norm solutions or the minimum Frobenius norm least-squares solutions over some constrained matrices, such as symmetric, generalized bisymmetric and (R,S)-symmetric matrices. Meanwhile, the optimal approximate solutions to the given matrices can be derived by solving the corresponding new generalized coupled Sylvester matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the present method.

A matrix LSQR iterative method to solve matrix equation AXB=C

This paper is a matrix iterative method presented to compute the solutions of the matrix equation, AXB=C, with unknown matrix X∈𝒮, where 𝒮 is the constrained matrices set like symmetric, symmetric-R-symmetric and (R, S)-symmetric. By this iterative method, for any initial matrix X 0∈𝒮, a solution X* can be obtained within finite iteration steps if exact arithmetics were used, and the solution X* with the minimum Frobenius norm can be obtained by choosing a special kind of initial matrix. The solution , which is nearest to a given matrix ˜X in Frobenius norm, can be obtained by first finding the minimum Frobenius norm solution of a new compatible matrix equation. The numerical examples given here show that the iterative method proposed in this paper has faster convergence and higher accuracy than the iterative methods proposed in [G.-X. Huang, F. Yin, and K. Guo, An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB=C , J. Comput. Appl. Math. 212 (2008), pp. 231–244; Y. Lei and A.-P. Liao, A minimal residual algorithm for the inconsistent matrix equation AXB=C over symmetric matrices, Appl. Math. Comput. 188 (2007), pp. 499–513; Z.-Y. Peng, An iterative method for the least squares symmetric solution of the linear matrix equation AXB=C , Appl. Math. Comput. 170 (2005), pp. 711–723; Y.-X. Peng, X.-Y. Hu, and L. Zhang, An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C , Appl. Math. Comput. 160 (2005), pp. 763–777].

Quasi-cyclic LDPC codes: an algebraic construction

This paper presents two new large classes of QC-LDPC codes, one binary and one non-binary. Codes in these two classes are constructed by array dispersions of row-distance constrained matrices formed based on additive subgroups of finite fields. Experimental results show that codes constructed perform very well over the AWGN channel with iterative decoding based on belief propagation. Codes of a subclass of the class of binary codes have large minimum distances comparable to finite geometry LDPC codes and they offer effective tradeoff between error performance and decoding complexity when decoded with low-complexity reliability-based iterative decoding algorithms such as binary message passing decoding algorithms. Non-binary codes decoded with a Fast-Fourier Transform based sum-product algorithm achieve significantly large coding gains over Reed-Solomon codes of the same lengths and rates decoded with either the hard-decision Berlekamp-Massey algorithm or the algebraic soft-decision Kotter-Vardy algorithm. They have potential to replace Reed-Solomon codes in some communication or storage systems where combinations of random and bursts of errors (or erasures) occur.

Regulation of Corneal Fibroblast Morphology and Collagen Reorganization by Extracellular Matrix Mechanical Properties

To investigate how extracellular matrix mechanical properties influence cell and matrix patterning in three-dimensional culture. Human corneal fibroblasts were seeded within 30 x 10 mm collagen matrices that were unconstrained (UN), fully constrained (CO) along the long axis by attaching the construct to two immobilized plastic bars, or partially constrained (PC) by allowing linear elastic displacement of one bar. After 24 hours, constructs were labeled with phalloidin and were imaged using fluorescent and reflected light (for collagen) confocal microscopy. Cell morphology and local collagen fibril density and alignment were measured using digital image processing. Corneal fibroblasts in UN matrices were less elongated (UN < PC < CO; P < 0.05) than those in constrained matrices. Cells were aligned parallel to the long axis in the anisotropic region of constrained matrices but were randomly aligned in unconstrained (isotropic) matrices (UN < PC = CO; P < 0.05). Both the local collagen density and the degree of cell/collagen coalignment were higher in constrained matrices (UN < PC < CO; P < 0.05). In regions of higher cell density, additional bands of aligned collagen were often observed between individual cells. These data suggest that cell spreading, alignment, and contractile force generation are directly influenced by the mechanical properties of the surrounding extracellular matrix (ECM). Corneal fibroblasts generally align and compact collagen parallel to the axis of greatest ECM stiffness. Mechanical cross-talk between adjacent cells leads to enhancement of matrix reorganization, and results in additional, more complex matrix patterning.

A bridging domain and strain computation method for coupled atomistic–continuum modelling of solids

AbstractWe present a multiscale method that couples atomistic models with continuum mechanics. The method is based on an overlapping domain‐decomposition scheme. Constraints are imposed by a Lagrange multiplier method to enforce displacement compatibility in the overlapping subdomain in which atomistic and continuum representations overlap. An efficient version of the method is developed for cases where the continuum can be modelled as a linear elastic material. An iterative scheme is utilized to optimize the coupled configuration. Conditions for the regularity of the constrained matrices are determined. A method for computing strain in atomistic models and handshake domains is formulated based on a moving least‐square approximation which includes both extensional and angle‐bending terms. It is shown that this method exactly computes the linear strain field. Applications to the fracture of defected single‐layer atomic sheets and nanotubes are given. Copyright © 2006 John Wiley &amp; Sons, Ltd.

Application's Method of Quadratic Programming for Optimization of Portfolio Selection

Investors or fund-managers face with optimization of portfolio selection, which means that determine the kind and the quantity of investment among several brands. We have developed a method to obtain optimal stock’s portfolio more rapidly from twice to three times than conventional method with efficient universal optimization.The method is characterized by quadratic matrix of utility function and constrained matrices divided into several sub-matrices by focusing on structure of these matrices.

Efficient design method of cosine-modulated QMF banks satisfying PR property

In this paper the design problem of perfect-reconstruction cosine-modulated QMF banks has been formulated as a quadratic-constrained least-squares (QCLS) minimization problem in which all constrained matrices of the QCLS optimization problem are symmetric and positive definite. A cost function which is a convex function of desired prototype filter coefficients is constructed so that this kind of QCLS optimization problems can be efficiently solved. So a global minimizer of this problem can be easily obtained. Results of two design examples are presented to support the derivations and analyses.

An efficient design method of cosine-modulated QMF banks satisfying PR property

In this paper the design problem of perfect-reconstruction cosine-modulated QMF banks has been formulated as a quadratic-constrained least-squares (QCLS) minimization problem in which all constrained matrices of the QCLS optimization problem are symmetric and positive definite. A cost function which is a convex function of desired prototype filter coefficients is constructed so that this kind of QCLS optimization problems can be efficiently solved. So a global minimizer of this problem can be easily obtained. Results of two design examples are presented to support the derivations and analyses.

Self-scaling fast rotations for stiff and equality-constrained linear least squares problems

We present algorithms which apply self-scaling fast plane rotations to the QR decomposition for stiff least squares problems. We show that both fast and standard Givens rotation-based algorithms produce accurate results, regardless of row sorting and even with extremely large weights, when equality-constrained least squares problems are solved by the weighting method. Numerical test results show that the Householder QR decomposition algorithm is sensitive to row sorting and produces less accurate results when the weights are large, and that the modified Gram-Schmidt algorithm is less sensitive to row sorting. This makes the fast plane rotation a method of choice for the QR decomposition of stiff matrices, since it is also competitive in computational complexity. Based on the above results, an efficient algorithm is also derived for the application where the least squares solutions are required for various constrained matrices for each fixed data matrix.

Optimal sensor placement in decentralized control systems

The problem of sensor placement for decentralized discrete systems is considered, A design technique is proposed by which the performance of decentralized regulators can be improved by optimizing the constrained observation matrices. Necessary conditions of optimality are derived.