Adaptive generalized matrix projective lag synchronization between two different complex networks with non-identical nodes and different dimensions

The adaptive generalized matrix projective lag synchronization between two different complex networks with non-identical nodes and different dimensions is investigated in this paper. Based on Lyapunov stability theory and Barbalat＇s lemma, generalized matrix projective lag synchronization criteria are derived by using the adaptive control method. Furthermore, each network can be undirected or directed, connected or disconnected, and nodes in either network may have identical or different dynamics. The proposed strategy is applicable to almost all kinds of complex networks. In addition, numerical simulation results are presented to illustrate the effectiveness of this method, showing that the synchronization speed is sensitively influenced by the adaptive law strength, the network size, and the network topological structure.

Pseudopolarity of Generalized Matrix Rings over a Local Ring

Pseudopolar rings are closely related to strongly -regular rings, uniquelystrongly clean rings and semiregular rings. In this paper, we investigate pseudopolar-ity of generalized matrix rings Ks（R） over a local ring R. We determine the conditionsunder which elements of Ks（R） are pseudopolar. Assume that R is a local ring. It isshown that A ∈ Ks（R） is pseudopolar if and only if A is invertible or A^2 ∈ J（Ks（R））or A is similar to a diagonal matrix [ u 0 0 j ]; where lu -rj and lj-ru are injectiveand u 2 U（R） and j ∈ J（R）. Furthermore, several equivalent conditions for Ks（R）over a local ring R to be pseudopolar are obtained.

Jordan Left Derivations of Generalized Matrix Algebras

In this paper, it is proved that under certain conditions, each Jordan left derivation on a generalized matrix algebra is zero and each generalized Jordan left derivation is a generalized left derivation.

Co-commuting Mappings of Generalized Matrix Algebras

Let G be a generalized matrix algebra over a commutative ring R and Z（G） be the center of G. Suppose that F, T ：G→G are two co-commuting R-linear mappings, i.e., F（x）x = xT（x） for all x ∈ G. In this note, we study the question of when co-commuting mappings on G are proper.

Positivity of Matrices with Generalized Matrix Functions

Using an elementary fact on matrices we show by a unified approach the positivity of a partitioned positive semidefinite matrix with each square block replaced by a compound matrix, an elementary symmetric function or a generalized matrix function. In addition, we present a refined version of the Thompson determinant compression theorem.

THE GENERALIZED REFLEXIVE SOLUTION FOR A CLASS OF MATRIX EQUATIONS (AX-B,XC=D)

In this article, the generalized reflexive solution of matrix equations （AX = B, XC = D） is considered. With special properties of generalized reflexive matrices, the necessary and sufficient conditions for the solvability and the general expression of the solution are obtained. Moreover, the related optimal approximation problem to a given matrix over the solution set is solved.

Solutions to the generalized Sylvester matrixequations by a singular value decomposition

In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are established by a singular value decomposition of a matrix with dimensions n × （n ＋ pr）. The algorithm proposed in this paper for the euqation AX - XF = BY does not require the controllability of matrix pair （A, B） and the restriction that A, F do not have common eigenvalues. Since singular value decomposition is adopted, the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations, and can perform important functions in many design problems in control systems theory.

Schmidt Decomposition of Quaternion Matrix and the Orthonormalization of Vectors in a Generalized Unitary Space

In this paper we derive a practical method of solving simultaneously the problem of Schmidt decomposition of quaternion matrix and the orthonormalization of vectors in a generalized unitary space by using elementary column operations on matrices over the quaternion field.

New Criteria for Judging Generalized Strictly Diagonally Dominant Matrix

Generalized strictly diagonally dominant matrices play a wide and important role in computational mathematics, mathematical physics, theory of dynamical systems, etc.But it is difficult to judge a matrix is or not generalized strictly diagonally dominant matrix.In this paper, by using the properties of α-chain diagonally dominant matrix, we obtain new criteria for judging generalized strictly diagonally dominant matrix, which enlarge the identification range.

Generalized bipositive semidefinite solutions to a system of matrix equations

In this paper, a system of complex matrix equations was studied. Necessary and sufficient conditions for the existence and the expression of generalized bipositive semidefinite solution to the system were given. In addition, a criterion for a matrix to be generalized bipositive semidefinite was determined.

Minimum Cycle of Row Vector of a Generalized Circulant Fuzzy Matrix

In this paper,we intreduce the concept and discuss the properties of minimum cycle of row vector in a generalized circulant Fuzzy matrix. We present a new expression for circulant Fuzzy matrix,and discuss some properties of the idempotent elements of the semigroup of generalized circulant Fuzzy matrixes in connection with minimum cycle of row vector.

Exact Computation of Parallel Robot's Generalized Inertia Matrix

According to the definition of the new hypothetical states which have obvious physical significance and are termed as no-gravity static and accelerated states, a method for exact computation of the parallel robot＇s generalized inertia matrix is presented. Based on the matrix theory, the generalized inertia matrix of the parallel robot can be computed on the assumption that the robot is in these new hypothetical states respectively. The approach is demonstrated by the Delta robot as an example. Based on the principle of the virtual work, the inverse dynamics model of the robot is formulized after the kinematics analysis. Finally, a numerical example is given and the element distribution of the Delta robot＇s inertia matrix in the workspace is studied. The method has computationa＇, advantage of numerical accuracy for the Delta robot and can be parallelized easily.

GENERALIZED INVERSE RATIONAL EXTRAPOLATION METHODS FOR MATRIX SEQUENCES

Assume that a convergent matrix sequence{An}:An→A（n→∞）, An,A∈C3×3.We want to form a new matrix sequence {Hn}, derived from {An}, which has also A aslimit and whose convergence is faster than the of {An}. Three rational extrapolation meth-ods for accelerating the convergence of matrix sequences {An} are presented in this paper.The underlying methods are based on the generalized inverse for matrices which is

Damage Identification under Incomplete Mode Shape Data Using Optimization Technique Based on Generalized Flexibility Matrix

A generalized flexibility–based objective function utilized for structure damage identification is constructed for solving the constrained nonlinear least squares optimized problem. To begin with, the generalized flexibility matrix (GFM) proposed to solve the damage identification problem is recalled and a modal expansion method is introduced. Next, the objective function for iterative optimization process based on the GFM is formulated, and the Trust-Region algorithm is utilized to obtain the solution of the optimization problem for multiple damage cases. And then for computing the objective function gradient, the sensitivity analysis regarding design variables is derived. In addition, due to the spatial incompleteness, the influence of stiffness reduction and incomplete modal measurement data is discussed by means of two numerical examples with several damage cases. Finally, based on the computational results, it is evident that the presented approach provides good validity and reliability for the large and complicated engineering structures.

Matrix Generator (MG): a Program for Creating 0/1 Matrix from Sized DNA Fragments

A computer program called Matrix Generator (MG) was developed for transforming sized DNA fragments into a presence/absence data matrix. Dynamic computation was run to avoid errors introduced using fixed-bin-width arithmetic. MG can be used with bin sized fragments from AFLP, ISSR, RAPD, RFLP, and other molecular markers. The accuracy of MG was tested using fAFLP data of Abelia and the results show that MG results in higher resolution of taxa and is more reliable than programs of the similar usage.

PERFORMANCE OF SIMPLE-ENCODING IRREGULAR LDPC CODES BASED ON SPARSE GENERATOR MATRIX

A new method for the construction of the high performance systematic irregular low-density paritycheck （LDPC） codes based on the sparse generator matrix （G-LDPC） is introduced. The code can greatly reduce the encoding complexity while maintaining the same decoding complexity as traditional regular LDPC （H-LDPC） codes defined by the sparse parity check matrix. Simulation results show that the performance of the proposed irregular LDPC codes can offer significant gains over traditional LDPC codes in low SNRs with a few decoding iterations over an additive white Gaussian noise （AWGN） channel.

Computation of Jordan Chains by Generalized Characteristic Matrices

In this paper, we introduce a method to define generalized characteristic matrices of a defective matrix by the common form of Jordan chains. The generalized characteristic matrices can be obtained by solving a system of linear equations and they can be used to compute Jordan basis.

Necessary and Sufficient Condition for Generalized Diagonal Dominance Matrices

In the paper,a necessary and sufficeent condition for generalized diagonal domiance matrices is given.Further, the relations among all generalized positive definite matrices are shown, also,some flaws and mistakes in the references are corrected.

The Least Square Solutions to the Quaternion Matrix Equation AX=B

A norm of a quaternion matrix is defined. The expressions of the least square solutions of the quaternion matrix equation AX = B and the equation with the constraint condition DX = E are given.

A MATRIX EQUATION FROM AN INVERSE PROBLEM OF VIBRATION THEORY

The symmetric,positive semidefinite,and positive definite real solutions of the matrix equation XA=YAD from an inverse problem of vibration theory are considered.When D=T the necessary and sufficient conditions for the existence of such solutions and their general forms are derived.