the common re nnd and re pd solutions to the matrix equations ax c and xb d abstract in this article we consider common re nnd and re pd solutions of the matrix equations ax c and xb d with respect to x where a b c and d are given matrices we give necessary and sufficient conditions for the existence of common re nnd and re pd solutions to the pair of the matrix equations and derive a representation of the common re nnd and re pd solutions to these two equations when they exist the presented examples show the advantage of the proposed approach

global exponential stability for complex valued recurrent neural networks with asynchronous time delays in this paper we investigate the global exponential stability for complex valued recurrent neural networks with asynchronous time delays by decomposing complex valued networks to real and imaginary parts and construct an equivalent real valued system the network model is described by a continuous time equation there are two main differences of this paper with previous works 1 time delays can be asynchronous i e delays between different nodes are different which make our model more general and 2 we prove the exponential convergence directly while the existence and uniqueness of the equilibrium point is just a direct consequence of the exponential convergence using three generalized norms we present some sufficient conditions for the uniqueness and global exponential stability of the equilibrium point for delayed complex valued neural networks these conditions in our results are less restrictive because of our consideration of the excitatory and inhibitory effects between neurons so previous works of other researchers can be extended finally some numerical simulations are given to demonstrate the correctness of our obtained results

multistability analysis of delayed quaternion valued neural networks with nonmonotonic piecewise nonlinear activation functions abstract this paper deals with the multistability problem of the quaternion valued neural networks qvnns with nonmonotonic piecewise nonlinear activation functions and unbounded time varying delays by virtue of the non commutativity of quaternion multiplication resulting from hamilton rules the qvnns can be separated into four real valued systems by using the fixed point theorem and other analytical tools some novel algebraic criteria are established to guarantee that the qvnns can have 54n equilibrium points 34n of which are locally stable some criteria that guarantee the multiple exponential stability multiple power stability multiple log stability multiple log log stability are also derived as special cases the obtained results reveal that the introduced qvnns in this paper can have larger storage capacity than the complex valued ones finally one numerical example is presented to clarify the validity of the theoretical results

re nnd generalized inverses abstract in this paper we give some necessary and sufficient conditions for the existence of re nnd and nonnegative definite 1 3 and 1 4 inverses of a matrix a c n n and completely described these sets moreover we prove that the existence of nonnegative definite 1 3 inverse of a matrix a is equivalent with the existence of its nonnegative definite 1 2 3 inverse and present the necessary and sufficient conditions for the existence of re nnd 1 3 4 inverse of a

stability analysis of continuous time and discrete time quaternion valued neural networks with linear threshold neurons this paper addresses the problem of stability for continuous time and discrete time quaternion valued neural networks qvnns with linear threshold neurons applying the semidiscretization technique to the continuous time qvnns the discrete time analogs are obtained which preserve the dynamical characteristics of their continuous time counterparts via the plural decomposition method of quaternion homeomorphic mapping theorem as well as lyapunov theorem some sufficient conditions on the existence uniqueness and global asymptotical stability of the equilibrium point are derived for the continuous time qvnns and their discrete time analogs respectively furthermore a uniform sufficient condition on the existence uniqueness and global asymptotical stability of the equilibrium point is obtained for both continuous time qvnns and their discrete time version finally two numerical examples are provided to substantiate the effectiveness of the proposed results

hopf bifurcation analysis of a complex valued neural network model with discrete and distributed delays abstract in this paper a class of complex valued neural network model with discrete and distributed delays is proposed regarding the discrete time delay as the bifurcating parameter the problem of hopf bifurcation in the newly proposed complex valued neural network model is investigated under the assumption that the activation function can be separated into its real and imaginary parts based on the normal form theory and center manifold theorem some sufficient conditions which determine the direction of the hopf bifurcation and the stability of the bifurcating periodic solutions are established finally a numerical example is given to illustrate the validity of the theoretical results

singular value and generalized singular value decompositions and the solution of linear matrix equations abstract the solution of the linear matrix equations i axb cyd e and ii axb fxg e h are considered new necessary and sufficient conditions for the consistency of the equations are derived some using the generalized singular value decomposition special cases iii ax yd e and iv axb e are treated using the singular value decomposition numerical algorithms for the solutions are also suggested

multistability analysis of quaternion valued neural networks with time delays this paper addresses the multistability issue for quaternion valued neural networks qvnns with time delays by using the inequality technique sufficient conditions are proposed for the boundedness and the global attractivity of delayed qvnns based on the geometrical properties of the activation functions several criteria are obtained to ensure the existence of 81 n equilibrium points 16 n of which are locally stable two numerical examples are provided to illustrate the effectiveness of the obtained results

lmi based criteria for global robust stability of bidirectional associative memory networks with time delay abstract in this paper several new sufficient conditions are given to ensure existence uniqueness and globally exponential robust stability of the equilibrium point for bidirectional associative memory bam networks with delays this novel approach based on the linear matrix inequality lmi technique removes some existing restrictions on the system s parameters and the derived conditions are easy to verify via the lmi toolbox in addition two examples are given to show the effectiveness and the advantage of the proposed results

global exponential periodicity and stability of a class of memristor based recurrent neural networks with multiple delays the paper presents theoretical results on the global exponential periodicity and stability of a class of memristor based recurrent neural networks with multiple delays the dynamic analysis in the paper employs the theory of differential equations with discontinuous right hand side as introduced by filippov by using the inequality techniques and a useful lyapunov functional some new testable algebraic criteria are obtained for ensuring the existence and global exponential stability of periodic solution of the system the model based on the memristor widens the application scope for the design of neural networks and the new effective results also enrich the toolbox for the qualitative analysis of neural networks

a pair of simultaneous linear matrix equations a1xb1 c1 a2xb2 c2 and a matrix programming problem necessary and sufficient conditions are obtained for a pair of matrix equations a1xb1 c1 a2xb2 c2 on a general field f to have a common solution along with the expression for a general common solution when certain conditions hold common solutions of minimum rank are described a matrix programming problem is solved en route

quasi synchronization and bifurcation results on fractional order quaternion valued neural networks in this article the quasi synchronization and hopf bifurcation issues are investigated for the fractional order quaternion valued neural networks qvnns with time delay in the presence of parameter mismatches on the basis of noncommutativity property of quaternion multiplication results the quaternion network has been split as four real valued networks a synchronization theorem for fractional order qvnns is derived by employing suitable lyapunov functional candidate furthermore the bifurcation behavior of the hub structured fractional order qvnns with time delay has been investigated finally two numerical examples are provided to demonstrate the effectiveness of the theoretical results

two bivariate geometrically defined reference regions with applications to male reproductive hormones and human growth two conceptually simple two dimensional reference charts are described the charts are characterized analytically but also by geometric properties that are easily communicated to subject matter researchers the users of such charts for both charts the proposed estimators are non parametric so no distributional assumptions are needed about the two dimensional response an important example that will serve to motivate and illustrate one of the charts is the study of the human reproductive hormones inhibin b and fsh a low inhibin b concentration combined with a high fsh concentration is an indication that something may be wrong in the seminal cells which produce semen and such measurements help in the process of distinguishing between different reasons for infertility the aim in the study of the human reproductive hormones was to give a two dimensional reference curve for the non normally distributed hormone measurements while taking into account that some measurements were left censored because of a detection limit in the assay used to measure the concentrations copyright 2003 john wiley sons ltd

leader following consensus of a class of stochastic delayed multi agent systems with partial mixed impulses in this paper the exponential leader following consensus problem is investigated for a class of nonlinear stochastic networked multi agent systems with partial mixed impulses and unknown time varying but bounded delays the main feature of partial mixed impulses is that time varying impulses are not only composed of synchronizing and desynchronizing impulses simultaneously but they are also injected into a fraction of nodes in multi agent systems three kinds of partial mixed impulses are proposed fixed partial mixed impulses periodic partial mixed impulses and try once discard like partial mixed impulses by means of the lyapunov function theory and the comparison principle conditions are derived for ensuring global exponential leader following consensus under the presented three kinds of partial mixed impulses simulations of leader following consensus of robotic systems are provided to validate the effectiveness of the proposed results and to show the advantages of the proposed partial mixed impulses

quaternion ica from second order statistics this paper addresses the independent component analysis ica of quaternion random vectors in particular we focus on the gaussian case and therefore only consider the quaternion second order statistics sos which are given by the covariance matrix and three complementary covariance matrices first we derive the necessary and sufficient conditions for the identifiability of the quaternion ica model which are based on the definition of the properness profile of a quaternion random variable and more specifically on the concept of rotationally equivalent properness profiles second we show that the maximum likelihood ml approach to the quaternion ica problem reduces to the approximated joint diagonalization ajd of the sample mean estimates of the covariance and complementary covariance matrices unlike the complex case these four matrices cannot be simultaneously diagonalized in general and we have to resort to a particular ajd algorithm the proposed technique which can be seen as a quasi newton method is based on the local approximation of the nonconvex ml ica cost function a measure of the entropy loss due to the residual correlation among the estimated quaternion sources and it provides a satisfactory solution of the quaternion ica model the performance of the proposed quaternion ml ica algorithm as well as its relationship to the identifiability conditions are illustrated by means of several numerical examples

stability analysis of quaternion valued neural networks decomposition and direct approaches in this paper we investigate the global stability of quaternion valued neural networks qvnns with time varying delays on one hand in order to avoid the noncommutativity of quaternion multiplication the qvnn is decomposed into four real valued systems based on hamilton rules ij ji k jk kj i ki ik j i 2 j 2 k 2 ijk 1 with the lyapunov function method some criteria are respectively presented to ensure the global mu stability and power stability of the delayed qvnn on the other hand by considering the noncommutativity of quaternion multiplication and time varying delays the qvnn is investigated directly by the techniques of the lyapunov krasovskii functional and the linear matrix inequality lmi where quaternion self conjugate matrices and quaternion positive definite matrices are used some new sufficient conditions in the form of quaternion valued lmi are respectively established for the global mu stability and exponential stability of the considered qvnn besides some assumptions are presented for the two different methods which can help to choose quaternion valued activation functions finally two numerical examples are given to show the feasibility and the effectiveness of the main results

the solutions to linear matrix equations ax b ya d with k involutory symmetries let rcmm and scnn be nontrivial k involutions if their minimal polynomials are both xk1 for some k2 i e rk1 r1i and sk1 s1i we say that acmn is r s symmetric if ras1 a and a is r s symmetric if ras a with 0 1 k1 and 0 let s be one of the subsets of all r s symmetric and r s symmetric matrices given xcnr ycsm bcmr and dcsn we characterize the matrices a in s that minimize axb2 yad2 frobenius norm under the assumption that r and s are unitary moreover among the set s x y b d s of the minimizers of axb2 yad2 min we find the optimal approximate matrix as x y b d that minimizes ag to a given unstructural matrix gcmn we also present the necessary and sufficient conditions such that ax b ya d is consistent in s if the conditions are satisfied we characterize the consistent solution set of all such a finally a numerical algorithm and some numerical examples are given to illustrate the proposed results

global µ stability criteria for quaternion valued neural networks with unbounded time varying delays in this paper we first propose quaternion valued neural networks qvnns with unbounded time varying delays some sufficient conditions on the global stability in the form of both complex valued and real valued linear matrix inequalities lmis are provided by solving two difficulties one is decomposing the qvnn into two complex valued systems with the plural decomposition method of quaternion which can reduce the complexity of calculations by avoiding the non commutativity of quaternion multiplication the other is choosing the appropriate lyapunov krasovskii functional in the form of hermitian matrices which is a big challenge finally two numerical examples are provided to verify the effectiveness of the obtained results

on the symmetric solutions of linear matrix equations abstract the symmetric solutions of linear matrix equations ax c and axb c are considered the necessary and sufficient conditions for the consistency of the equations with a symmetric condition on solutions are derived using the singular value decomposition and the generalized singular value decomposition numerical algorithms for solving the equations are suggested

a representation of the general common solution to the matrix equations a1xb1 c1 and a2xb2 c2 with applications we give new necessary and sufficient conditions for the existence of a common solution to the pair of linear matrix equations a1xb1 c1 and a2xb2 c2 and derive a new representation of the general common solution to these two equations we apply this result to determine new necessary and sufficient conditions for the existence of an hermitian solution and a representation of the general hermitian solution to the matrix equation axb c

design of state dependent switching laws for stability of switched stochastic neural networks with time delays we study the stability properties of switched stochastic neural networks ssnns with time varying delays whose subsystem is not necessarily stable we introduce state dependent switching sds as a tool for stability analysis some sds laws for asymptotic stability and p th moment exponentially stable are designed by employing lyapunov krasovskii l k functional and lyapunov razumikhin l r method respectively it is shown that the stability of ssnns with time varying delays composed of unstable subsystems can be achieved by using sds law the control gains in the designed sds laws can be derived by solving the lmis in derived stability criteria two numerical examples are provided to demonstrate the effectiveness of the proposed sds laws

the re nonnegative definite and re positive definite solutions to the matrix equation axb d in this paper the re nonnegative definite re nnd and re positive definite re pd solutions of the matrix equation axb d are considered the necessary and sufficient conditions for the existence of re nnd and re pd solutions to the equation are provided and the explicit representations of the general re nnd and re pd solutions are given when it is solvable

positive solutions to the equations ax c and xb d for hilbert space operators abstract the paper studies the equation a x c for bounded linear operators between hilbert spaces gives conditions for the existence of hermitian solutions and positive solutions and obtains the formula for the general form of these solutions then the common hermitian and positive solutions to the equations a x c and x b d are studied and new representations of the general solutions are given many results for matrices are recovered as special cases and the results of phadke and thakare s v phadke n k thakare generalized inverses and operator equations linear algebra appl 23 1979 191 199 are corrected

stability analysis for stochastic cohen grossberg neural networks with mixed time delays in this letter the global asymptotic stability analysis problem is considered for a class of stochastic cohen grossberg neural networks with mixed time delays which consist of both the discrete and distributed time delays based on an lyapunov krasovskii functional and the stochastic stability analysis theory a linear matrix inequality lmi approach is developed to derive several sufficient conditions guaranteeing the global asymptotic convergence of the equilibrium point in the mean square it is shown that the addressed stochastic cohen grossberg neural networks with mixed delays are globally asymptotically stable in the mean square if two lmis are feasible where the feasibility of lmis can be readily checked by the matlab lmi toolbox it is also pointed out that the main results comprise some existing results as special cases a numerical example is given to demonstrate the usefulness of the proposed global stability criteria

pinning stabilization of linearly coupled stochastic neural networks via minimum number of controllers pinning stabilization problem of linearly coupled stochastic neural networks lcsnns is studied in this paper a minimum number of controllers are used to force the lcsnns to the desired equilibrium point by fully utilizing the structure of the network in order to pinning control the lcsnns to a certain desired state only one controller is required for strongly connected network topology and m controllers which will be shown to be the minimum number are needed for lcsnns with m reducible coupling matrix the isolate node of the lcsnns can be stable periodic or even chaotic the coupling laplacian matrix of the lcsnns can be symmetric irreducible asymmetric irreducible or m reducible which means that the network topology can be strongly connected weakly connected or even unconnected there is no constraint on the network topology some criteria are derived to judge whether the lcsnns can be controlled in mean square by using designed controllers the given criteria are expressed in terms of strict linear matrix inequalities which can be easily checked by resorting to recently developed algorithm moreover numerical examples including small world and scale free networks are also given to demonstrate that our theoretical results are valid and efficient for large systems

brief paper a unified synchronization criterion for impulsive dynamical networks this paper focuses on the problem of globally exponential synchronization of impulsive dynamical networks two types of impulses are considered synchronizing impulses and desynchronizing impulses in previous literature all of the results are devoted to investigating these two kinds of impulses separately thus a natural question arises is there any unified synchronization criterion which is simultaneously effective for synchronizing impulses and desynchronizing impulses in this paper a unified synchronization criterion is derived for directed impulsive dynamical networks by proposing a concept named average impulsive interval the derived criterion is theoretically and numerically proved to be less conservative than existing results numerical examples including scale free and small world structures are given to show that our results are applicable to large scale networks

an approach to multivariate covariate dependent quantile contours with application to bivariate conditional growth charts multivariate quantile contours are useful in numerous applications and have been studied in different contexts however no easy solutions exist when dynamic and conditional quantile contours are needed without strong distributional assumptions in this article we propose a new form of bivariate quantile contours and a two stage estimation procedure to take time effect into account the proposed procedure relies on quantile regression for longitudinal data and is flexible to include potentially important covariates as necessary in addition we propose a visual model assessment tool and discuss a practical guideline for model selection the performance of the proposed methodology is demonstrated by a simulation study as well as an application to joint height weight screening of young children in the united states we construct bivariate growth charts by a nested sequence of age dependent and covariate varying quantile contours of height and weight and use it to locate an individual subject s percentile

stability and stabilization of quaternion valued neural networks with uncertain time delayed impulses direct quaternion method abstract in this paper the exponential stability of impulsive quaternion valued neural networks iqvnns with time delays is investigated both the differential equation and impulses are affected by time varying delays which may be nonidentical uncertain parameters in impulses are also considered firstly the existence and uniqueness of the equilibrium of the iqvnns are discussed with the help of homeomorphic mapping theorem then by utilizing lyapunov functions and a newly developed inequality several sufficient criteria are established in the form of quaternion valued linear matrix inequalities lmis it should be noted that compared to most of existing results which are obtained by decomposing quaternion valued neural networks qvnns into four real valued neural networks rvnns or two complex valued neural networks cvnns our results are easier to be verified since the quaternion valued lmis exhibit lower dimensions then based on obtained results one impulsive controller is designed to realize the stabilization of delayed qvnns finally two numerical examples are presented to verify the effectiveness and merits of the theoretical analysis

linear matrix equations from an inverse problem of vibration theory abstract the symmetric positive semidefinite and positive definite real solutions of matrix equations a t xa d and a t xa xa y ad d 0 are considered necessary and sufficient conditions for the existence of such solutions and their general forms are derived using the singular value decomposition the theory is motivated and illustrated with a problem of vibration theory

the matrix equations ax c xb d for the pair of matrix equations ax c xb d this paper gives common solutions of minimum possible rank and also other feasible specified ranks

hypercomplex fourier transforms of color images fourier transforms are a fundamental tool in signal and image processing yet until recently there was no definition of a fourier transform applicable to color images in a holistic manner in this paper hypercomplex numbers specifically quaternions are used to define a fourier transform applicable to color images the properties of the transform are developed and it is shown that the transform may be computed using two standard complex fast fourier transforms the resulting spectrum is explained in terms of familiar phase and modulus concepts and a new concept of hypercomplex axis a method for visualizing the spectrum using color graphics is also presented finally a convolution operational formula in the spectral domain is discussed

upper and lower bounds for ranks of matrix expressions using generalized inverses abstract the maximal and minimal ranks of the matrix expression a 1 b 1 x 1 c 1 b 2 x 2 c 2 with respect to x 1 and x 2 are presented as applications the maximal and minimal ranks of a 1 b 1 xc 1 subject to a consistent matrix equation b 2 xc 2 a 2 are also determined in addition the maximal and minimal ranks of the schur complement d ca b with respect to the generalized inverse a of a and their various consequences are also considered

quaternionic analysis representation theory and physics we develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group we first review the cauchy fueter and poisson formulas and explain their representation theoretic meaning the requirement of unitarity of representations leads us to the extensions of these formulas in the minkowski space which can be viewed as another real form of quaternions representation theory also suggests a quaternionic version of the cauchy formula for the second order pole remarkably the derivative appearing in the complex case is replaced by the maxwell equations in the quaternionic counterpart we also uncover the connection between quaternionic analysis and various structures in quantum mechanics and quantum field theory such as the spectrum of the hydrogen atom polarization of vacuum one loop feynman integrals we also make some further conjectures the main goal of this and our subsequent paper is to revive quaternionic analysis and to show profound relations between quaternionic analysis representation theory and four dimensional physics

comments on the common re nnd and re pd solutions to the matrix equations ax c and xb d abstract the purpose of this paper is to present a critical review on the paper the common re nnd and re pd solutions to the matrix equations ax c and xb d by authors xiong and qin 2011 1 we point out that the conditions given by theorem 2 1 are not enough to ensure the existence of common re nnd solution to ax c and xb d and the proofs are unacceptable finally the corrected results are given

bivariate extensions of the boxplot the boxplot has proven to be a very useful tool for summarizing univariate data several options of bivariate boxplot type constructions are discussed these include both elliptic and asymmetric plots an inner region contains 50 of the data and a fence identifies potential outliers such a robust plot shows location scale correlation and a resistant regression line alternative constructions are compared in terms of efficiency of the relevant parameters additional properties are given and recommendations made emphasis is given to the bivariate biweight m estimator several practical examples illustrate that standard least squares ellipsoids can give graphically misleading summaries

quaternion music for vector sensor array processing this paper considers the problem of direction of arrival doa and polarization parameters estimation in the case of multiple polarized sources impinging on a vector sensor array the quaternion model is used and a data covariance model is proposed using quaternion formalism a comparison between long vector orthogonality and quaternion vector orthogonality is also performed and its implications for signal subspace estimation are discussed consequently a music like algorithm is presented allowing estimation of wave s doas and polarization parameters the algorithm is tested in numerical simulations and performance analysis is conducted when compared with other music like algorithms for vector sensor array the newly proposed algorithm results in a reduction by half of memory requirements for representation of data covariance model and reduces the computational effort for equivalent performance this paper also illustrates a compact and elegant way of dealing with multicomponent complex valued data

the reflexive solutions of the matrix equation ax b c in this paper we study the existence of a reflexive with respect to the generalized reflection matrix p solution of the matrix equation ax b c for the special case when b i we get the result of peng and hu 1