condensed cramer rule for some restricted quaternion linear equations abstract in this paper we derive an alternative more condensed cramer rule for the unique solution of some restricted left and right systems of quaternion linear equations the findings of this paper extend some known results in the literature

a new representation for at s 2 3 a new representation of a t s 2 3 of a having prescribed range space t and null space s is derived using this representation the well known generalized inverses such as the moore penrose inverse the group inverse the drazin inverse the bott duffin inverse and the generalized bott duffin inverse are computed three numerical examples are worked out to demonstrate the computation of the moore penrose inverse the drazin inverse and the 2 3 inverse to show the efficacy of our approach

singular perturbation analysis of competitive neural networks with different time scales the dynamics of complex neural networks must include the aspects of long and short term memory the behavior of the network is characterized by an equation of neural activity as a fast phenomenon and an equation of synaptic modification as a slow part of the neural system the main idea of this paper is to apply a stability analysis method of fixed points of the combined activity and weight dynamics for a special class of competitive neural networks we present a quadratic type lyapunov function for the flow of a competitive neural system with fast and slow dynamic variables as a global stability method and a modality of detecting the local stability behavior around individual equilibrium points

subproper and regular splittings for restricted rectangular linear system iterative schemes for approximating a solution to restricted rectangular but consistent linear system of equations ax b email protected t are considered the methods are based upon so called subproper splitting a m n which is a generalization of the concept of subproper splitting introduced by neumann and studied further by berman and neumann and others we give a necessary and sufficient condition on the splitting such that the iterative sequence converges to a solution of ax b in the case of email protected at for every x 0 where email protected c m x n and t is a subspace of c n monotonicity and the concept of regular subproper splitting are used to determine a necessary and a sufficient condition for the convergence of the iterative scheme finally we present two numerical examples to verify our conclusions

solving constrained matrix equations and cramer rule this paper presents the solution of a general constrained matrix equation using generalized inverses and gives an explicit expression for the elements of the solution matrix using cramer rule

multistability of second order competitive neural networks with nondecreasing saturated activation functions in this paper second order interactions are introduced into competitive neural networks nns and the multistability is discussed for second order competitive nns socnns with nondecreasing saturated activation functions firstly based on decomposition of state space cauchy convergence principle and inequality technique some sufficient conditions ensuring the local exponential stability of 2n equilibrium points are derived secondly some conditions are obtained for ascertaining equilibrium points to be locally exponentially stable and to be located in any designated region thirdly the theory is extended to more general saturated activation functions with 2r corner points and a sufficient criterion is given under which the socnns can have r 1 n locally exponentially stable equilibrium points even if there is no second order interactions the obtained results are less restrictive than those in some recent works finally three examples with their simulations are presented to verify the theoretical analysis

a cramer rule for minimum norm t least squares s solution of inconsistent linear equations abstract in this paper the minimum norm t least squares s solution of inconsistent linear equations ax y a c m n r y r a is given a determinantal form and reduces to common cramer s rule if a is nonsingular

complete discrete 2 d gabor transforms by neural networks for image analysis and compression a three layered neural network is described for transforming two dimensional discrete signals into generalized nonorthogonal 2 d gabor representations for image analysis segmentation and compression these transforms are conjoint spatial spectral representations which provide a complete image description in terms of locally windowed 2 d spectral coordinates embedded within global 2 d spatial coordinates in the present neural network approach based on interlaminar interactions involving two layers with fixed weights and one layer with adjustable weights the network finds coefficients for complete conjoint 2 d gabor transforms without restrictive conditions in wavelet expansions based on a biologically inspired log polar ensemble of dilations rotations and translations of a single underlying 2 d gabor wavelet template image compression is illustrated with ratios up to 20 1 also demonstrated is image segmentation based on the clustering of coefficients in the complete 2 d gabor transform

the hermite transform applications it is demonstrated how the hermite transform can be used for image coding and analysis hierarchical coding structures based on increasingly specified basic patterns i e general 2 d patterns general 1 d patterns and specific 1 d patterns such as edges and corners are presented in the image coding application the relation with existing pyramid coders is described a new coding scheme based on local one dimensional image approximations is introduced in the image analysis application the relation between the hermite transform and existing line edge detection schemes is described it is shown that by concentrating on more specific patterns the coding efficiency can be increased since fewer coefficients have to be coded meanwhile sufficient descriptive power can be maintained for approximating the most interesting features in natural images

global exponential stability of bidirectional associative memory neural networks with time delays in this paper we consider delayed bidirectional associative memory bam neural networks nns with lipschitz continuous activation functions by applying young s inequality and holder s inequality techniques together with the properties of monotonic continuous functions global exponential stability criteria are established for bam nns with time delays this is done through the use of a new lyapunov functional and an m matrix the results obtained in this paper extend and improve previous results

generalized schultz iterative methods for the computation of outer inverses abstract we consider a general matrix iterative method of the type x k 1 x k p a x k for computing an outer inverse a r g n g 2 for given matrices a c m n and g c n m such that a r g n g c m here p x is an arbitrary polynomial of degree d the convergence of the method is proven under certain necessary conditions and the characterization of all methods having order r is given the obtained results provide a direct generalization of all known iterative methods of the same type moreover we introduce one new method and show a procedure how to improve the convergence order of existing methods this procedure is demonstrated on one concrete method and the improvement is confirmed by numerical examples

exponential stability of impulsive high order hopfield type neural networks with time varying delays this paper considers the problems of global exponential stability and exponential convergence rate for impulsive high order hopfield type neural networks with time varying delays by using the method of lyapunov functions some sufficient conditions for ensuring global exponential stability of these networks are derived and the estimated exponential convergence rate is also obtained as an illustration an numerical example is worked out using the results obtained

hadamard s theorem for locally lipschitzian maps if the derivativef p of a continuously differentiable mapffrom r into itself is an invertible linear transformation for each p then by the classical inverse function theorem f is a local homeomorphism every p has a neighborhood u mapped homeomorphically onto its image f u a natural and intriguing question under what further conditions will f be a homeomorphism of r onto itself was answered in one way by hadamard 8 in 1906 using a condition on the norms b max bh h i of the inverses b f p

iterative solution of linear systems in the 20th century this paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century although iterative methods for solving linear systems nd their origin in the early 19th century work by gauss the eld has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences the past ve decades have been particularly rich in new developments ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientic and industrial computational models as in any other scientic area research in iterative methods has been a journey characterized by a chain of contributions building on each other it is the aim of this paper not only to sketch the most signicant of these contributions during the past century but also to relate them to one another c 2000 elsevier science b v all rights reserved

iterative methods for solving restricted linear equations in this paper the iterative methods for computing some solution of the general restricted linear equations ax b x et are established and the iterative methods for finding the solution of several special linear equations considered are constructed

the design and use of steerable filters the authors present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters allowing one to adaptively steer a filter to any orientation and to determine analytically the filter output as a function of orientation steerable filters may be designed in quadrature pairs to allow adaptive control over phase as well as orientation the authors show how to design and steer the filters and present examples of their use in the analysis of orientation and phase angularly adaptive filtering edge detection and shape from shading one can also build a self similar steerable pyramid representation the same concepts can be generalized to the design of 3 d steerable filters

new families of fourier eigenfunctions for steerable filtering a new diadic family of eigenfunctions of the 2 d fourier transform has been discovered specifically new wavelets are derived by steering the elongated hermite gauss filters with respect to rotations thus obtaining a natural generalization of the laguerre gauss harmonics interestingly these functions are also proportional to their 2 d fourier transform their analytical expression is provided in a compact and treatable form by means of a new ad hoc matrix notation in which the cases of even and odd orders of the hermite polynomials are unified moreover these functions can be efficiently implemented by means of a recursive formula that is derived in this paper the proposed filters are applied to the problem of gradient estimation to improve the theoretical canny tradeoff of position accuracy versus noise rejection that occurs in edge detection experimental results show considerable improvements in using the new wavelets over both isotropic gaussian derivatives and other elongated steerable filters more recently introduced finally being the proposed wavelets a set of fourier eigenfunctions they can be of interest in other fields of science such as optics and quantum mechanics

a comment on on equilibria stability and instability of hopfield neural networks and reply it is pointed out that the main analysis results about the existence uniqueness and global asymptotic stability of the equilibrium of a continuous time hopfield type neural network given in the paper by zhi hong guan et al 2000 are special cases of relevant ones previously obtained in the literature in reply the original authors consider the reasoning of xue bin liang s comments and state that their analysis method is in fact different to existing ones

optical flow using spatiotemporal filters a model is presented consonant with current views regarding the neurophysiology and psychophysics of motion perception that combines the outputs of a set of spatiotemporal motion energy filters to estimate image velocity a parallel implementation computes a distributed representation of image velocity a measure of image flow uncertainty is formulated preliminary results indicate that this uncertainty measure may be used to recognize ambiguity due to the aperture problem the model appears to deal with the aperture problem as well as the human visual system since it extracts the correct velocity for some patterns that have large differences in contrast at different spatial orientations

computing generalized inverses of matrices by iterative methods based on splittings of matrices in this paper we study the iterative methods for computing the generalized inverses of the form a 2 t s we prove the result that if a has the splitting a m n and if the generalized inverses a 2 t s and m 2 t s exist then the iterationx j 1 m 2 t snx j m 2 t sconverges to a 2 t s for any x 0 if and only ift t s s email protected 2 t sn 1 or equivalently m 2 t email protected 2 t sn 1 the splitting a m n is called t s splitting of a if m satisfies email protected s c m where t and s are subspaces of c n and c m respectively with dimt dims we also present a specific method to construct convergent t s splitting of a apply these two results to the most of commonly used generalized inverses such as a a d a d w a h k we can get criteria of convergence for the corresponding iterations many old and new splittings and specific choice of the related convergent splittings

existence and global exponential stability of almost periodic solution for cellular neural networks with variable coefficients and time varying delays in this paper we study cellular neural networks with almost periodic variable coefficients and time varying delays by using the existence theorem of almost periodic solution for general functional differential equations introducing many real parameters and applying the lyapunov functional method and the technique of young inequality we obtain some sufficient conditions to ensure the existence uniqueness and global exponential stability of almost periodic solution the results obtained in this paper are new useful and extend and improve the existing ones in previous literature

the representation and approximation for the generalized inverse at s 2 we present a unified representation theorem for the generalized inverses a t s 2 which are 2 or outer inverses having the prescribed range t and null space s specific expressions and computational procedures for the generalized inverses a t s 2 can be uniformly derived

deriving sufficient conditions for global asymptotic stability of delayed neural networks via nonsmooth analysis ii in this paper we obtain new sufficient conditions ensuring existence uniqueness and global asymptotic stability gas of the equilibrium point for a general class of delayed neural networks dnns via nonsmooth analysis which makes full use of the lipschitz property of functions defining dnns based on this new tool of nonsmooth analysis we first obtain a couple of general results concerning the existence and uniqueness of the equilibrium point then those results are applied to show that existence assumptions on the equilibrium point in some existing sufficient conditions ensuring gas are actually unnecessary and some strong assumptions such as the boundedness of activation functions in some other existing sufficient conditions can be actually dropped finally we derive some new sufficient conditions which are easy to check comparison with some related existing results is conducted and advantages are illustrated with examples throughout our paper spectral properties of the matrix a a sup spl tau play an important role which is a distinguished feature from previous studies here a and a sup spl tau are respectively the feedback and the delayed feedback matrix defining the neural network under consideration

an iterative method for solving general restricted linear equations the newton iterative method for computing outer inverses with prescribed range and null space is used in the non stationary richardson iterative method to develop an iterative method for solving general restricted linear equations starting with any suitably chosen initial iterate our method generates a sequence of iterates converging to the solution the necessary and sufficient conditions for the convergence along with the error bounds are established the applications of the iterative method for solving some special linear equations are also discussed a number of numerical examples are worked out they include singular square rectangular randomly generated rank deficient matrices full rank matrices and a set of singular matrices given in matrix computation toolbox mctoolbox with the condition numbers ranging from order 1016 to 1050 the mean cpu time mct and the error bounds are the performance measures used our results when compared with the results obtained by chen 1997 leads to substantial improvement in terms of both computational speed and accuracy

subband coding of images subband coding has become quite popular for the source encoding of speech this paper presents a simple yet efficient extension of this concept to the source coding of images we specify the constraints for a set of two dimensional quadrature mirror filters qmf s for a particular frequency domain partition and show that these constraints are satisfied by a separable combination of one dimensional qmf s bits are then optimally allocated among the subbands to minimize the mean squared error for dpcm coding of the subbands also an adaptive technique is developed to allocate the bits within each subband by means of a local variance mask optimum quantization is employed with quantizers matched to the laplacian distribution subband coded images are presented along with their signal to noise ratios snr s the snr performance of the subband coder is compared to that of the adaptive discrete cosine transform dct vector quantization and differential vector quantization for bit rates of 0 67 1 0 and 2 0 bits per pixel for 256 256 monochrome images the adaptive subband coder has the best snr performance

a least squares framework for component analysis over the last century component analysis ca methods such as principal component analysis pca linear discriminant analysis lda canonical correlation analysis cca locality preserving projections lpp and spectral clustering sc have been extensively used as a feature extraction step for modeling classification visualization and clustering ca techniques are appealing because many can be formulated as eigen problems offering great potential for learning linear and nonlinear representations of data in closed form however the eigen formulation often conceals important analytic and computational drawbacks of ca techniques such as solving generalized eigen problems with rank deficient matrices e g small sample size problem lacking intuitive interpretation of normalization factors and understanding commonalities and differences between ca methods this paper proposes a unified least squares framework to formulate many ca methods we show how pca lda cca lpp sc and its kernel and regularized extensions correspond to a particular instance of least squares weighted kernel reduced rank regression ls wkrrr the ls wkrrr formulation of ca methods has several benefits 1 provides a clean connection between many ca techniques and an intuitive framework to understand normalization factors 2 yields efficient numerical schemes to solve ca techniques 3 overcomes the small sample size problem 4 provides a framework to easily extend ca methods we derive weighted generalizations of pca lda sc and cca and several new ca techniques

a condensed cramer s rule for the minimum norm least squares solution of linear equations abstract in this paper we will derive the condensed cramer s rule of werner for minimal norm least squares solution of linear equations ax b from the cramer s rule of ben israel and verghese in addition a new condensed cramer s rule will be obtained in this paper

global exponential stability and global convergence in finite time of delayed neural networks with infinite gain this paper introduces a general class of neural networks with arbitrary constant delays in the neuron interconnections and neuron activations belonging to the set of discontinuous monotone increasing and possibly unbounded functions the discontinuities in the activations are an ideal model of the situation where the gain of the neuron amplifiers is very high and tends to infinity while the delay accounts for the finite switching speed of the neuron amplifiers or the finite signal propagation speed it is known that the delay in combination with high gain nonlinearities is a particularly harmful source of potential instability the goal of this paper is to single out a subclass of the considered discontinuous neural networks for which stability is instead insensitive to the presence of a delay more precisely conditions are given under which there is a unique equilibrium point of the neural network which is globally exponentially stable for the states with a known convergence rate the conditions are easily testable and independent of the delay moreover global convergence in finite time of the state and output is investigated in doing so new interesting dynamical phenomena are highlighted with respect to the case without delay which make the study of convergence in finite time significantly more difficult the obtained results extend previous work on global stability of delayed neural networks with lipschitz continuous neuron activations and neural networks with discontinuous neuron activations but without delays

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

efficient deformable filter banks this article describes efficient schemes for the computation of a large number of differently scaled oriented filtered versions of an image we generalize the well known steerable scalable deformable filter bank structure by imposing x y separability on the basis filters the resulting systems designed by an iterative projections technique achieve substantial reduction of the computational cost to reduce the memory requirement we adopt a multirate implementation due to the inner sampling rate alteration the resulting structure is not shift invariant we introduce a design criterion for multirate deformable structures that jointly controls the approximation error and the shift variance

robust local stability of multilayer recurrent neural networks we derive a condition for robust local stability of multilayer recurrent neural networks with two hidden layers the stability condition follows from linking theories about linearization robustness analysis of linear systems under nonlinear perturbation and matrix inequalities a characterization of the basin of attraction of the origin is given in terms of the level set of a quadratic lyapunov function similar to the nl sub q theory the local stability is imposed around the origin and the apparent basin of attraction is made large by applying the criterion while the proven basin of attraction is relatively small due to conservatism of the criterion modification of the dynamic backpropagation by the new stability condition is discussed and illustrated by simulation examples

global exponential stability of multitime scale competitive neural networks with nonsmooth functions in this paper we study the global exponential stability of a multitime scale competitive neural network model with nonsmooth functions which models a literally inhibited neural network with unsupervised hebbian learning the network has two types of state variables one corresponds to the fast neural activity and another to the slow unsupervised modification of connection weights based on the nonsmooth analysis techniques we prove the existence and uniqueness of equilibrium for the system and establish some new theoretical conditions ensuring global exponential stability of the unique equilibrium of the neural network numerical simulations are conducted to illustrate the effectiveness of the derived conditions in characterizing stability regions of the neural network

a cramer rule for finding the solution of a class of singular equations a cramer rule for finding the unique solution x r ak of singular equations ax b ind a k b r ak is given and reduces to the common cramer rule if a is a nonsingular

design of steerable filters for feature detection using canny like criteria we propose a general approach for the design of 2d feature detectors from a class of steerable functions based on the optimization of a canny like criterion in contrast with previous computational designs our approach is truly 2d and provides filters that have closed form expressions it also yields operators that have a better orientation selectivity than the classical gradient or hessian based detectors we illustrate the method with the design of operators for edge and ridge detection we present some experimental results that demonstrate the performance improvement of these new feature detectors we propose computationally efficient local optimization algorithms for the estimation of feature orientation we also introduce the notion of shape adaptable feature detection and use it for the detection of image corners

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

deformable kernels for early vision early vision algorithms often have a first stage of linear filtering that extracts from the image information at multiple scales of resolution and multiple orientations a common difficulty in the design and implementation of such schemes is that one feels compelled to discretize coarsely the space of scales and orientations in order to reduce computation and storage costs a technique is presented that allows 1 computing the best approximation of a given family using linear combinations of a small number of basis functions and 2 describing all finite dimensional families i e the families of filters for which a finite dimensional representation is possible with no error the technique is based on singular value decomposition and may be applied to generating filters in arbitrary dimensions and subject to arbitrary deformations the relevant functional analysis results are reviewed and precise conditions for the decomposition to be feasible are stated experimental results are presented that demonstrate the applicability of the technique to generating multiorientation multi scale 2d edge detection kernels the implementation issues are also discussed

multiperiodicity and attractivity of delayed recurrent neural networks with unsaturating piecewise linear transfer functions this paper studies multiperiodicity and attractivity for a class of recurrent neural networks rnns with unsaturating piecewise linear transfer functions and variable delays using local inhibition conditions for boundedness and global attractivity are established these conditions allow coexistence of stable and unstable trajectories moreover multiperiodicity of the network is investigated by using local invariant sets it shows that under some interesting conditions there exists one periodic trajectory in each invariant set which exponentially attracts all trajectories in that region correspondingly simulations are carried out to illustrate the theories

full rank representations of outer inverses based on the qr decomposition abstract an efficient algorithm for computing a t s 2 inverses of a given constant matrix a based on the qr decomposition of an appropriate matrix w is presented correlations between the derived representation of outer inverses and corresponding general representation based on arbitrary full rank factorization are derived in particular cases we derive representations of 2 4 and 2 3 inverses numerical examples on different test matrices dense or sparse are presented as well as the comparison with several well known methods for computing the moore penrose inverse and the drazin inverse

linear shape from shading in many situations the reflectance function of a surface is approximately linear and there is an effielent closed form solution to the shape from shading problem when boundary conditions e g edges singular points are not available good estimates of shape may still be extracted by using the assumption of general viewing position an improved method for estimating the illuminant direction is also presented

local shading analysis local analysis of image shading in the absence of prior knowledge about the viewed scene may be used to provide information about the scene the following has been proved every image point has the same image intensity and first and second derivatives as the image of some point on a lambertian surface with principal curvatures of equal magnitude further if the principal curvatures are assumed to be equal there is a unique combination of image formation parameters up to a mirror reversal that will produce a particular set of image intensity and first and second derivatives a solution for the unique combination of surface orientation etc is presented this solution has been extended to natural imagery by using general position and regional constraints to obtain estimates of the following surface orientation at each image point the qualitative type of the surface i e whether the surface is planar cylindrical convex concave or saddle the illuminant direction within a region algorithms to recover illuminant direction and estimate surface orientation have been evaluated on both natural and synthesized images and have been found to produce useful information about the scene

the hermite transform theory the author introduces a scheme for the local processing of visual information called the hermite transform the problem is addressed from the point of view of image coding and therefore the scheme is presented as an analysis resynthesis system the objectives of the present work however are not restricted to coding the analysis part is designed so that it can also serve applications in the area of computer vision indeed derivatives of gaussians which have found widespread application in feature detection over the past few years play a central role in the hermite analysis it is also argued that the proposed processing scheme is in close agreement with current insight into the image processing that is carried out by the human visual system in particular it is demonstrated that the hermite transform is in better agreement with human visual modeling than gabor expansions

a class of numerical algorithms for computing outer inverses the aim of this study is to present a class of numerical algorithms for finding outer inverses with prescribed range and null space general convergence theorems are proven some particular cases of the general algorithm are introduced and demonstrated theoretical convergence speed and computational efficiency will further be supported by employing some numerical tests

the representation and computation of generalized inverse at s 2 this paper presents a novel representation for the generalized inverse a t s 2 based on this we give an algorithm to compute this generalized inverse as an application we use gauss jordan elimination to compute the weighted moore penrose inverse a m n and the drazin inverse a d

explicit expressions of the generalized inverses and condensed cramer rules in this paper we obtain an explicit representation of the 2 inverse a 2 t s of a matrix a c m n with the prescribed range t and null space s as special cases new expressions for the moore penrose inverse a and drazin inverse a d are derived through explicit expressions we re derive the condensed cramer rules of werner for minimal norm least squares solution of linear equations ax b and propose two new condensed cramer rules for the unique solution of a class of singular system ax b x r a k b r a k k ind a finally condensed determinantal expressions for a a d a a a a a ndaa d are also presented

learning continuous trajectories in recurrent neural networks with time dependent weights the paper is concerned with a general learning with arbitrary criterion and state dependent constraints of continuous trajectories by means of recurrent neural networks with time varying weights the learning process is transformed into an optimal control framework where the weights to be found are treated as controls a learning algorithm based on a variational formulation of pontryagin s maximum principle is proposed this algorithm is shown to converge under reasonable conditions to an optimal solution the neural networks with time dependent weights make it possible to efficiently find an admissible solution i e initial weights satisfying state constraints which then serves as an initial guess to carry out a proper minimization of a given criterion the proposed methodology may be directly applicable to both classification of temporal sequences and to optimal tracking of nonlinear dynamic systems numerical examples are also given which demonstrate the efficiency of the approach presented

gauss jordan elimination method for computing outer inverses this paper deals with the algorithm for computing outer inverse with prescribed range and null space based on the choice of an appropriate matrix g and gauss jordan elimination of the augmented matrix g i the advantage of such algorithms is the fact that one can compute various generalized inverses using the same procedure for different input matrices in particular we derive representations of the moore penrose inverse the weighted moore penrose inverse the drazin inverse as well as 2 4 and 2 3 inverses numerical examples on different test matrices are presented as well as the comparison with well known methods for generalized inverses computation

global exponential stability of competitive neural networks with different time scales the dynamics of cortical cognitive maps developed by self organization must include the aspects of long and short term memory the behavior of such a neural network is characterized by an equation of neural activity as a fast phenomenon and an equation of synaptic modification as a slow part of the neural system we present a new method of analyzing the dynamics of a biological relevant system with different time scales based on the theory of flow invariance we are able to show the conditions under which the solutions of such a system are bounded being less restrictive than with the k monotone theory singular perturbation theory or those based on supervised synaptic learning we prove the existence and the uniqueness of the equilibrium a strict lyapunov function for the flow of a competitive neural system with different time scales is given and based on it we are able to prove the global exponential stability of the equilibrium point

stability analysis for the generalized cohen grossberg neural networks with inverse lipschitz neuron activations in this paper by using nonsmooth analysis approach linear matrix inequality lmi technique topological degree theory and lyapunov krasovskii function method the issue of global exponential stability is investigated for a class of generalized cohen grossberg neural networks possessing inverse lipschitz neuron activations and nonsmooth behaved functions several novel delay dependent sufficient conditions are established towards the existence uniqueness and global exponential stability of the equilibrium point which are shown in terms of lmis it is noted that the results above require neither the lipschitz continuity of the activation functions nor the smoothness of the behaved functions also for the case of the activation function that satisfies not only the inverse lipschitz conditions but also the lipschitz conditions some conditions are derived which generalize the previous results finally two examples with their simulations are given to show the effectiveness of the theoretical results

explicit formulas for determinantal representations of the drazin inverse solutions of some matrix and differential matrix equations the drazin inverse solutions of the matrix equations ax b xa b and axb d are considered in this paper we use both the determinantal representations of the drazin inverse obtained earlier by the author and in the paper we get analogs of the cramer rule for the drazin inverse solutions of these matrix equations and using their for determinantal representations of solutions of some differential matrix equations x ax b and x xa b where the matrix a is singular

adaptive lag synchronization for competitive neural networks with mixed delays and uncertain hybrid perturbations this paper investigates the problem of adaptive lag synchronization for a kind of competitive neural network with discrete and distributed delays mixed delays as well as uncertain nonlinear external and stochastic perturbations hybrid perturbations a simple but robust adaptive controller is designed such that the response system can lag synchronize with a drive system based on the lyapunov stability theory and some suitable lyapunov krasovskii functionals several sufficient conditions ensuring the lag synchronization are developed our synchronization criteria are easily verified and do not need to solve any linear matrix inequality some existing results are improved and extended moreover the designed adaptive controller has better anti interference capacity and is more practical than the usual adaptive controller numerical simulations are exploited to show the effectiveness of the theoretical results

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed