The purpose of this paper is to study some famous inequalities in Euclidean space.We are able to reveal an elegant relation between the famous Selberg inequality and Bessel inequality in Euclidean space.
In order to carry out tensor analysis in a neighborhood of a reference surface,the principal-direction orthogonal basis accompanying with Lame s coefficients or general curvilinear coordinate systems are widely used.A...In order to carry out tensor analysis in a neighborhood of a reference surface,the principal-direction orthogonal basis accompanying with Lame s coefficients or general curvilinear coordinate systems are widely used.A novel kind of field theory termed as the nonholonomic theory of the Principal-Direction Orthonormal Basis(PDOB)is presented systematically in the present paper,in which the formal Christoffel symbols are related directly to the principal and geodesic curvatures with respect to the principal directions of the surface.Furthermore,a systematic and simple way to determine the curvatures of the surface are presented with some examples.It provides a way to recognize qualitatively the bending property of a surface.展开更多
A two-degree-of-freedom vibro-impact system having symmetrical rigid stops and subjected to periodic excitation is investigated in this paper. By introducing local maps between different stages of motion in the whole ...A two-degree-of-freedom vibro-impact system having symmetrical rigid stops and subjected to periodic excitation is investigated in this paper. By introducing local maps between different stages of motion in the whole impact process, the Poincare map of the system is constructed. Using the Poincare map and the Gram Schmidt orthonormalization, a method of calculating the spectrum of Lyapunov exponents of the above vibro-impact system is presented. Then the phase portraits of periodic and chaotic attractors for the system and the corresponding convergence diagrams of the spectrum of Lyapunov exponents are given out through the numerical simulations. To further identify the validity of the aforementioned computation method, the bifurcation diagram of the system with respect to the bifurcation parameter and the corresponding largest Lyapunov exponents are shown.展开更多
Suppose that Φ(x)∈L 2(R) with compact support and V= span{Φ(x-k)|k∈Z}. In this note, we prove that if {Φ(x-k)k|k∈Z} is tight frame with bound 1 in V, then {Φ(x-k)|k∈Z} must be an orthonormal basis of V.
Recently, wavelet neural networks have become a popular tool for non-linear functional approximation. Wavelet neural networks, which basis functions are orthonormal scalling functions, are more suitable in approximati...Recently, wavelet neural networks have become a popular tool for non-linear functional approximation. Wavelet neural networks, which basis functions are orthonormal scalling functions, are more suitable in approximating to function. Based on it, approximating to NLAR(p) with wavelet neural networks is studied.展开更多
A parameter-free method based on orthonormal wavelet transforms is recommended for calculating the principal time scale of coherent structures in atmospheric boundary-layer measurements. First, the atmospheric turbule...A parameter-free method based on orthonormal wavelet transforms is recommended for calculating the principal time scale of coherent structures in atmospheric boundary-layer measurements. First, the atmospheric turbulent signal is decomposed into the small scale vortex that has approximate isotropy and the large scale vortex with the digital filter. Then, the large scale vortex is used to detect coherent structures with this method. The principal time scale and profile of coherent structures for velocity components (u, v, w above rice fields are obtained. In order to testify the validity of this method, the correlation of coherent structures and non-coherent structures are also calculated.展开更多
Motivated by representing multidimensional periodic nonlinear and non-stationary signals （functions）, we study a class of orthonormal exponential basis for L2（Id） with I ：= [0, 1）, whose exponential parts are pi...Motivated by representing multidimensional periodic nonlinear and non-stationary signals （functions）, we study a class of orthonormal exponential basis for L2（Id） with I ：= [0, 1）, whose exponential parts are piecewise linear spectral sequences with p-knots. It is widely applied in time-frequency analysis.展开更多
The modal wave number tomography approach is used to obtain sound speed profile of water column in deep ocean. The approach consists of estimation of the local modal eigenvalues from complex pressure field and use of ...The modal wave number tomography approach is used to obtain sound speed profile of water column in deep ocean. The approach consists of estimation of the local modal eigenvalues from complex pressure field and use of these data as input to modal perturbative inversion method for obtaining the local sound speed profile. The empirical orthonormal function (EOF) is applied to reduce the parameter search space. The ocean environment used for numerical simulations includes the Munk profile as the unperturbed background speed profile and a weak Gaussian eddy as the sound speed profile perturbation. The results of numerical simulations show the method is capable of monitoring the oceanic interior structure.展开更多
In this paper we show how to construct a scaling function and an orthonor-mal wavelet basis from a multiresolution approximation using an operator theoreticmethod.
We present a simple demonstration on the orthonormality of Volkov solutions with emphasizing on the suffcient condition to the orthonormality. Properly aligning the external electromagnetic wave along the third-axis, ...We present a simple demonstration on the orthonormality of Volkov solutions with emphasizing on the suffcient condition to the orthonormality. Properly aligning the external electromagnetic wave along the third-axis, the Volkov solutions are eigenfunctions of the hermitian momentum ■1, ■2 and the light-cone hamiltonian operators with real eigenvalues, which can lead to a verification of the orthonormality in the context of quantum mechanics when the x3-integration of the external potential is not singularity as severe as δ(0). The hermiticity of the fermion field fourmomentum operators validates the application of the demonstration to the intense field quantum electrodynamic. The proof based on a direct calculation to the inner products of the solutions is recapitulated as well in a general manner without dependence on explicit representation of the Dirac matrices and spinors, which can be conducive to understand the suffcient condition and to the study of the polarized electron production where a convenient representation is selected elaborately to project out the spin-polarization.展开更多
In this paper,we propose a new linear multiuser receiver for synchronous code-division multiple-access (CDMA) systems,referred to as the orthogonal multiuser(OMU) receiver.Unlike the linear minimum mean-squared error(...In this paper,we propose a new linear multiuser receiver for synchronous code-division multiple-access (CDMA) systems,referred to as the orthogonal multiuser(OMU) receiver.Unlike the linear minimum mean-squared error(MMSE) receiver,the OMU receiver depends only on the signature vectors and does not require knowledge of the received amplitudes or the channel signal-to-noise ratio(SNR).Here we develop methods that construct an optimal set of vectors with a specified inner product structure,from a given set of vectors in a complex Hilbert space.The optimal vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors.An algorithm has been developed using the principles of quantum parameters and some of its axioms and constraints.In place of the classical matched filter(MF) receiver we propose a modified receiver.This approach assumes that improving the accuracy will necessarily result in im- proved performance.The simulation results provided here suggests that in certain cases the OMU and POMU receivers can significantly increase the probability of correct detection with low error rate over the MF receiver.展开更多
This paper is concerned with the problem of system identification using expansions on generalized orthonormal bases(GOB). Three algorithms are proposed to optimize the poles of such a basis. The first two algorithms d...This paper is concerned with the problem of system identification using expansions on generalized orthonormal bases(GOB). Three algorithms are proposed to optimize the poles of such a basis. The first two algorithms determine a GOB with optimal real poles while the third one determines a GOB with optimal real and complex poles. These algorithms are based on the estimation of the dominant mode associated with a residual signal obtained by iteratively filtering the output of the process to be modelled. These algorithms are iterative and based on the quadratic error between the linear process output and the GOB based model output. They present the advantage to be very simple to implement. No numerical optimization technique is needed, and in consequence there is no problem of local minima as is the case for other algorithms in the literature. The convergence of the proposed algorithms is proved by demonstrating that the modeling quadratic error between the process output and the GOB based model is decreasing at each iteration of the algorithm. The performance of the proposed pole selection algorithms are based on the quadratic error criteria and illustrated by means of simulation results.展开更多
A new kind of k-quantum nonlinear coherent states,i.e.,the k eigenstates of the k-th power~k (k≥3) of the generalized annihilation operator=1/f() of f-oscillators,are obtained and their properties are discussed.The c...A new kind of k-quantum nonlinear coherent states,i.e.,the k eigenstates of the k-th power~k (k≥3) of the generalized annihilation operator=1/f() of f-oscillators,are obtained and their properties are discussed.The completeness of the k states is investigated.An alternative method to construct them is proposed.It is shown that these states may form a complete Hilbert space,and all of them can be generated by a linear superposition of k Roy-type nonlinear coherent states.Physically,they can be generated by a linear superposition of the time-dependent Roy-type nonlinear coherent states at different instants.展开更多
Routh stability test is covered in almost all undergraduate control texts. It determines the stability or, a little beyond, the number of unstable roots of a polynomial in terms of the signs of certain entries of the ...Routh stability test is covered in almost all undergraduate control texts. It determines the stability or, a little beyond, the number of unstable roots of a polynomial in terms of the signs of certain entries of the Routh table constructed from the coefficients of the polynomial. The use of the Routh table, as far as the common textbooks show, is only limited to this function. We will show that the Routh table can actually be used to construct an orthonormal basis in the space of strictly proper rational functions with a common stable denominator. This orthonormal basis can then be used for many other purposes, including the computation of the H2 norm, the Hankel singular values and singular vectors, model reduction, H∞ optimization, etc. Keywords Routh stablity criterion - Orthonormal basis - Root-mean-squared value - Hankel operater - Nehari problem - Model reduction This work was supported by the Hong Kong Research Grants Council.展开更多
A continuous-time Model Predictive Controller was proposed using Kautz function in order to improve the performance of Load Frequency Control(LFC).A dynamic model of an interconnected power system was used for Model P...A continuous-time Model Predictive Controller was proposed using Kautz function in order to improve the performance of Load Frequency Control(LFC).A dynamic model of an interconnected power system was used for Model Predictive Controller(MPC)design.MPC predicts the future trajectory of the dynamic model by calculating the optimal closed loop feedback gain matrix.In this paper,the optimal closed loop feedback gain matrix was calculated using Kautz function.Being an Orthonormal Basis Function(OBF),Kautz function has an advantage of solving complex pole-based nonlinear system.Genetic Algorithm(GA)was applied to optimally tune the Kautz function-based MPC.A constraint based on phase plane analysis was implemented with the cost function in order to improve the robustness of the Kautz function-based MPC.The proposed method was simulated with three area interconnected power system and the efficiency of the proposed method was measured and exhibited by comparing with conventional Proportional and Integral(PI)controller and Linear Quadratic Regulation(LQR).展开更多
There are many forms of tensor theory which are quite different. Discussions are put forward and their relationships are found. The differences between them depend on whether there is metric on the space and the basis...There are many forms of tensor theory which are quite different. Discussions are put forward and their relationships are found. The differences between them depend on whether there is metric on the space and the basis is orthonormal.展开更多
The scope of this paper broadly spans in two areas： system identification of resonant system and design of an efficient control scheme suitable for resonant systems. Use of filters based on orthogonal basis functions...The scope of this paper broadly spans in two areas： system identification of resonant system and design of an efficient control scheme suitable for resonant systems. Use of filters based on orthogonal basis functions （OBF） have been advocated for modelling of resonant process. Kautz filter has been identified as best suited OBF for this purpose. A state space based system identification technique using Kautz filters, viz. Kautz model, has been demonstrated. Model based controllers are believed to be more efficient than classical controllers because explicit use of process model is essential with these modelling techniques. Extensive literature search concludes that very few reports are available which explore use of the model based control studies on resonant system. Two such model based controllers are considered in this work, viz. model predictive controller and internal model controller. A model predictive control algorithm has been developed using the Kautz model. The efficacy of the model and the controller has been verified by two case studies, viz. linear second order underdamped process and a mildly nonlinear magnetic ball suspension system. Comparative assessment of performances of these controllers in those case studies have been carried out.展开更多
Suppose M and N are two r×r and s×s dilation matrices,respectively.LetΓM andΓN represent the complete sets of representatives of distinct cosets of the quotient groups M-T Zr/Zr and N-T Zs/Zs,respectively....Suppose M and N are two r×r and s×s dilation matrices,respectively.LetΓM andΓN represent the complete sets of representatives of distinct cosets of the quotient groups M-T Zr/Zr and N-T Zs/Zs,respectively.Two methods for constructing nonseparable Ω-filter banks from M-filter banks and N-filter banks are presented,where Ω is a（r＋s） ×（r＋s） dilation matrix such that one of its complete sets of representatives of distinct cosets of the quotient groups Ω-T Zr＋s/Zr＋s areΓΩ={[γT h,ζ T q] T：γh∈ΓM,ζq∈ΓN}.Specially,Ω can be [MΘ0N],whereΘis a r×s integer matrix with M-1Θbeing also an integer matrix.Moreover,if the constructed filter bank satisfies Lawton＇s condition,which can be easy to verify,then it generates an orthonormal nonseparable Ω-wavelet basis for L2（Rr＋s）.Properties,including Lawton＇s condition,vanishing moments and regularity of the new Ω-filter banks or new Ω-wavelet basis are discussed then.Finally,a class of nonseparable Ω-wavelet basis for L2（Rr＋1） are constructed and three other examples are given to illustrate the results.In particular,when M=N=2,all results obtained in this paper appeared in[1].展开更多
基金Supported by Shandong Provincial Natural Science Foundation(ZR2024MA053).
文摘The purpose of this paper is to study some famous inequalities in Euclidean space.We are able to reveal an elegant relation between the famous Selberg inequality and Bessel inequality in Euclidean space.
基金Project supported by the National Natural Science Foundation of China(11972120,11472082,12032016)。
文摘In order to carry out tensor analysis in a neighborhood of a reference surface,the principal-direction orthogonal basis accompanying with Lame s coefficients or general curvilinear coordinate systems are widely used.A novel kind of field theory termed as the nonholonomic theory of the Principal-Direction Orthonormal Basis(PDOB)is presented systematically in the present paper,in which the formal Christoffel symbols are related directly to the principal and geodesic curvatures with respect to the principal directions of the surface.Furthermore,a systematic and simple way to determine the curvatures of the surface are presented with some examples.It provides a way to recognize qualitatively the bending property of a surface.
基金supported by the National Natural Science Foundation of China (Grant No. 10972059)the Natural Science Foundation of the Guangxi Zhuang Autonmous Region of China (Grant Nos. 0640002 and 2010GXNSFA013110)+1 种基金the Guangxi Youth Science Foundation of China (Grant No. 0832014)the Project of Excellent Innovating Team of Guangxi University
文摘A two-degree-of-freedom vibro-impact system having symmetrical rigid stops and subjected to periodic excitation is investigated in this paper. By introducing local maps between different stages of motion in the whole impact process, the Poincare map of the system is constructed. Using the Poincare map and the Gram Schmidt orthonormalization, a method of calculating the spectrum of Lyapunov exponents of the above vibro-impact system is presented. Then the phase portraits of periodic and chaotic attractors for the system and the corresponding convergence diagrams of the spectrum of Lyapunov exponents are given out through the numerical simulations. To further identify the validity of the aforementioned computation method, the bifurcation diagram of the system with respect to the bifurcation parameter and the corresponding largest Lyapunov exponents are shown.
文摘Suppose that Φ(x)∈L 2(R) with compact support and V= span{Φ(x-k)|k∈Z}. In this note, we prove that if {Φ(x-k)k|k∈Z} is tight frame with bound 1 in V, then {Φ(x-k)|k∈Z} must be an orthonormal basis of V.
文摘Recently, wavelet neural networks have become a popular tool for non-linear functional approximation. Wavelet neural networks, which basis functions are orthonormal scalling functions, are more suitable in approximating to function. Based on it, approximating to NLAR(p) with wavelet neural networks is studied.
基金supported by the National Natural Science Foundation of China (61071189)Innovation Scientists and Technicians Troop Construction of Henan Province of China (084100510012)the Natural Science Foundation for the Education Department of Henan Province of China (2008B510001)
文摘In this paper, a characterization of orthonormal wavelet families in Sobolev spaces H s （R） is established.
基金Acknowledgments. This research is supported by the Knowledge Innovative Foundation of Chinese Academy of Science (No. KZCX2-204, No. KZ-CX-SW-01-01B), and the National Natural Science Foundation of China (No. 40035010). The authors thank Professors Huang
文摘A parameter-free method based on orthonormal wavelet transforms is recommended for calculating the principal time scale of coherent structures in atmospheric boundary-layer measurements. First, the atmospheric turbulent signal is decomposed into the small scale vortex that has approximate isotropy and the large scale vortex with the digital filter. Then, the large scale vortex is used to detect coherent structures with this method. The principal time scale and profile of coherent structures for velocity components (u, v, w above rice fields are obtained. In order to testify the validity of this method, the correlation of coherent structures and non-coherent structures are also calculated.
基金Supported in part by the President Fund of GUCASSupported in part by National Natural Foundation of China(Grant No.10631080)National Natural Foundation of Beijing (Grant No.1092004)
文摘Motivated by representing multidimensional periodic nonlinear and non-stationary signals （functions）, we study a class of orthonormal exponential basis for L2（Id） with I ：= [0, 1）, whose exponential parts are piecewise linear spectral sequences with p-knots. It is widely applied in time-frequency analysis.
文摘The modal wave number tomography approach is used to obtain sound speed profile of water column in deep ocean. The approach consists of estimation of the local modal eigenvalues from complex pressure field and use of these data as input to modal perturbative inversion method for obtaining the local sound speed profile. The empirical orthonormal function (EOF) is applied to reduce the parameter search space. The ocean environment used for numerical simulations includes the Munk profile as the unperturbed background speed profile and a weak Gaussian eddy as the sound speed profile perturbation. The results of numerical simulations show the method is capable of monitoring the oceanic interior structure.
基金The first author got support in part from the fund provided by the University of North Carolina at Charlotte.The second author got support from the National Natural Science Foundation of China and the Science Foundation of Zhejiang Province.
文摘In this paper we show how to construct a scaling function and an orthonor-mal wavelet basis from a multiresolution approximation using an operator theoreticmethod.
基金Supported in part by the National Natural Science Foundation of China under Grants Nos.11475258,11205242,and 11675263
文摘We present a simple demonstration on the orthonormality of Volkov solutions with emphasizing on the suffcient condition to the orthonormality. Properly aligning the external electromagnetic wave along the third-axis, the Volkov solutions are eigenfunctions of the hermitian momentum ■1, ■2 and the light-cone hamiltonian operators with real eigenvalues, which can lead to a verification of the orthonormality in the context of quantum mechanics when the x3-integration of the external potential is not singularity as severe as δ(0). The hermiticity of the fermion field fourmomentum operators validates the application of the demonstration to the intense field quantum electrodynamic. The proof based on a direct calculation to the inner products of the solutions is recapitulated as well in a general manner without dependence on explicit representation of the Dirac matrices and spinors, which can be conducive to understand the suffcient condition and to the study of the polarized electron production where a convenient representation is selected elaborately to project out the spin-polarization.
文摘In this paper,we propose a new linear multiuser receiver for synchronous code-division multiple-access (CDMA) systems,referred to as the orthogonal multiuser(OMU) receiver.Unlike the linear minimum mean-squared error(MMSE) receiver,the OMU receiver depends only on the signature vectors and does not require knowledge of the received amplitudes or the channel signal-to-noise ratio(SNR).Here we develop methods that construct an optimal set of vectors with a specified inner product structure,from a given set of vectors in a complex Hilbert space.The optimal vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors.An algorithm has been developed using the principles of quantum parameters and some of its axioms and constraints.In place of the classical matched filter(MF) receiver we propose a modified receiver.This approach assumes that improving the accuracy will necessarily result in im- proved performance.The simulation results provided here suggests that in certain cases the OMU and POMU receivers can significantly increase the probability of correct detection with low error rate over the MF receiver.
文摘This paper is concerned with the problem of system identification using expansions on generalized orthonormal bases(GOB). Three algorithms are proposed to optimize the poles of such a basis. The first two algorithms determine a GOB with optimal real poles while the third one determines a GOB with optimal real and complex poles. These algorithms are based on the estimation of the dominant mode associated with a residual signal obtained by iteratively filtering the output of the process to be modelled. These algorithms are iterative and based on the quadratic error between the linear process output and the GOB based model output. They present the advantage to be very simple to implement. No numerical optimization technique is needed, and in consequence there is no problem of local minima as is the case for other algorithms in the literature. The convergence of the proposed algorithms is proved by demonstrating that the modeling quadratic error between the process output and the GOB based model is decreasing at each iteration of the algorithm. The performance of the proposed pole selection algorithms are based on the quadratic error criteria and illustrated by means of simulation results.
基金The project supported by National Natural Science Foundation of China under Grant No.10074072the Natural Science Foundation of Shandong Province of China under Grant No.Y2002A05
文摘A new kind of k-quantum nonlinear coherent states,i.e.,the k eigenstates of the k-th power~k (k≥3) of the generalized annihilation operator=1/f() of f-oscillators,are obtained and their properties are discussed.The completeness of the k states is investigated.An alternative method to construct them is proposed.It is shown that these states may form a complete Hilbert space,and all of them can be generated by a linear superposition of k Roy-type nonlinear coherent states.Physically,they can be generated by a linear superposition of the time-dependent Roy-type nonlinear coherent states at different instants.
文摘Routh stability test is covered in almost all undergraduate control texts. It determines the stability or, a little beyond, the number of unstable roots of a polynomial in terms of the signs of certain entries of the Routh table constructed from the coefficients of the polynomial. The use of the Routh table, as far as the common textbooks show, is only limited to this function. We will show that the Routh table can actually be used to construct an orthonormal basis in the space of strictly proper rational functions with a common stable denominator. This orthonormal basis can then be used for many other purposes, including the computation of the H2 norm, the Hankel singular values and singular vectors, model reduction, H∞ optimization, etc. Keywords Routh stablity criterion - Orthonormal basis - Root-mean-squared value - Hankel operater - Nehari problem - Model reduction This work was supported by the Hong Kong Research Grants Council.
文摘A continuous-time Model Predictive Controller was proposed using Kautz function in order to improve the performance of Load Frequency Control(LFC).A dynamic model of an interconnected power system was used for Model Predictive Controller(MPC)design.MPC predicts the future trajectory of the dynamic model by calculating the optimal closed loop feedback gain matrix.In this paper,the optimal closed loop feedback gain matrix was calculated using Kautz function.Being an Orthonormal Basis Function(OBF),Kautz function has an advantage of solving complex pole-based nonlinear system.Genetic Algorithm(GA)was applied to optimally tune the Kautz function-based MPC.A constraint based on phase plane analysis was implemented with the cost function in order to improve the robustness of the Kautz function-based MPC.The proposed method was simulated with three area interconnected power system and the efficiency of the proposed method was measured and exhibited by comparing with conventional Proportional and Integral(PI)controller and Linear Quadratic Regulation(LQR).
基金Sponsored by the National Natural Science Foundation of China(10372074)
文摘There are many forms of tensor theory which are quite different. Discussions are put forward and their relationships are found. The differences between them depend on whether there is metric on the space and the basis is orthonormal.
文摘The scope of this paper broadly spans in two areas： system identification of resonant system and design of an efficient control scheme suitable for resonant systems. Use of filters based on orthogonal basis functions （OBF） have been advocated for modelling of resonant process. Kautz filter has been identified as best suited OBF for this purpose. A state space based system identification technique using Kautz filters, viz. Kautz model, has been demonstrated. Model based controllers are believed to be more efficient than classical controllers because explicit use of process model is essential with these modelling techniques. Extensive literature search concludes that very few reports are available which explore use of the model based control studies on resonant system. Two such model based controllers are considered in this work, viz. model predictive controller and internal model controller. A model predictive control algorithm has been developed using the Kautz model. The efficacy of the model and the controller has been verified by two case studies, viz. linear second order underdamped process and a mildly nonlinear magnetic ball suspension system. Comparative assessment of performances of these controllers in those case studies have been carried out.
基金Supported by the National Natural Science Foundation of China(11071152)the Natural Science Foundation of Guangdong Province(10151503101000025)
文摘Suppose M and N are two r×r and s×s dilation matrices,respectively.LetΓM andΓN represent the complete sets of representatives of distinct cosets of the quotient groups M-T Zr/Zr and N-T Zs/Zs,respectively.Two methods for constructing nonseparable Ω-filter banks from M-filter banks and N-filter banks are presented,where Ω is a（r＋s） ×（r＋s） dilation matrix such that one of its complete sets of representatives of distinct cosets of the quotient groups Ω-T Zr＋s/Zr＋s areΓΩ={[γT h,ζ T q] T：γh∈ΓM,ζq∈ΓN}.Specially,Ω can be [MΘ0N],whereΘis a r×s integer matrix with M-1Θbeing also an integer matrix.Moreover,if the constructed filter bank satisfies Lawton＇s condition,which can be easy to verify,then it generates an orthonormal nonseparable Ω-wavelet basis for L2（Rr＋s）.Properties,including Lawton＇s condition,vanishing moments and regularity of the new Ω-filter banks or new Ω-wavelet basis are discussed then.Finally,a class of nonseparable Ω-wavelet basis for L2（Rr＋1） are constructed and three other examples are given to illustrate the results.In particular,when M=N=2,all results obtained in this paper appeared in[1].
