The invariant subspace method is used to construct the explicit solution of a nonlinear evolution equation. The second-order nonlinear differential operators that possess invariant subspaces of submaximal dimension ar...The invariant subspace method is used to construct the explicit solution of a nonlinear evolution equation. The second-order nonlinear differential operators that possess invariant subspaces of submaximal dimension are described. There are second-order nonlinear differential operators, including cubic operators and quadratic operators, which preserve an invariant subspace of submaximal dimension. A full. description, of the second-order cubic operators with constant coefficients admitting a four-dimensional invariant subspace is given. It is shown that the maximal dimension of invaxiant subspaces preserved by a second-order cubic operator is four. Several examples are given for the construction of the exact solutions to nonlinear evolution equations with cubic nonlinearities. These solutions blow up in a finite展开更多
In this paper, third-order nonlinear differential operators are studied. It is shown that they are quadratic forms when they preserve invariant subspaces of maximal dimension. A complete description of third-order qua...In this paper, third-order nonlinear differential operators are studied. It is shown that they are quadratic forms when they preserve invariant subspaces of maximal dimension. A complete description of third-order quadratic operators with constant coefficients is obtained. One example is given to derive special solutions for evolution equations with third-order quadratic operators.展开更多
Let Mφ be the operator of multiplication by φ on a Hilbert space of functions analytic on the open unit disk. For an invariant subspace F for the multiplication operator Mz, we derive some spectral properties of the...Let Mφ be the operator of multiplication by φ on a Hilbert space of functions analytic on the open unit disk. For an invariant subspace F for the multiplication operator Mz, we derive some spectral properties of the multiplication operator Mφ : F→F. We characterize norm, spectrum, essential norm and essential spectrum of such operators when F has the codimension n property with n∈{1,2,...,+∞}.展开更多
Invariant subspace method is exploited to obtain exact solutions of the two- component b-family system. It is shown that the two-component b-family system admits the generalized functional separable solutions. Further...Invariant subspace method is exploited to obtain exact solutions of the two- component b-family system. It is shown that the two-component b-family system admits the generalized functional separable solutions. Furthermore, blow up and behavior of those exact solutions are also investigated.展开更多
In this paper, the invariant subspaces of the generalized strongly dispersive DGH equation are given, and the exact solutions of the strongly dispersive DGH equation are obtained. Firstly, transform nonlinear partial ...In this paper, the invariant subspaces of the generalized strongly dispersive DGH equation are given, and the exact solutions of the strongly dispersive DGH equation are obtained. Firstly, transform nonlinear partial differential Equation (PDE) into ordinary differential Equation (ODE) systems by using the invariant subspace method. Secondly, combining with the dynamical system method, we use the invariant subspaces which have been obtained to construct the exact solutions of the equation. In the end, the figures of the exact solutions are given.展开更多
We consider a Hamiltonian of a system of two fermions on a three-dimensional lattice Z<sup>3</sup> with special potential <img alt="" src="Edit_56564354-6d65-4104-9126-d4657fa750af.png&qu...We consider a Hamiltonian of a system of two fermions on a three-dimensional lattice Z<sup>3</sup> with special potential <img alt="" src="Edit_56564354-6d65-4104-9126-d4657fa750af.png" />. The corresponding Shrödinger operator <em>H</em>(<strong>k</strong>) of the system has an invariant subspac <span style="white-space:nowrap;"><span><em>L</em></span><sup>-</sup><sub style="margin-left:-10px;">123</sub>(T<sup>3</sup>)</span> , where we study the eigenvalues and eigenfunctions of its restriction <span style="white-space:nowrap;"><span><em>H</em></span><sup>-</sup><sub style="margin-left:-10px;">123</sub></span><span style="white-space:nowrap;">(<strong>k</strong>)</span>. Moreover, there are shown that <span style="white-space:nowrap;"><span><em>H</em></span><sup>-</sup><sub style="margin-left:-10px;">123</sub>(<em>k</em><sub>1</sub>, <em>k</em><sub>2</sub>, π)</span> has also infinitely many invariant subspaces <img alt="" src="Edit_4955ffad-4b18-434a-8c99-ff14779f2812.bmp" />, where the eigenvalues and eigenfunctions of eigenvalue problem <img alt="" src="Edit_01b218d2-fa3e-4f39-bc2d-ce736205db93.bmp" />are explicitly found.展开更多
In this research,invariant subspaces and exact solutions for the governing equation are obtained through the invariant subspace method,and the generalized second-order Kudryashov-Sinelshchikov equation is used to desc...In this research,invariant subspaces and exact solutions for the governing equation are obtained through the invariant subspace method,and the generalized second-order Kudryashov-Sinelshchikov equation is used to describe pressure waves in a liquid with bubbles.The governing equations are classified and transformed into a system of ordinary differential equations,and the exact solutions of the classified equation are obtained by solving the system of ordinary differential equations.The method gives logarithmic,polynomial,exponential,and trigonometric solutions for equations.The primary accomplishments of this work are displaying how to obtain the exact solutions of the classified equations and giving the stability analysis of reduced ordinarydifferential equations.展开更多
For a backward shift invariant subspace N in H^2(Г^2), the operators Sz and Sw on N are defined by Sz = PNTz|N and Sw, = PNTw|N, where PN is the orthogonal projection from L^2(Г^2) onto N. We give a characteri...For a backward shift invariant subspace N in H^2(Г^2), the operators Sz and Sw on N are defined by Sz = PNTz|N and Sw, = PNTw|N, where PN is the orthogonal projection from L^2(Г^2) onto N. We give a characterization of N satisfying rank [Sz, Sw^*] = 1.展开更多
Let F and G be closed subspaces of the complex Hilbert spaceH, and U and V be closed subspaces of F- and G, respectively. In this paper, using the technique of operator block, we present the necessary and sufficient c...Let F and G be closed subspaces of the complex Hilbert spaceH, and U and V be closed subspaces of F- and G, respectively. In this paper, using the technique of operator block, we present the necessary and sufficient conditions under which (U, V) is a pair of (strictly, non-degenerate) principal invariant subspaces for (F, G).展开更多
The key purpose of the present research is to derive the exact solutions of nonlinear water wave equations(NLWWEs)in oceans through the invariant subspace scheme(ISS).In this respect,the NLWWEs which describe specific...The key purpose of the present research is to derive the exact solutions of nonlinear water wave equations(NLWWEs)in oceans through the invariant subspace scheme(ISS).In this respect,the NLWWEs which describe specific nonlinear waves are converted to a number of systems of ordinary differential equations(ODEs)such that the resulting systems can be efficiently handled by computer algebra systems.As an accomplishment,the performance of the well-designed ISS in extracting a group of exact solutions is formally confirmed.In the end,the stability analysis for the NLWWE is investigated through the linear stability scheme.展开更多
We propose a quadratically convergent algorithm for computing the invariant subspaces of an Hermitian matrix. Each iteration of the algorithm consists of one matrix-matrix multiplication and one QR decomposition. We p...We propose a quadratically convergent algorithm for computing the invariant subspaces of an Hermitian matrix. Each iteration of the algorithm consists of one matrix-matrix multiplication and one QR decomposition. We present an accurate convergence analysis of the algorithm without using the big O notation. We also propose a general framework based on implicit rational transformations which allows us to make connections with several existing algorithms and to derive classes of extensions to our basic algorithm with faster convergence rates. Several numerical examples are given which compare some aspects of the existing algorithms and the new algorithms.展开更多
Let{T(t)}_(t≥0) be a C_(0)-semigroupon an infinite-dimensional separable Hilbert space;a suitable definition of near{T(t)^(*)}_(t≥0) invariance of a subspace is presented in this paper.A series of prototypical examp...Let{T(t)}_(t≥0) be a C_(0)-semigroupon an infinite-dimensional separable Hilbert space;a suitable definition of near{T(t)^(*)}_(t≥0) invariance of a subspace is presented in this paper.A series of prototypical examples for minimal nearly{S(t)^(*)}_(t≥0) invariant subspaces for the shift semigroup{S(t)}_(t≥0) on L^(2)(0,∞)are demonstrated,which have close links with near T_(θ)^(*)invariance on Hardy spaces of the unit disk for an inner functionθ.Especially,the corresponding subspaces on Hardy spaces of the right half-plane and the unit disk are related to model spaces.This work further includes a discussion on the structure of the closure of certain subspaces related to model spaces in Hardy spaces.展开更多
In this paper, we prove that every operator in a class of contraction operators on a Banach space whose spectrum contains the unit circle has a nontrivial hyperinvariant subspace.
In this paper, for an invariant subspace I of the weighted Bergman space, the weighted root operator is defined. We study the weighted root operator and obtain its fundamental properties when the invariant subspace I ...In this paper, for an invariant subspace I of the weighted Bergman space, the weighted root operator is defined. We study the weighted root operator and obtain its fundamental properties when the invariant subspace I has finite index. Furthermore, we give some examples of the root operator and estimate ranks of the operators.展开更多
In the present paper,invariant subspace method has been extended for solving systems of multi-term fractional partial differential equations(FPDEs)involving both time and space fractional derivatives.Further,the metho...In the present paper,invariant subspace method has been extended for solving systems of multi-term fractional partial differential equations(FPDEs)involving both time and space fractional derivatives.Further,the method has also been employed for solving multi-term fractional PDEs in(1+n)dimensions.A diverse set of examples is solved to illustrate the method.展开更多
Let F be any commutative field. Let v be an integer≥1 and be a fixed 2v × 2v nonsingular alternate matrix over F. Define Sp(F)={T: 2v×2v matrix over F|TKT~T=K}. It is well-known that Sp(F) is a group with r...Let F be any commutative field. Let v be an integer≥1 and be a fixed 2v × 2v nonsingular alternate matrix over F. Define Sp(F)={T: 2v×2v matrix over F|TKT~T=K}. It is well-known that Sp(F) is a group with respect to the matrix multiplication and is called the symplectic group of degree 2v over F展开更多
In this paper, we show that for log(2/3)/2log2≤ β ≤1/2, suppose S is an invariant subspace of the Hardy-Sobolev spaces H_β~2(D^n) for the n-tuple of multiplication operators(M_(z_1),...,M_(z_n)). If(M_(z_1)|S,...,...In this paper, we show that for log(2/3)/2log2≤ β ≤1/2, suppose S is an invariant subspace of the Hardy-Sobolev spaces H_β~2(D^n) for the n-tuple of multiplication operators(M_(z_1),...,M_(z_n)). If(M_(z_1)|S,..., M_(z_n)|S) is doubly commuting, then for any non-empty subset α = {α_1,..., α_k} of {1,..., n}, W_α~S is a generating wandering subspace for M_α|_S =(M_(z_(α_1))|_S,..., M_(z_(α_k))|_S), that is, [W_α~S]_(M_(α |S))= S, where W_α~S=■(S ■ z_(α_i)S).展开更多
In most literature about joint direction of arrival(DOA) and polarization estimation, the case that sources possess different power levels is seldom discussed. However, this case exists widely in practical applicati...In most literature about joint direction of arrival(DOA) and polarization estimation, the case that sources possess different power levels is seldom discussed. However, this case exists widely in practical applications, especially in passive radar systems. In this paper, we propose a joint DOA and polarization estimation method for unequal power sources based on the reconstructed noise subspace. The invariance property of noise subspace(IPNS) to power of sources has been proved an effective method to estimate DOA of unequal power sources. We develop the IPNS method for joint DOA and polarization estimation based on a dual polarized array. Moreover, we propose an improved IPNS method based on the reconstructed noise subspace, which has higher resolution probability than the IPNS method. It is theoretically proved that the IPNS to power of sources is still valid when the eigenvalues of the noise subspace are changed artificially. Simulation results show that the resolution probability of the proposed method is enhanced compared with the methods based on the IPNS and the polarimetric multiple signal classification(MUSIC) method. Meanwhile, the proposed method has approximately the same estimation accuracy as the IPNS method for the weak source.展开更多
In this paper, we define the ({A,E},B)-invariant subspace pair contained in Ker C for singular systems, rigorously justifying the name and demonstrating the existence of the supremal ({A,E},B)-invariant;subspace pair ...In this paper, we define the ({A,E},B)-invariant subspace pair contained in Ker C for singular systems, rigorously justifying the name and demonstrating the existence of the supremal ({A,E},B)-invariant;subspace pair contained in Ker C, we show how the supremal ({A,E},B)-invariant subspace pair contained in Ker C can be computed via some subspace recursions, We provide necessary and sufficient condition for the existence of a state feedback that achieves disturbance localization in a linear time-invariant singular system.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant No.10926082)the Natural Science Foundation of Anhui Province of China(Grant No.KJ2010A128)the Fund for Youth of Anhui Normal University,China(Grant No.2009xqn55)
文摘The invariant subspace method is used to construct the explicit solution of a nonlinear evolution equation. The second-order nonlinear differential operators that possess invariant subspaces of submaximal dimension are described. There are second-order nonlinear differential operators, including cubic operators and quadratic operators, which preserve an invariant subspace of submaximal dimension. A full. description, of the second-order cubic operators with constant coefficients admitting a four-dimensional invariant subspace is given. It is shown that the maximal dimension of invaxiant subspaces preserved by a second-order cubic operator is four. Several examples are given for the construction of the exact solutions to nonlinear evolution equations with cubic nonlinearities. These solutions blow up in a finite
基金supported by the National Natural Science Foundation of China(Grant No.11371293)the Civil Military Integration Research Foundation of Shaanxi Province,China(Grant No.13JMR13)+2 种基金the Natural Science Foundation of Shaanxi Province,China(Grant No.14JK1246)the Mathematical Discipline Foundation of Shaanxi Province,China(Grant No.14SXZD015)the Basic Research Project Foundation of Weinan City,China(Grant No.2013JCYJ-4)
文摘In this paper, third-order nonlinear differential operators are studied. It is shown that they are quadratic forms when they preserve invariant subspaces of maximal dimension. A complete description of third-order quadratic operators with constant coefficients is obtained. One example is given to derive special solutions for evolution equations with third-order quadratic operators.
文摘Let Mφ be the operator of multiplication by φ on a Hilbert space of functions analytic on the open unit disk. For an invariant subspace F for the multiplication operator Mz, we derive some spectral properties of the multiplication operator Mφ : F→F. We characterize norm, spectrum, essential norm and essential spectrum of such operators when F has the codimension n property with n∈{1,2,...,+∞}.
基金supported by NSFC(11471260)the Foundation of Shannxi Education Committee(12JK0850)
文摘Invariant subspace method is exploited to obtain exact solutions of the two- component b-family system. It is shown that the two-component b-family system admits the generalized functional separable solutions. Furthermore, blow up and behavior of those exact solutions are also investigated.
文摘In this paper, the invariant subspaces of the generalized strongly dispersive DGH equation are given, and the exact solutions of the strongly dispersive DGH equation are obtained. Firstly, transform nonlinear partial differential Equation (PDE) into ordinary differential Equation (ODE) systems by using the invariant subspace method. Secondly, combining with the dynamical system method, we use the invariant subspaces which have been obtained to construct the exact solutions of the equation. In the end, the figures of the exact solutions are given.
文摘We consider a Hamiltonian of a system of two fermions on a three-dimensional lattice Z<sup>3</sup> with special potential <img alt="" src="Edit_56564354-6d65-4104-9126-d4657fa750af.png" />. The corresponding Shrödinger operator <em>H</em>(<strong>k</strong>) of the system has an invariant subspac <span style="white-space:nowrap;"><span><em>L</em></span><sup>-</sup><sub style="margin-left:-10px;">123</sub>(T<sup>3</sup>)</span> , where we study the eigenvalues and eigenfunctions of its restriction <span style="white-space:nowrap;"><span><em>H</em></span><sup>-</sup><sub style="margin-left:-10px;">123</sub></span><span style="white-space:nowrap;">(<strong>k</strong>)</span>. Moreover, there are shown that <span style="white-space:nowrap;"><span><em>H</em></span><sup>-</sup><sub style="margin-left:-10px;">123</sub>(<em>k</em><sub>1</sub>, <em>k</em><sub>2</sub>, π)</span> has also infinitely many invariant subspaces <img alt="" src="Edit_4955ffad-4b18-434a-8c99-ff14779f2812.bmp" />, where the eigenvalues and eigenfunctions of eigenvalue problem <img alt="" src="Edit_01b218d2-fa3e-4f39-bc2d-ce736205db93.bmp" />are explicitly found.
基金supported by a training program for key young teachers of colleges and universities in Henan Province(No.2019GGJS143)the Natural Science Foundation of Shannxi Province of China(No.2021JM-521)+2 种基金key research projects of Henan higher education institutions(No.21A110026)research team development project of Zhongyuan University of Technology(No.K2020TD004)the Natural Science of Foundation of Zhongyuan University of Technology(No.K2023MS002).
文摘In this research,invariant subspaces and exact solutions for the governing equation are obtained through the invariant subspace method,and the generalized second-order Kudryashov-Sinelshchikov equation is used to describe pressure waves in a liquid with bubbles.The governing equations are classified and transformed into a system of ordinary differential equations,and the exact solutions of the classified equation are obtained by solving the system of ordinary differential equations.The method gives logarithmic,polynomial,exponential,and trigonometric solutions for equations.The primary accomplishments of this work are displaying how to obtain the exact solutions of the classified equations and giving the stability analysis of reduced ordinarydifferential equations.
基金supported by Grant-in-Aid for Scientific Research (No. 16340037)Japan Society for the Promotion of Science
文摘For a backward shift invariant subspace N in H^2(Г^2), the operators Sz and Sw on N are defined by Sz = PNTz|N and Sw, = PNTw|N, where PN is the orthogonal projection from L^2(Г^2) onto N. We give a characterization of N satisfying rank [Sz, Sw^*] = 1.
基金supported by National Natural Science Foundation of China(Grant No.11326107)supported by National Natural Science Foundation of China(Grant No.11071188)Special Foundation for Excellent Young College and University Teachers(Grant No.405ZK12YQ21-ZZyyy12021)
文摘Let F and G be closed subspaces of the complex Hilbert spaceH, and U and V be closed subspaces of F- and G, respectively. In this paper, using the technique of operator block, we present the necessary and sufficient conditions under which (U, V) is a pair of (strictly, non-degenerate) principal invariant subspaces for (F, G).
文摘The key purpose of the present research is to derive the exact solutions of nonlinear water wave equations(NLWWEs)in oceans through the invariant subspace scheme(ISS).In this respect,the NLWWEs which describe specific nonlinear waves are converted to a number of systems of ordinary differential equations(ODEs)such that the resulting systems can be efficiently handled by computer algebra systems.As an accomplishment,the performance of the well-designed ISS in extracting a group of exact solutions is formally confirmed.In the end,the stability analysis for the NLWWE is investigated through the linear stability scheme.
文摘We propose a quadratically convergent algorithm for computing the invariant subspaces of an Hermitian matrix. Each iteration of the algorithm consists of one matrix-matrix multiplication and one QR decomposition. We present an accurate convergence analysis of the algorithm without using the big O notation. We also propose a general framework based on implicit rational transformations which allows us to make connections with several existing algorithms and to derive classes of extensions to our basic algorithm with faster convergence rates. Several numerical examples are given which compare some aspects of the existing algorithms and the new algorithms.
基金supported by National Natural Science Foundation of China(Grant No.11701422)。
文摘Let{T(t)}_(t≥0) be a C_(0)-semigroupon an infinite-dimensional separable Hilbert space;a suitable definition of near{T(t)^(*)}_(t≥0) invariance of a subspace is presented in this paper.A series of prototypical examples for minimal nearly{S(t)^(*)}_(t≥0) invariant subspaces for the shift semigroup{S(t)}_(t≥0) on L^(2)(0,∞)are demonstrated,which have close links with near T_(θ)^(*)invariance on Hardy spaces of the unit disk for an inner functionθ.Especially,the corresponding subspaces on Hardy spaces of the right half-plane and the unit disk are related to model spaces.This work further includes a discussion on the structure of the closure of certain subspaces related to model spaces in Hardy spaces.
基金the Natural Science Foundation of P.R.China (No.10771039)
文摘In this paper, we prove that every operator in a class of contraction operators on a Banach space whose spectrum contains the unit circle has a nontrivial hyperinvariant subspace.
基金Supported by the National Natural Science Foundation of China (Grant Nos.10671028 10971020)
文摘In this paper, for an invariant subspace I of the weighted Bergman space, the weighted root operator is defined. We study the weighted root operator and obtain its fundamental properties when the invariant subspace I has finite index. Furthermore, we give some examples of the root operator and estimate ranks of the operators.
文摘In the present paper,invariant subspace method has been extended for solving systems of multi-term fractional partial differential equations(FPDEs)involving both time and space fractional derivatives.Further,the method has also been employed for solving multi-term fractional PDEs in(1+n)dimensions.A diverse set of examples is solved to illustrate the method.
文摘Let F be any commutative field. Let v be an integer≥1 and be a fixed 2v × 2v nonsingular alternate matrix over F. Define Sp(F)={T: 2v×2v matrix over F|TKT~T=K}. It is well-known that Sp(F) is a group with respect to the matrix multiplication and is called the symplectic group of degree 2v over F
基金supported by the Natural Science Foundation of China(11271092,11471143)the key research project of Nanhu College of Jiaxing University(N41472001-18)
文摘In this paper, we show that for log(2/3)/2log2≤ β ≤1/2, suppose S is an invariant subspace of the Hardy-Sobolev spaces H_β~2(D^n) for the n-tuple of multiplication operators(M_(z_1),...,M_(z_n)). If(M_(z_1)|S,..., M_(z_n)|S) is doubly commuting, then for any non-empty subset α = {α_1,..., α_k} of {1,..., n}, W_α~S is a generating wandering subspace for M_α|_S =(M_(z_(α_1))|_S,..., M_(z_(α_k))|_S), that is, [W_α~S]_(M_(α |S))= S, where W_α~S=■(S ■ z_(α_i)S).
基金supported by the National Natural Science Foundation of China(61501142)the China Postdoctoral Science Foundation(2015M571414)+3 种基金the Fundamental Research Funds for the Central Universities(HIT.NSRIF.2016102)Shandong Provincial Natural Science Foundation(ZR2014FQ003)the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology(HIT.NSRIF 2013130HIT(WH)XBQD 201022)
文摘In most literature about joint direction of arrival(DOA) and polarization estimation, the case that sources possess different power levels is seldom discussed. However, this case exists widely in practical applications, especially in passive radar systems. In this paper, we propose a joint DOA and polarization estimation method for unequal power sources based on the reconstructed noise subspace. The invariance property of noise subspace(IPNS) to power of sources has been proved an effective method to estimate DOA of unequal power sources. We develop the IPNS method for joint DOA and polarization estimation based on a dual polarized array. Moreover, we propose an improved IPNS method based on the reconstructed noise subspace, which has higher resolution probability than the IPNS method. It is theoretically proved that the IPNS to power of sources is still valid when the eigenvalues of the noise subspace are changed artificially. Simulation results show that the resolution probability of the proposed method is enhanced compared with the methods based on the IPNS and the polarimetric multiple signal classification(MUSIC) method. Meanwhile, the proposed method has approximately the same estimation accuracy as the IPNS method for the weak source.
文摘In this paper, we define the ({A,E},B)-invariant subspace pair contained in Ker C for singular systems, rigorously justifying the name and demonstrating the existence of the supremal ({A,E},B)-invariant;subspace pair contained in Ker C, we show how the supremal ({A,E},B)-invariant subspace pair contained in Ker C can be computed via some subspace recursions, We provide necessary and sufficient condition for the existence of a state feedback that achieves disturbance localization in a linear time-invariant singular system.