Necessary and sufficient conditions for the exact controllability and exact observability of a descriptor infinite dimensional system are obtained in the sense of distributional solution.These general results are used...Necessary and sufficient conditions for the exact controllability and exact observability of a descriptor infinite dimensional system are obtained in the sense of distributional solution.These general results are used to examine the exact controllability and exact observability of the Dzektser equation in the theory of seepage and the exact controllability of wave equation.展开更多
Let {(Xi, Si, μi) : i ℃ N} be a sequence of probability measure spaces and (*Xi, L(*Si), L(*μi)) be the Loeb measure space with respect to (Xi, Si, μi) for i ℃ N. Let X =× Xi, S = ×Si,μ = ×μi. We...Let {(Xi, Si, μi) : i ℃ N} be a sequence of probability measure spaces and (*Xi, L(*Si), L(*μi)) be the Loeb measure space with respect to (Xi, Si, μi) for i ℃ N. Let X =× Xi, S = ×Si,μ = ×μi. We prove that × L(*Si) CL(*S) and in embedding meaning.展开更多
In this paper, the ascent of 2 × 2 infinite dimensional Hamiltonian operators and a class of 4 × 4 infinite dimensional Hamiltonian operators are studied, and the conditions under which the ascent of 2 ×...In this paper, the ascent of 2 × 2 infinite dimensional Hamiltonian operators and a class of 4 × 4 infinite dimensional Hamiltonian operators are studied, and the conditions under which the ascent of 2 × 2 infinite dimensional Hamiltonian operator is 1 and the ascent of a class of 4 × 4 infinite dimensional Hamiltonian operators that arises in study of elasticity is2 are obtained. Concrete examples are given to illustrate the effectiveness of criterions.展开更多
Non-isomorphic two dimensional indecomposable modules over infinite dimensional hereditary path algebras are described. We infer that none of them can be determined by their dimension vectors.
A multilayer flow is a stratified fluid composed of a finite number of layers with densities homogeneous within one layer but different from each other. It is an intermediate system between the single-layer barotropic...A multilayer flow is a stratified fluid composed of a finite number of layers with densities homogeneous within one layer but different from each other. It is an intermediate system between the single-layer barotropic model and the continuously stratified baroclinic model. Since this system can simulate the baroclinic effect simply, it is widely used to study the large-scale dynamic process in atmosphere and ocean. The present paper is concerned with the linear stability of the multilayer quasi-geostrophic flow, and the associated linear stability criteria are established. Firstly, the nonlinear model is turned into the form of a Hamiltonian system, and a basic flow is defined. But it cannot be an extreme point of the Hamiltonian function since the system is an infinite-dimensional one. Therefore, it is necessary to reconstruct a new Hamiltonian function so that the basic flow becomes an extreme point of it. Secondly, the linearized equations of disturbances in the multilayer quasi-geostrophic flow are derived by introducing infinitesimal disturbances superposed on the basic flows. Finally, the properties of the linearized system are discussed, and the linear stability criteria in the sense of Liapunov are derived under two different conditions with respect to certain norms.展开更多
In modern mathematics, geometric and algebraic properties of space can be applied by calculus in which the concept of gradually increasings of dimensions is existence (such as zero-dimension, one-dimension, two-dimens...In modern mathematics, geometric and algebraic properties of space can be applied by calculus in which the concept of gradually increasings of dimensions is existence (such as zero-dimension, one-dimension, two-dimension, and three-dimension, etc). However, this is not fact because some new concepts have been put forward in this paper where there is only a concept of infinitely great that is one quantitative continuum implied by the change of direction. The accurate description of this one quantitative continuum is that its parts are connected each other as a unity at the infinite distance (infinitely great) relative to any orientation (all orientations) of our existence. It is unity in which its random parts are these infinitely great quantities and thus we call this unity as infinite quantities of infinite dimensions.展开更多
In this paper,we study Sobolev spaces in infinite dimensions and the corresponding embedding theorems.Our underlying spaces areℓ^(r)for r ∈[1,∞),equipped with corresponding probability measures.For the weighted Sobo...In this paper,we study Sobolev spaces in infinite dimensions and the corresponding embedding theorems.Our underlying spaces areℓ^(r)for r ∈[1,∞),equipped with corresponding probability measures.For the weighted Sobolev space W_(b)^(1,p)(ℓ^(r),γa)with a weight a∈ℓ^(r)of the Gaussian measureγa and a gradient weight b∈l^(∞),we characterize the relation between the weights(a and b)and the continuous(resp.compact)log-Sobolev embedding for p∈[1,∞)(resp.p∈(1,∞)).Several counterexamples are also constructed,which are of independent interest.展开更多
The properties of eigenvalues and eigenfunctions of the infinite dimensional Hamiltonian operators are studied, and the sufficient conditions of the completeness in the sense of Cauchy principal value of the eigenfunc...The properties of eigenvalues and eigenfunctions of the infinite dimensional Hamiltonian operators are studied, and the sufficient conditions of the completeness in the sense of Cauchy principal value of the eigenfunction systems of the infinite dimensional Hamiltonian operators are given. In the end, concrete examples are constructed to justify the effectiveness of the criterion.展开更多
This paper deals with the structure of the spectrum of infinite dimensional Hamiltonian operators.It is shown that the spectrum,the union of the point spectrum and residual spectrum,and the continuous spectrum are all...This paper deals with the structure of the spectrum of infinite dimensional Hamiltonian operators.It is shown that the spectrum,the union of the point spectrum and residual spectrum,and the continuous spectrum are all symmetric with respect to the imaginary axis of the complex plane. Moreover,it is proved that the residual spectrum does not contain any pair of points symmetric with respect to the imaginary axis;and a complete characterization of the residual spectrum in terms of the point spectrum is then given.As applications of these structure results,we obtain several necessary and sufficient conditions for the residual spectrum of a class of infinite dimensional Hamiltonian operators to be empty.展开更多
For the off-diagonal infinite dimensional Hamiltonian operators, which have at most countable eigenvalues, a necessary and sufficient condition of the eigenfunction systems to be complete in the sense of Cauchy princi...For the off-diagonal infinite dimensional Hamiltonian operators, which have at most countable eigenvalues, a necessary and sufficient condition of the eigenfunction systems to be complete in the sense of Cauchy principal value is presented by using the spectral symmetry and new orthogonal relationship of the operators. Moreover, the above result is extended to a more general case. At last, the completeness of eigenfunction systems for the operators arising from the isotropic plane magnetoelectroelastic solids is described to illustrate the effectiveness of the criterion. The whole results offer theoretical guarantee for separation of variables in Hamiltonian system for some mechanics equations.展开更多
Given a real-valued separable M-type 2 Banach space X with the p-th power of the norm of C2-class, the almost sure asymptotic upper bounds of the solutions of the Ornstein-Uhlenbeck Processes described by the followin...Given a real-valued separable M-type 2 Banach space X with the p-th power of the norm of C2-class, the almost sure asymptotic upper bounds of the solutions of the Ornstein-Uhlenbeck Processes described by the following equations {dz(t)=A(t,z(t))dt+σ(t,z(t))dW(t),z(0)=z0∈X,are investigated. This study generalizes the corresponding well-known finite dimensional results of Lapeyre (1989) and Mao (1992).展开更多
In this paper,we construct certain irreducible infinite dimensional representations of algebraic groups with Frobenius maps.In particular,a few classical results of Steinberg and Deligne&Lusztig on complex represe...In this paper,we construct certain irreducible infinite dimensional representations of algebraic groups with Frobenius maps.In particular,a few classical results of Steinberg and Deligne&Lusztig on complex representations of finite groups of Lie type are extended to reductive algebraic groups with Frobenius maps.展开更多
In this paper, we gave a proof for the local continuity modulus theorem of the Wiener process, i. e., lim t→0 sup 0≤s≤t |W(s)|/(2s log log(1/s))<sup>1/2</sup>=1 a.s. This result was given by Csrg...In this paper, we gave a proof for the local continuity modulus theorem of the Wiener process, i. e., lim t→0 sup 0≤s≤t |W(s)|/(2s log log(1/s))<sup>1/2</sup>=1 a.s. This result was given by Csrg and Revesz (1981), but the proof gets them nowhere. We also gave a similar local continuity modulus result for the infinite dimensional OU processes.展开更多
In this paper, by using characterization of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H, a necessary and sufficient condition is obtained on the symmetry of σP(A) and σ1/...In this paper, by using characterization of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H, a necessary and sufficient condition is obtained on the symmetry of σP(A) and σ1/P(-A^*) with respect to the imaginary axis. Then the symmetry of the point spectrum of H is given, and several examples are presented to illustrate the results.展开更多
An explicit differencescheme is described,analyzed and tested for numer-ically approximating stochastic elastic equation driven by infinite dimensional noise.The noise processes are approximated by piecewise constant ...An explicit differencescheme is described,analyzed and tested for numer-ically approximating stochastic elastic equation driven by infinite dimensional noise.The noise processes are approximated by piecewise constant random processes and the integral formula of the stochastic elastic equation is approximated by a truncated series.Error analysis of the numerical method yields estimate of convergence rate.The rate of convergence is demonstrated with numerical experiments.展开更多
This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H. Note that the point spectrum of H can be described as σp(H)...This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H. Note that the point spectrum of H can be described as σp(H) = σp (A) U σp1 (-A*). Using the characteristic of the set σp1(-A*), we divide the point spectrum σp (d) of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which σp1(-A*) and one part of σp(A) are symmetric with respect to the real axis each other. Based on this result, the symmetry of σp(H) is completely given. Moreover, the above result is applied to thin plates on elastic foundation, plane elasticity problems and harmonic equations.展开更多
This paper is addressed to develop an approximate method to solve a class of infinite dimensional LQ optimal regulator problems over infinite time horizon. Our algorithm is based on a construction of approximate solut...This paper is addressed to develop an approximate method to solve a class of infinite dimensional LQ optimal regulator problems over infinite time horizon. Our algorithm is based on a construction of approximate solutions which solve some finite dimensional LQ optimal regulator problems over finite time horizon, and it is shown that these approximate solutions converge strongly to the desired solution in the double limit sense.展开更多
This paper deals with the output regulation problem for a class of SISO infinite dimensional systems with an uncertain exosystem.For these systems,a concept of relative degree is firstly introduced and used to constru...This paper deals with the output regulation problem for a class of SISO infinite dimensional systems with an uncertain exosystem.For these systems,a concept of relative degree is firstly introduced and used to construct a transformation which leads to the canonical form of output feedback systems.Then,based on this canonical form,by means of an internal model and a recursive adaptive control,the authors obtain an adaptive regulator which solves the problem.It should be pointed out that the proposed regulator is finite dimensional while it is usually infinite dimensional in existing literatures.展开更多
Let X be the solution of the SDE:dX_t=σ(X_t)dB_t+b(X_t)dt,with σ and b ∈C_B~∞(R) such that σ≥λ>0 for some constant λ,and B a real Brownian motion.Let μ be the law of X on E=C([0,1],R)and k ∈E~*-{0},where ...Let X be the solution of the SDE:dX_t=σ(X_t)dB_t+b(X_t)dt,with σ and b ∈C_B~∞(R) such that σ≥λ>0 for some constant λ,and B a real Brownian motion.Let μ be the law of X on E=C([0,1],R)and k ∈E~*-{0},where E~* is the topological dual space of E.Consider the classical form:where u and v are smooth functions on E.We prove that,if is closable for any k in a dense subset of E~* and if the smooth functions are contained in the domain of the generator of the closure of ε_k,σ must be a constant function.展开更多
In this paper we study the infinite dimensional Malliavin calculus, and apply it to determingwhen the solution of an infinite dimensional stochastic differential equation has the property thatits finite dimensional di...In this paper we study the infinite dimensional Malliavin calculus, and apply it to determingwhen the solution of an infinite dimensional stochastic differential equation has the property thatits finite dimensional distributions possess smooth density.展开更多
基金This work was supported by the National Natural Science Foundation of China(11926402,61973338).
文摘Necessary and sufficient conditions for the exact controllability and exact observability of a descriptor infinite dimensional system are obtained in the sense of distributional solution.These general results are used to examine the exact controllability and exact observability of the Dzektser equation in the theory of seepage and the exact controllability of wave equation.
基金The Special Science Foundation (00jk207) of the Educational Committee of Shaanxi Province.
文摘Let {(Xi, Si, μi) : i ℃ N} be a sequence of probability measure spaces and (*Xi, L(*Si), L(*μi)) be the Loeb measure space with respect to (Xi, Si, μi) for i ℃ N. Let X =× Xi, S = ×Si,μ = ×μi. We prove that × L(*Si) CL(*S) and in embedding meaning.
基金supported by the National Natural Science Foundation of China(Grant Nos.11101200 and 11371185)the Natural Science Foundation of Inner Mongolia Autonomous Region,China(Grant No.2013ZD01)
文摘In this paper, the ascent of 2 × 2 infinite dimensional Hamiltonian operators and a class of 4 × 4 infinite dimensional Hamiltonian operators are studied, and the conditions under which the ascent of 2 × 2 infinite dimensional Hamiltonian operator is 1 and the ascent of a class of 4 × 4 infinite dimensional Hamiltonian operators that arises in study of elasticity is2 are obtained. Concrete examples are given to illustrate the effectiveness of criterions.
基金The NSF(11371307)of ChinaResearch Culture Funds(2014xmpy11)of Anhui Normal University
文摘Non-isomorphic two dimensional indecomposable modules over infinite dimensional hereditary path algebras are described. We infer that none of them can be determined by their dimension vectors.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.41575026,41275113,and 41475021)
文摘A multilayer flow is a stratified fluid composed of a finite number of layers with densities homogeneous within one layer but different from each other. It is an intermediate system between the single-layer barotropic model and the continuously stratified baroclinic model. Since this system can simulate the baroclinic effect simply, it is widely used to study the large-scale dynamic process in atmosphere and ocean. The present paper is concerned with the linear stability of the multilayer quasi-geostrophic flow, and the associated linear stability criteria are established. Firstly, the nonlinear model is turned into the form of a Hamiltonian system, and a basic flow is defined. But it cannot be an extreme point of the Hamiltonian function since the system is an infinite-dimensional one. Therefore, it is necessary to reconstruct a new Hamiltonian function so that the basic flow becomes an extreme point of it. Secondly, the linearized equations of disturbances in the multilayer quasi-geostrophic flow are derived by introducing infinitesimal disturbances superposed on the basic flows. Finally, the properties of the linearized system are discussed, and the linear stability criteria in the sense of Liapunov are derived under two different conditions with respect to certain norms.
文摘In modern mathematics, geometric and algebraic properties of space can be applied by calculus in which the concept of gradually increasings of dimensions is existence (such as zero-dimension, one-dimension, two-dimension, and three-dimension, etc). However, this is not fact because some new concepts have been put forward in this paper where there is only a concept of infinitely great that is one quantitative continuum implied by the change of direction. The accurate description of this one quantitative continuum is that its parts are connected each other as a unity at the infinite distance (infinitely great) relative to any orientation (all orientations) of our existence. It is unity in which its random parts are these infinitely great quantities and thus we call this unity as infinite quantities of infinite dimensions.
基金supported by National Natural Science Foundation of China(Grant Nos.11931011 and 11901407)New Cornerstone Investigator Program and the Science Development Project of Sichuan University(Grant No 2020SCUNL201).
文摘In this paper,we study Sobolev spaces in infinite dimensions and the corresponding embedding theorems.Our underlying spaces areℓ^(r)for r ∈[1,∞),equipped with corresponding probability measures.For the weighted Sobolev space W_(b)^(1,p)(ℓ^(r),γa)with a weight a∈ℓ^(r)of the Gaussian measureγa and a gradient weight b∈l^(∞),we characterize the relation between the weights(a and b)and the continuous(resp.compact)log-Sobolev embedding for p∈[1,∞)(resp.p∈(1,∞)).Several counterexamples are also constructed,which are of independent interest.
基金supported by the National Natural Science Foundation of China (Grant No. 10562002)Colleges and Universities Doctoral Subject Research Funds (Grant No. 20070126002)the Natural Science Foundation of Inner Mongolia (Grant No. 200508010103)
文摘The properties of eigenvalues and eigenfunctions of the infinite dimensional Hamiltonian operators are studied, and the sufficient conditions of the completeness in the sense of Cauchy principal value of the eigenfunction systems of the infinite dimensional Hamiltonian operators are given. In the end, concrete examples are constructed to justify the effectiveness of the criterion.
基金the National Natural Science Foundation of China (Grant No.10562002) the Natural Science Foundation of Inner Mongolia (Grant Nos.200508010103,200711020106)
文摘This paper deals with the structure of the spectrum of infinite dimensional Hamiltonian operators.It is shown that the spectrum,the union of the point spectrum and residual spectrum,and the continuous spectrum are all symmetric with respect to the imaginary axis of the complex plane. Moreover,it is proved that the residual spectrum does not contain any pair of points symmetric with respect to the imaginary axis;and a complete characterization of the residual spectrum in terms of the point spectrum is then given.As applications of these structure results,we obtain several necessary and sufficient conditions for the residual spectrum of a class of infinite dimensional Hamiltonian operators to be empty.
基金Supported by the National Natural Science Foundation of China under Grant No. 10962004the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20070126002+1 种基金the Natural Science Foundation of Inner Mongolia under Grant No. 20080404MS0104the Research Foundation for Talented Scholars of Inner Mongolia University under Grant No. 207066
文摘For the off-diagonal infinite dimensional Hamiltonian operators, which have at most countable eigenvalues, a necessary and sufficient condition of the eigenfunction systems to be complete in the sense of Cauchy principal value is presented by using the spectral symmetry and new orthogonal relationship of the operators. Moreover, the above result is extended to a more general case. At last, the completeness of eigenfunction systems for the operators arising from the isotropic plane magnetoelectroelastic solids is described to illustrate the effectiveness of the criterion. The whole results offer theoretical guarantee for separation of variables in Hamiltonian system for some mechanics equations.
基金the National Natural Science Foundation of China under Grant No.50579022the Foundation of Pre-973 Program of China under Grant No.2004CCA02500+1 种基金the SRF for the ROCS,SEMthe Talent Recruitment Foundation of HUST
文摘Given a real-valued separable M-type 2 Banach space X with the p-th power of the norm of C2-class, the almost sure asymptotic upper bounds of the solutions of the Ornstein-Uhlenbeck Processes described by the following equations {dz(t)=A(t,z(t))dt+σ(t,z(t))dW(t),z(0)=z0∈X,are investigated. This study generalizes the corresponding well-known finite dimensional results of Lapeyre (1989) and Mao (1992).
基金supported by National Natural Science Foundation of China (Grant No. 11321101)
文摘In this paper,we construct certain irreducible infinite dimensional representations of algebraic groups with Frobenius maps.In particular,a few classical results of Steinberg and Deligne&Lusztig on complex representations of finite groups of Lie type are extended to reductive algebraic groups with Frobenius maps.
基金Supported by the National Natural Science FundZhejiang Provincial Natural Science Foundation.
文摘In this paper, we gave a proof for the local continuity modulus theorem of the Wiener process, i. e., lim t→0 sup 0≤s≤t |W(s)|/(2s log log(1/s))<sup>1/2</sup>=1 a.s. This result was given by Csrg and Revesz (1981), but the proof gets them nowhere. We also gave a similar local continuity modulus result for the infinite dimensional OU processes.
基金Foundation item: the National Natural Science Foundation of China (No. 10562002) the Natural Science Foundation of Inner Mongolia (Nos. 200508010103+2 种基金 200711020106) the Specialized Research Fund of the Doctoral Program of Higher Education of China (No. 20070126002) Research Foundation for Talented Scholars of Inner Mongolia University (No. 206029).
文摘In this paper, by using characterization of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H, a necessary and sufficient condition is obtained on the symmetry of σP(A) and σ1/P(-A^*) with respect to the imaginary axis. Then the symmetry of the point spectrum of H is given, and several examples are presented to illustrate the results.
基金supported by the Innovation Foundation of BUAA for PhD Graduates and the National Natural Science Foundation of China under grant 61271010.
文摘An explicit differencescheme is described,analyzed and tested for numer-ically approximating stochastic elastic equation driven by infinite dimensional noise.The noise processes are approximated by piecewise constant random processes and the integral formula of the stochastic elastic equation is approximated by a truncated series.Error analysis of the numerical method yields estimate of convergence rate.The rate of convergence is demonstrated with numerical experiments.
基金Supported by the National Natural Science Foundation of China (No. 11061019, 10962004, 11101200)the Chunhui Program of Ministry of Education of China (No. Z2009-1-01010)+1 种基金the Natural Science Foundation of Inner Mongolia (No. 2010MS0110, 2009BS0101)the Cultivation of Innovative Talent of ‘211 Project’ of Inner Mongolia University
文摘This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H. Note that the point spectrum of H can be described as σp(H) = σp (A) U σp1 (-A*). Using the characteristic of the set σp1(-A*), we divide the point spectrum σp (d) of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which σp1(-A*) and one part of σp(A) are symmetric with respect to the real axis each other. Based on this result, the symmetry of σp(H) is completely given. Moreover, the above result is applied to thin plates on elastic foundation, plane elasticity problems and harmonic equations.
基金This work was partially supported by the National Natural Science Foundation of China(Grant Nos.10171059,10371084&10525105)the National Natural Science Foundation of Spain(Grant No.MTM2005-00714)+1 种基金the Program for New Century Excellent Talents in University of China(Grant No.NCET-04-0882)the Foundation for the Author of National Excellent Doctoral Dissertation of China(Grant Nos.200119&200218).
文摘This paper is addressed to develop an approximate method to solve a class of infinite dimensional LQ optimal regulator problems over infinite time horizon. Our algorithm is based on a construction of approximate solutions which solve some finite dimensional LQ optimal regulator problems over finite time horizon, and it is shown that these approximate solutions converge strongly to the desired solution in the double limit sense.
基金supported by the National Natural Science Foundation of China under Grant No.61273090
文摘This paper deals with the output regulation problem for a class of SISO infinite dimensional systems with an uncertain exosystem.For these systems,a concept of relative degree is firstly introduced and used to construct a transformation which leads to the canonical form of output feedback systems.Then,based on this canonical form,by means of an internal model and a recursive adaptive control,the authors obtain an adaptive regulator which solves the problem.It should be pointed out that the proposed regulator is finite dimensional while it is usually infinite dimensional in existing literatures.
文摘Let X be the solution of the SDE:dX_t=σ(X_t)dB_t+b(X_t)dt,with σ and b ∈C_B~∞(R) such that σ≥λ>0 for some constant λ,and B a real Brownian motion.Let μ be the law of X on E=C([0,1],R)and k ∈E~*-{0},where E~* is the topological dual space of E.Consider the classical form:where u and v are smooth functions on E.We prove that,if is closable for any k in a dense subset of E~* and if the smooth functions are contained in the domain of the generator of the closure of ε_k,σ must be a constant function.
基金This project is supported by the National Natural Science Foundation of China
文摘In this paper we study the infinite dimensional Malliavin calculus, and apply it to determingwhen the solution of an infinite dimensional stochastic differential equation has the property thatits finite dimensional distributions possess smooth density.