A kind of cone separation theorems is established, by which the extension theorems for cone linear continuous operators are developed. As an application, the extension theorem for positive linear continuous operators ...A kind of cone separation theorems is established, by which the extension theorems for cone linear continuous operators are developed. As an application, the extension theorem for positive linear continuous operators is given.展开更多
In this paper classical linear elastic variational principles are systematically derivedfrom the reciprocal theorem and mixed variational principles of variations of boundaryconditions are given.
In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations （△-e/et）u（x,t）＋q（x,t）u^p（x,t）=0 and nonlinear parabolic equations （△-e/et）u（x,...In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations （△-e/et）u（x,t）＋q（x,t）u^p（x,t）=0 and nonlinear parabolic equations （△-e/et）u（x,t）＋h（x,t）u^p（x,t）=0（p 〉 1） on Riemannian manifolds.As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang （[1], Bull. London Math. Soc. 38（2006）, 1045-1053） and the author （[2], Nonlinear Anal. 74 （2011）, 5141-5146）.展开更多
This article considers the equation △2u = f（x,u）with boundary conditions either u｜aΩ = au/an｜aΩ = 0 or u｜aΩ = △u｜aΩ = 0, where f（x, t） is asymptotically linear with respect to t at infinity, and Ω is a ...This article considers the equation △2u = f（x,u）with boundary conditions either u｜aΩ = au/an｜aΩ = 0 or u｜aΩ = △u｜aΩ = 0, where f（x, t） is asymptotically linear with respect to t at infinity, and Ω is a smooth bounded domain in R^N, N 〉 4. By a variant version of Mountain Pass Theorem, it is proved that the above problems have a nontrivial solution under suitable assumptions of f（x, t）.展开更多
New objects characterizing the structure of complex linear transformations areintroduced. These new objects yield a new result for the decomposition of complexvector spaces relative to complex lrnear transformations a...New objects characterizing the structure of complex linear transformations areintroduced. These new objects yield a new result for the decomposition of complexvector spaces relative to complex lrnear transformations and provide all Jordan basesby which the Jordan canonical form is constructed. Accordingly, they can result in thecelebrated Jordan theorem and the third decomposition theorem of space directly. and,moreover, they can give a new deep insight into the exquisite and subtle structure ofthe Jordan form. The latter indicates that the Jordan canonical form of a complexlinear transformation is an invariant structure associated with double arbitrary. choices.展开更多
Wigner theorem is the cornerstone of the mathematical formula of quan-tum mechanics, it has promoted the research of basic theory of quantum mechanics. In this article, we give a certain pair of functional equations b...Wigner theorem is the cornerstone of the mathematical formula of quan-tum mechanics, it has promoted the research of basic theory of quantum mechanics. In this article, we give a certain pair of functional equations between two real spaces s or two real sn(H), that we called “phase isometry”. It is obtained that all such solutions are phase equivalent to real linear isometries in the space s and the space sn(H).展开更多
The aim of this brief paper is to give several results concerning the regional controllability of distributed systems governed by semi-linear parabolic equations. We concentrate on the determination of a control achie...The aim of this brief paper is to give several results concerning the regional controllability of distributed systems governed by semi-linear parabolic equations. We concentrate on the determination of a control achieving internal and boundary regional controllability. The approach is based on an extension of the Hilbert Uniqueness Method (HUM) and Schauder’s fixed point theorem. We give a numerical example developed in internal and boundary sub region. These numerical illustrations show the efficiency of the approach and lead to conjectures.展开更多
用5个定理给出最小一乘线性回归的相关性质,为其工程应用奠定了基础。文中首先证明了“由“最小一乘”准则确定的直线y=b1x1+ b2x2经过其两个样本点”以及“由最小一乘准则确定的直线y=b1x1+ b2x2+a经过其三个样本点”。然后应用数学归...用5个定理给出最小一乘线性回归的相关性质,为其工程应用奠定了基础。文中首先证明了“由“最小一乘”准则确定的直线y=b1x1+ b2x2经过其两个样本点”以及“由最小一乘准则确定的直线y=b1x1+ b2x2+a经过其三个样本点”。然后应用数学归纳法得到如下定理:设有n(n>P)个样本点(x1i, x2i, ? xP i, yi,),则由最小一乘准则确定的线性非奇次模型y=b1x1+b2x2+?bPxP+a经过其P+1个样本点,而相应的奇次模型必经过其P个样本点。通过大量工程实例证实了最小一乘具有较强的稳健性,同时也证实了定理的正确性。展开更多
文摘A kind of cone separation theorems is established, by which the extension theorems for cone linear continuous operators are developed. As an application, the extension theorem for positive linear continuous operators is given.
文摘In this paper classical linear elastic variational principles are systematically derivedfrom the reciprocal theorem and mixed variational principles of variations of boundaryconditions are given.
基金supported by the National Science Foundation of China(41275063 and 11401575)
文摘In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations （△-e/et）u（x,t）＋q（x,t）u^p（x,t）=0 and nonlinear parabolic equations （△-e/et）u（x,t）＋h（x,t）u^p（x,t）=0（p 〉 1） on Riemannian manifolds.As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang （[1], Bull. London Math. Soc. 38（2006）, 1045-1053） and the author （[2], Nonlinear Anal. 74 （2011）, 5141-5146）.
基金This work was supported by NSFC(10571174,10631030)and CAS(KJCX3-SYW-S03)
文摘This article considers the equation △2u = f（x,u）with boundary conditions either u｜aΩ = au/an｜aΩ = 0 or u｜aΩ = △u｜aΩ = 0, where f（x, t） is asymptotically linear with respect to t at infinity, and Ω is a smooth bounded domain in R^N, N 〉 4. By a variant version of Mountain Pass Theorem, it is proved that the above problems have a nontrivial solution under suitable assumptions of f（x, t）.
文摘New objects characterizing the structure of complex linear transformations areintroduced. These new objects yield a new result for the decomposition of complexvector spaces relative to complex lrnear transformations and provide all Jordan basesby which the Jordan canonical form is constructed. Accordingly, they can result in thecelebrated Jordan theorem and the third decomposition theorem of space directly. and,moreover, they can give a new deep insight into the exquisite and subtle structure ofthe Jordan form. The latter indicates that the Jordan canonical form of a complexlinear transformation is an invariant structure associated with double arbitrary. choices.
文摘Wigner theorem is the cornerstone of the mathematical formula of quan-tum mechanics, it has promoted the research of basic theory of quantum mechanics. In this article, we give a certain pair of functional equations between two real spaces s or two real sn(H), that we called “phase isometry”. It is obtained that all such solutions are phase equivalent to real linear isometries in the space s and the space sn(H).
文摘The aim of this brief paper is to give several results concerning the regional controllability of distributed systems governed by semi-linear parabolic equations. We concentrate on the determination of a control achieving internal and boundary regional controllability. The approach is based on an extension of the Hilbert Uniqueness Method (HUM) and Schauder’s fixed point theorem. We give a numerical example developed in internal and boundary sub region. These numerical illustrations show the efficiency of the approach and lead to conjectures.
文摘用5个定理给出最小一乘线性回归的相关性质,为其工程应用奠定了基础。文中首先证明了“由“最小一乘”准则确定的直线y=b1x1+ b2x2经过其两个样本点”以及“由最小一乘准则确定的直线y=b1x1+ b2x2+a经过其三个样本点”。然后应用数学归纳法得到如下定理:设有n(n>P)个样本点(x1i, x2i, ? xP i, yi,),则由最小一乘准则确定的线性非奇次模型y=b1x1+b2x2+?bPxP+a经过其P+1个样本点,而相应的奇次模型必经过其P个样本点。通过大量工程实例证实了最小一乘具有较强的稳健性,同时也证实了定理的正确性。
