In this paper, we discuss the relations between the 2-harmornic totally real submsnifold and the minimal totall real submanifold in the complex protective spsace, and obtain the pinching conductions for the second fu...In this paper, we discuss the relations between the 2-harmornic totally real submsnifold and the minimal totall real submanifold in the complex protective spsace, and obtain the pinching conductions for the second fundamental form and the Rieci curature of the 2-harmornic totally real submanifold in the complex projective space.展开更多
In the present paper, the authors study totally real 2-harmonic submanifolds in a quasi constant holomorphic sectional curvature space and obtain a Simons＇ type inte- gral inequality of compact submanifoids as well a...In the present paper, the authors study totally real 2-harmonic submanifolds in a quasi constant holomorphic sectional curvature space and obtain a Simons＇ type inte- gral inequality of compact submanifoids as well as some pinching theorems on＇the second fundamental form.展开更多
In the present paper,the authors study totally real 2-harmonic submanifolds in a complex space form and obtain a Simons' type integral inequality of compact submanifolds as well as some relevant conclusions.
An extension theorem for holomorphic functions with L^2 growth condition is strengthened for the case of the extension from hypersurfaces with isolated singularities.
Abstract This paper studies the problem of minimizing a homogeneous polynomial （form） f（x） over the unit sphere Sn-1 = {x ∈ R^n： ｜｜X｜｜2 = 1}. The problem is NP-hard when f（x） has degree 3 or higher. Denote...Abstract This paper studies the problem of minimizing a homogeneous polynomial （form） f（x） over the unit sphere Sn-1 = {x ∈ R^n： ｜｜X｜｜2 = 1}. The problem is NP-hard when f（x） has degree 3 or higher. Denote by fmin （resp. fmax） the minimum （resp. maximum） value of f（x） on S^n-1. First, when f（x） is an even form of degree 2d, we study the standard sum of squares （SOS） relaxation for finding a lower bound of the minimum .fmin ：max γ s.t.f（x）-γ.｜｜x｜｜2^2d is SOS.Let fos be be the above optimal value. Then we show that for all n ≥ 2d,Here, the constant C（d） is independent of n. Second, when f（x） is a multi-form and ^-1 becomes a muilti-unit sphere, we generalize the above SOS relaxation and prove a similar bound. Third, when f（x） is sparse, we prove an improved bound depending on its sparsity pattern; when f（x） is odd, we formulate the problem equivalently as minimizing a certain even form, and prove a similar bound. Last, for minimizing f（x） over a hypersurface H（g） = {x E lRn： g（x） = 1} defined by a positive definite form g（x）, we generalize the above SOS relaxation and prove a similar bound.展开更多
文摘In this paper, we discuss the relations between the 2-harmornic totally real submsnifold and the minimal totall real submanifold in the complex protective spsace, and obtain the pinching conductions for the second fundamental form and the Rieci curature of the 2-harmornic totally real submanifold in the complex projective space.
基金Foundation item： Supported by the National Natural Science Foundation of China（ll071005） Supported by the Natural Science Foundation of Anhui Province Education Department（KJ2008A05zC）
文摘In the present paper, the authors study totally real 2-harmonic submanifolds in a quasi constant holomorphic sectional curvature space and obtain a Simons＇ type inte- gral inequality of compact submanifoids as well as some pinching theorems on＇the second fundamental form.
基金Natural Science Foundation of Education Department of Anhui Province （No. 2004kj166zd）.
文摘In the present paper,the authors study totally real 2-harmonic submanifolds in a complex space form and obtain a Simons' type integral inequality of compact submanifolds as well as some relevant conclusions.
文摘An extension theorem for holomorphic functions with L^2 growth condition is strengthened for the case of the extension from hypersurfaces with isolated singularities.
文摘Abstract This paper studies the problem of minimizing a homogeneous polynomial （form） f（x） over the unit sphere Sn-1 = {x ∈ R^n： ｜｜X｜｜2 = 1}. The problem is NP-hard when f（x） has degree 3 or higher. Denote by fmin （resp. fmax） the minimum （resp. maximum） value of f（x） on S^n-1. First, when f（x） is an even form of degree 2d, we study the standard sum of squares （SOS） relaxation for finding a lower bound of the minimum .fmin ：max γ s.t.f（x）-γ.｜｜x｜｜2^2d is SOS.Let fos be be the above optimal value. Then we show that for all n ≥ 2d,Here, the constant C（d） is independent of n. Second, when f（x） is a multi-form and ^-1 becomes a muilti-unit sphere, we generalize the above SOS relaxation and prove a similar bound. Third, when f（x） is sparse, we prove an improved bound depending on its sparsity pattern; when f（x） is odd, we formulate the problem equivalently as minimizing a certain even form, and prove a similar bound. Last, for minimizing f（x） over a hypersurface H（g） = {x E lRn： g（x） = 1} defined by a positive definite form g（x）, we generalize the above SOS relaxation and prove a similar bound.
