In this paper we introduce the concept of pairwise singular sets and pairwise singular maps between pairwise locally compact and pairwise hausdorff spaces and study the properties of pairwise singular maps.
New Sobolev type embeddings for some weighted Banach spaces are established. Using such embeddings and the singular positive radial entire solutions, we construct singular positive weak solutions with a prescribed sin...New Sobolev type embeddings for some weighted Banach spaces are established. Using such embeddings and the singular positive radial entire solutions, we construct singular positive weak solutions with a prescribed singular set for a weighted elliptic equation. Our main results in this paper also provide positive weak solutions with a prescribed singular set to an equation with Hardy potential.展开更多
The author proves that the on the singular set of a local solution to existence of an optimal control problem. right-hand term of a p-Laplace equation is zero the equation. Such a result is used to study the
Let X∈Alex^(n)(−1)be an n-dimensional Alexandrov space with curvature≥−1.Let the r-scale(k,ε)-singular set S_(ε,r)^(k)(X)be the collection of x∈X so that B_(r)(x)is notr-close to a ball in any splitting spaceℝ^(k...Let X∈Alex^(n)(−1)be an n-dimensional Alexandrov space with curvature≥−1.Let the r-scale(k,ε)-singular set S_(ε,r)^(k)(X)be the collection of x∈X so that B_(r)(x)is notr-close to a ball in any splitting spaceℝ^(k+1)×Z.We show that there exists C(n,ε)>0 and𝛽(n,ε)>0,independent of the volume,so that for any disjoint collection{B_(ri)(xi)∶x_(i)∈S_(ε,βri)^(k)(X)∩B_(1),r_(i)≤1,the packing estimateΣr_(i)^(k)≤C holds.Consequently,we obtain the Hausdorff measure estimates H^(k)(S_(ε)^(k)(X))∩B_(1))≤C and H^(n)(B_(r)(S_(ε,βri)^(k))∩B_(1))≤C rn−k.This answers an open question in Kapovitch et al.(Metric-measure boundary and geodesic flow on Alexandrov spaces.arXiv:1705.04767(2017)).We also show that the k-singular set S^(k)(X)=⋃ε>0⋂r>0 S_(ε,r)^(k)𝜖,ris k-rectifiable and construct examples to show that such a structure is sharp.For instance,in the k=1 case we can build for any closed set T⊆S^(1)andε>0 a space Y∈Alex^(3)(0)with S_(ε)^(1)(Y)=Ф(T),whereФ∶S^(1)→Y is a bi-Lipschitz embedding.Taking T to be a Cantor set it gives rise to an example where the singular set is a 1-rectifiable,1-Cantor set with positive 1-Hausdorff measure.展开更多
【目的】为提高轮腿式移动机器人的地形适应能力,简化驱动与轮腿模式切换的控制,提出一种零耦合度且部分运动解耦的并联式变拓扑机械腿。【方法】首先,基于变拓扑机构的结构组成原理,设计了一种变拓扑运动副;其次,基于方位特征(Position...【目的】为提高轮腿式移动机器人的地形适应能力,简化驱动与轮腿模式切换的控制,提出一种零耦合度且部分运动解耦的并联式变拓扑机械腿。【方法】首先,基于变拓扑机构的结构组成原理,设计了一种变拓扑运动副;其次,基于方位特征(Position and Orientation Characteristics, POC)集的拓扑设计理论,设计并分析了变拓扑机械腿;再次,基于拓扑特征运动学方法,求解出机械腿的符号式位置正解与逆解,并对机械腿的工作空间与奇异性进行分析;最后,由变拓扑机械腿构造了四足机器人,在陡坡和沟壑地形下进行了步态规划并测试其越障能力。【结果】变拓扑机械腿具有轮腿两种运动模式,在腿模式下的POC集为2R1T,在轮模式下的POC集为3T1R;基于变拓扑运动副实现的运动模式切换可使模式切换所需时间短、控制简单;越障测试表明,变拓扑机械腿对于陡坡和沟壑等复杂地形具有较强的通过能力。展开更多
In order to overcome the difficulties caused by singular optima, in the present paper, a new method for the solutions of structural topology optimization problems is proposed. The distinctive feature of this method is...In order to overcome the difficulties caused by singular optima, in the present paper, a new method for the solutions of structural topology optimization problems is proposed. The distinctive feature of this method is that instead of solving the original optimization problem directly, we turn to seeking the solutions of a sequence of approximated problems which are formulated by relaxing the constraints of the original problem to some extent. The approximated problem can be solved efficiently by employing the algorithms developed for sizing optimization problems because its solution is not singular. It can also be proved that when the relaxation parameter is tending to zero, the solution of the approximated problem will converge to the solution of the original problem uniformly. Numerical examples illustrate the effectiveness and validity of the present approach. Results are also compared with those obtained by traditional methods.展开更多
文摘In this paper we introduce the concept of pairwise singular sets and pairwise singular maps between pairwise locally compact and pairwise hausdorff spaces and study the properties of pairwise singular maps.
基金supported by National Natural Science Foundation of China (Grant Nos. 11171092 and 11571093)
文摘New Sobolev type embeddings for some weighted Banach spaces are established. Using such embeddings and the singular positive radial entire solutions, we construct singular positive weak solutions with a prescribed singular set for a weighted elliptic equation. Our main results in this paper also provide positive weak solutions with a prescribed singular set to an equation with Hardy potential.
基金the National Natural Science Foundation of China (No. 10671040) the Foundationfor the Author of National Excellent Doctoral Dissertation of China (No. 200522)the Program forNew Century Excellent Talents in University of China (No. 06-0359)
文摘The author proves that the on the singular set of a local solution to existence of an optimal control problem. right-hand term of a p-Laplace equation is zero the equation. Such a result is used to study the
文摘Let X∈Alex^(n)(−1)be an n-dimensional Alexandrov space with curvature≥−1.Let the r-scale(k,ε)-singular set S_(ε,r)^(k)(X)be the collection of x∈X so that B_(r)(x)is notr-close to a ball in any splitting spaceℝ^(k+1)×Z.We show that there exists C(n,ε)>0 and𝛽(n,ε)>0,independent of the volume,so that for any disjoint collection{B_(ri)(xi)∶x_(i)∈S_(ε,βri)^(k)(X)∩B_(1),r_(i)≤1,the packing estimateΣr_(i)^(k)≤C holds.Consequently,we obtain the Hausdorff measure estimates H^(k)(S_(ε)^(k)(X))∩B_(1))≤C and H^(n)(B_(r)(S_(ε,βri)^(k))∩B_(1))≤C rn−k.This answers an open question in Kapovitch et al.(Metric-measure boundary and geodesic flow on Alexandrov spaces.arXiv:1705.04767(2017)).We also show that the k-singular set S^(k)(X)=⋃ε>0⋂r>0 S_(ε,r)^(k)𝜖,ris k-rectifiable and construct examples to show that such a structure is sharp.For instance,in the k=1 case we can build for any closed set T⊆S^(1)andε>0 a space Y∈Alex^(3)(0)with S_(ε)^(1)(Y)=Ф(T),whereФ∶S^(1)→Y is a bi-Lipschitz embedding.Taking T to be a Cantor set it gives rise to an example where the singular set is a 1-rectifiable,1-Cantor set with positive 1-Hausdorff measure.
文摘【目的】为提高轮腿式移动机器人的地形适应能力,简化驱动与轮腿模式切换的控制,提出一种零耦合度且部分运动解耦的并联式变拓扑机械腿。【方法】首先,基于变拓扑机构的结构组成原理,设计了一种变拓扑运动副;其次,基于方位特征(Position and Orientation Characteristics, POC)集的拓扑设计理论,设计并分析了变拓扑机械腿;再次,基于拓扑特征运动学方法,求解出机械腿的符号式位置正解与逆解,并对机械腿的工作空间与奇异性进行分析;最后,由变拓扑机械腿构造了四足机器人,在陡坡和沟壑地形下进行了步态规划并测试其越障能力。【结果】变拓扑机械腿具有轮腿两种运动模式,在腿模式下的POC集为2R1T,在轮模式下的POC集为3T1R;基于变拓扑运动副实现的运动模式切换可使模式切换所需时间短、控制简单;越障测试表明,变拓扑机械腿对于陡坡和沟壑等复杂地形具有较强的通过能力。
基金The project supported by the National Natural Science Foundation of China under project No.19572023
文摘In order to overcome the difficulties caused by singular optima, in the present paper, a new method for the solutions of structural topology optimization problems is proposed. The distinctive feature of this method is that instead of solving the original optimization problem directly, we turn to seeking the solutions of a sequence of approximated problems which are formulated by relaxing the constraints of the original problem to some extent. The approximated problem can be solved efficiently by employing the algorithms developed for sizing optimization problems because its solution is not singular. It can also be proved that when the relaxation parameter is tending to zero, the solution of the approximated problem will converge to the solution of the original problem uniformly. Numerical examples illustrate the effectiveness and validity of the present approach. Results are also compared with those obtained by traditional methods.
