This work reports on the author's recent study about regularity and the singular set of a C 1 smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg group.As a differential equation...This work reports on the author's recent study about regularity and the singular set of a C 1 smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg group.As a differential equation,this is a degenerate hyperbolic and elliptic PDE of second order,arising from the study of CR geometry.Assuming only the p-mean curvature H ∈ C 0,it is shown that any characteristic curve is C 2 smooth and its (line) curvature equals-H.By introducing special coordinates and invoking the jump formulas along characteristic curves,it is proved that the Legendrian (horizontal) normal gains one more derivative.Therefore the seed curves are C 2 smooth.This work also obtains the uniqueness of characteristic and seed curves passing through a common point under some mild conditions,respectively.In an on-going project,it is shown that the p-area element is in fact C 2 smooth along any characteristic curve and satisfies a certain ordinary differential equation of second order.Moreover,this ODE is analyzed to study the singular set.展开更多
In this paper, we establish the first variational formula and its Euler-Lagrange equation for the total 2p-th mean curvature functional .M2p of a submanifold Mn in a general Riemannian manifold gn^n＋m for p = 0, 1,.....In this paper, we establish the first variational formula and its Euler-Lagrange equation for the total 2p-th mean curvature functional .M2p of a submanifold Mn in a general Riemannian manifold gn^n＋m for p = 0, 1,..., [n/2]. As an example, we prove that closed complex submanifolds in complex projective spaces are critical points of the functional M2p, called relatively 2p-minimal submanifolds, for all p. At last, we discuss the relations between relatively 2p-minimal submanifoIds and austere submanifolds in real space forms, as well as a special variational problem.展开更多
In this paper, a sufficient condition is obtained to ensure the stable recovery(ε≠ 0) or exact recovery(ε = 0) of all r-rank matrices X ∈ Rm×nfrom b = A(X) + z via nonconvex Schatten p-minimization for anyδ4...In this paper, a sufficient condition is obtained to ensure the stable recovery(ε≠ 0) or exact recovery(ε = 0) of all r-rank matrices X ∈ Rm×nfrom b = A(X) + z via nonconvex Schatten p-minimization for anyδ4r∈ [3～(1/2))2, 1). Moreover, we determine the range of parameter p with any given δ4r∈ [(3～(1/2))/22, 1). In fact, for any given δ4r∈ [3～(1/2))2, 1), p ∈(0, 2(1- δ4r)] suffices for the stable recovery or exact recovery of all r-rank matrices.展开更多
Recently,the l_(p)minimization problem(p∈(0,1))for sparse signal recovery has been studied a lot because of its efficiency.In this paper,we propose a general smoothing algorithmic framework based on the entropy funct...Recently,the l_(p)minimization problem(p∈(0,1))for sparse signal recovery has been studied a lot because of its efficiency.In this paper,we propose a general smoothing algorithmic framework based on the entropy function for solving a class of l_(p)minimization problems,which includes the well-known unconstrained l_(2)-l_(p)problem as a special case.We show that any accumulation point of the sequence generated by the proposed algorithm is a stationary point of the l_(p)minimization problem,and derive a lower bound for the nonzero entries of the stationary point of the smoothing problem.We implement a specific version of the proposed algorithm which indicates that the entropy function-based algorithm is effective.展开更多
Weighted ■_(p)(0<p<l)minimization has been extensively studied as an effective way to reconstruct a sparse signal from compressively sampled measurements when some prior support information of the signal is ava...Weighted ■_(p)(0<p<l)minimization has been extensively studied as an effective way to reconstruct a sparse signal from compressively sampled measurements when some prior support information of the signal is available.In this paper,we consider the recovery guarantees of Κ-sparse signals via the weighted ■_(p)(0<P<1)minimization when arbitrarily many support priors are given.Our analysis enables an extension to existing works that assume only a single support prior is used.展开更多
Accurate reconstruction from a reduced data set is highly essential for computed tomography in fast and/or low dose imaging applications. Conventional total variation（TV）-based algorithms apply the L1 norm-based pen...Accurate reconstruction from a reduced data set is highly essential for computed tomography in fast and/or low dose imaging applications. Conventional total variation（TV）-based algorithms apply the L1 norm-based penalties, which are not as efficient as Lp（0〈p〈1） quasi-norm-based penalties. TV with a p-th power-based norm can serve as a feasible alternative of the conventional TV, which is referred to as total p-variation（TpV）. This paper proposes a TpV-based reconstruction model and develops an efficient algorithm. The total p-variation and Kullback-Leibler（KL） data divergence, which has better noise suppression capability compared with the often-used quadratic term, are combined to build the reconstruction model. The proposed algorithm is derived by the alternating direction method（ADM） which offers a stable, efficient, and easily coded implementation. We apply the proposed method in the reconstructions from very few views of projections（7 views evenly acquired within 180°）. The images reconstructed by the new method show clearer edges and higher numerical accuracy than the conventional TV method. Both the simulations and real CT data experiments indicate that the proposed method may be promising for practical applications.展开更多
基金supported by the "Science Council" of Taiwan 11529,China (Grant No. 97-2115-M-001-016-MY3)
文摘This work reports on the author's recent study about regularity and the singular set of a C 1 smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg group.As a differential equation,this is a degenerate hyperbolic and elliptic PDE of second order,arising from the study of CR geometry.Assuming only the p-mean curvature H ∈ C 0,it is shown that any characteristic curve is C 2 smooth and its (line) curvature equals-H.By introducing special coordinates and invoking the jump formulas along characteristic curves,it is proved that the Legendrian (horizontal) normal gains one more derivative.Therefore the seed curves are C 2 smooth.This work also obtains the uniqueness of characteristic and seed curves passing through a common point under some mild conditions,respectively.In an on-going project,it is shown that the p-area element is in fact C 2 smooth along any characteristic curve and satisfies a certain ordinary differential equation of second order.Moreover,this ODE is analyzed to study the singular set.
基金supported by National Natural Science Foundation of China(Grant No.11001016)the Specialized Research Fund for the Doctoral Program of Higher Education(Grant No. 20100003120003)the Program for Changjiang Scholars and Innovative Research Team in University
文摘In this paper, we establish the first variational formula and its Euler-Lagrange equation for the total 2p-th mean curvature functional .M2p of a submanifold Mn in a general Riemannian manifold gn^n＋m for p = 0, 1,..., [n/2]. As an example, we prove that closed complex submanifolds in complex projective spaces are critical points of the functional M2p, called relatively 2p-minimal submanifolds, for all p. At last, we discuss the relations between relatively 2p-minimal submanifoIds and austere submanifolds in real space forms, as well as a special variational problem.
基金supported by National Natural Science Foundation of China(Grant Nos.11271050 and 11371183)Beijing Center for Mathematics and Information Interdisciplinary Sciences
文摘In this paper, a sufficient condition is obtained to ensure the stable recovery(ε≠ 0) or exact recovery(ε = 0) of all r-rank matrices X ∈ Rm×nfrom b = A(X) + z via nonconvex Schatten p-minimization for anyδ4r∈ [3～(1/2))2, 1). Moreover, we determine the range of parameter p with any given δ4r∈ [(3～(1/2))/22, 1). In fact, for any given δ4r∈ [3～(1/2))2, 1), p ∈(0, 2(1- δ4r)] suffices for the stable recovery or exact recovery of all r-rank matrices.
基金supported by the National Natural Science Foundation of China(Nos.11171252,11431002).
文摘Recently,the l_(p)minimization problem(p∈(0,1))for sparse signal recovery has been studied a lot because of its efficiency.In this paper,we propose a general smoothing algorithmic framework based on the entropy function for solving a class of l_(p)minimization problems,which includes the well-known unconstrained l_(2)-l_(p)problem as a special case.We show that any accumulation point of the sequence generated by the proposed algorithm is a stationary point of the l_(p)minimization problem,and derive a lower bound for the nonzero entries of the stationary point of the smoothing problem.We implement a specific version of the proposed algorithm which indicates that the entropy function-based algorithm is effective.
基金supported by the NSF of China(Nos.11871109,11901037 and 11801509)NSAF(Grant No.U1830107),CAEP Foundation(Grant No.CX20200027).
文摘Weighted ■_(p)(0<p<l)minimization has been extensively studied as an effective way to reconstruct a sparse signal from compressively sampled measurements when some prior support information of the signal is available.In this paper,we consider the recovery guarantees of Κ-sparse signals via the weighted ■_(p)(0<P<1)minimization when arbitrarily many support priors are given.Our analysis enables an extension to existing works that assume only a single support prior is used.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61372172 and 61601518)
文摘Accurate reconstruction from a reduced data set is highly essential for computed tomography in fast and/or low dose imaging applications. Conventional total variation（TV）-based algorithms apply the L1 norm-based penalties, which are not as efficient as Lp（0〈p〈1） quasi-norm-based penalties. TV with a p-th power-based norm can serve as a feasible alternative of the conventional TV, which is referred to as total p-variation（TpV）. This paper proposes a TpV-based reconstruction model and develops an efficient algorithm. The total p-variation and Kullback-Leibler（KL） data divergence, which has better noise suppression capability compared with the often-used quadratic term, are combined to build the reconstruction model. The proposed algorithm is derived by the alternating direction method（ADM） which offers a stable, efficient, and easily coded implementation. We apply the proposed method in the reconstructions from very few views of projections（7 views evenly acquired within 180°）. The images reconstructed by the new method show clearer edges and higher numerical accuracy than the conventional TV method. Both the simulations and real CT data experiments indicate that the proposed method may be promising for practical applications.
