Let S be a finite linear space, and let G be a group of automorphisms of S. If G is soluble and line-transitive, then for a given k but a finite number of pairs of ( S, G), S has v= pn points and G≤AΓ L(1,pn).
In this paper, we introduce the m-Cartan matrix and observe that some properties of the quadratic form associated to the Cartan matrix of an Euclidean diagram can be generalized to the m-Cartan matrix of a McKay quive...In this paper, we introduce the m-Cartan matrix and observe that some properties of the quadratic form associated to the Cartan matrix of an Euclidean diagram can be generalized to the m-Cartan matrix of a McKay quiver. We also describe the McKay quiver for a finite abelian subgroup of a special linear group.展开更多
Diffraction intensities of the 3D ptychographic iterative engine(3PIE)were written as a set of linear equations of the selfcorrelations of Fourier components of all sample slices,and an effective computing method was ...Diffraction intensities of the 3D ptychographic iterative engine(3PIE)were written as a set of linear equations of the selfcorrelations of Fourier components of all sample slices,and an effective computing method was developed to solve these linear equations for the transmission functions of all sample slices analytically.With both theoretical analysis and numerical simulations,this study revealed the underlying physics and mathematics of 3PIE and demonstrated for the first time,to our knowledge,that 3PIE can generate mathematically unique reconstruction even with noisy data.展开更多
We consider a class of nonsmooth convex optimization problems where the objective function is the composition of a strongly convex differentiable function with a linear mapping,regularized by the sum of both l1-norm a...We consider a class of nonsmooth convex optimization problems where the objective function is the composition of a strongly convex differentiable function with a linear mapping,regularized by the sum of both l1-norm and l2-norm of the optimization variables.This class of problems arise naturally from applications in sparse group Lasso,which is a popular technique for variable selection.An effective approach to solve such problems is by the Proximal Gradient Method(PGM).In this paper we prove a local error bound around the optimal solution set for this problem and use it to establish the linear convergence of the PGM method without assuming strong convexity of the overall objective function.展开更多
文摘Let S be a finite linear space, and let G be a group of automorphisms of S. If G is soluble and line-transitive, then for a given k but a finite number of pairs of ( S, G), S has v= pn points and G≤AΓ L(1,pn).
基金supported by National Natural Science Foundation of China (Grant No. 10671061)theResearch Foundation for Doctor Programme (Grant No. 200505042004)
文摘In this paper, we introduce the m-Cartan matrix and observe that some properties of the quadratic form associated to the Cartan matrix of an Euclidean diagram can be generalized to the m-Cartan matrix of a McKay quiver. We also describe the McKay quiver for a finite abelian subgroup of a special linear group.
基金This work was supported by the National Natural Science Foundation of China(No.61827816).
文摘Diffraction intensities of the 3D ptychographic iterative engine(3PIE)were written as a set of linear equations of the selfcorrelations of Fourier components of all sample slices,and an effective computing method was developed to solve these linear equations for the transmission functions of all sample slices analytically.With both theoretical analysis and numerical simulations,this study revealed the underlying physics and mathematics of 3PIE and demonstrated for the first time,to our knowledge,that 3PIE can generate mathematically unique reconstruction even with noisy data.
基金This work was partially supported by the National Natural Science Foundation of China(Nos.61179033,DMS-1015346)。
文摘We consider a class of nonsmooth convex optimization problems where the objective function is the composition of a strongly convex differentiable function with a linear mapping,regularized by the sum of both l1-norm and l2-norm of the optimization variables.This class of problems arise naturally from applications in sparse group Lasso,which is a popular technique for variable selection.An effective approach to solve such problems is by the Proximal Gradient Method(PGM).In this paper we prove a local error bound around the optimal solution set for this problem and use it to establish the linear convergence of the PGM method without assuming strong convexity of the overall objective function.
