A class of biholomorphic mappings named “quasi-convex mapping” is introduced in the unit ball of a complex Banach space. It is proved that this class of mappings is a proper subset of the class of starlike mappings ...A class of biholomorphic mappings named “quasi-convex mapping” is introduced in the unit ball of a complex Banach space. It is proved that this class of mappings is a proper subset of the class of starlike mappings and contains the class of convex mappings properly, and it has the same growth and covering theorems as the convex mappings. Furthermore, when the Banach space is confined to ?n, the “quasi-convex mapping” is exactly the “quasi-convex mapping of type A” introduced by K. A. Roper and T. J. Suffridge.展开更多
By a ball-covering B of a Banach space X, we mean that it is a collection of open balls off the origin whose union contains the sphere of the unit ball of X. The space X is said to have a ball-covering property, if it...By a ball-covering B of a Banach space X, we mean that it is a collection of open balls off the origin whose union contains the sphere of the unit ball of X. The space X is said to have a ball-covering property, if it admits a ball-covering consisting of countably many balls. This paper, by constructing the equivalent norms on l~∞, shows that ball-covering property is not invariant under isomorphic mappings, though it is preserved under such mappings if X is a Gateaux differentiability space; presents that this property of X is not heritable by its closed subspaces; and the property is also not preserved under quotient mappings.展开更多
This paper presents two counterexamples about ball-coverings of Banach spaces and shows a new characterization of uniformly non-square Banach spaces via ball-coverings.
This paper presents learning rates for the least-square regularized regression algorithms with polynomial kernels. The target is the error analysis for the regression problem in learning theory. A regularization schem...This paper presents learning rates for the least-square regularized regression algorithms with polynomial kernels. The target is the error analysis for the regression problem in learning theory. A regularization scheme is given, which yields sharp learning rates. The rates depend on the dimension of polynomial space and polynomial reproducing kernel Hilbert space measured by covering numbers. Meanwhile, we also establish the direct approximation theorem by Bernstein-Durrmeyer operators in $ L_{\rho _X }^2 $ with Borel probability measure.展开更多
Several theorems on closed (resp. open) covering properties of H-spaces are obtained which improve and generalize the corresponding results of Sperner, Klee, Alexandroff-Pasynkoff, Berge, Ghouila-Houri, Danzer-Grunbau...Several theorems on closed (resp. open) covering properties of H-spaces are obtained which improve and generalize the corresponding results of Sperner, Klee, Alexandroff-Pasynkoff, Berge, Ghouila-Houri, Danzer-Grunbaum-Klee, Ky Fan, Shih-Tan, Horvath and Lassonde. As application an almost fixed point theorem for lower semi-continuous map in l.c.-spaces and a generalization of Tychonoffs fixed point theorem are proved in l.c.-spaces which improve those results of Ky Fan and Horvath.展开更多
By a ball-covering B of a Banach space X, we mean that B is a collection of open （or closed） balls off the origin whose union contains the unit sphere of X; and X is said to have the ball-covering property provided ...By a ball-covering B of a Banach space X, we mean that B is a collection of open （or closed） balls off the origin whose union contains the unit sphere of X; and X is said to have the ball-covering property provided it admits a ball-covering of countably many balls. This paper shows that universal finite representability and B-convexity of X can be characterized by properties of ball-coverings of its finite dimensional subspaces.展开更多
Partially observable Markov decision processes （POMDPs） provide a rich mathematical framework for planning tasks in partially observable stochastic environments. The notion of the covering number, a metric of captur...Partially observable Markov decision processes （POMDPs） provide a rich mathematical framework for planning tasks in partially observable stochastic environments. The notion of the covering number, a metric of capturing the search space size of a POMDP planning problem, has been proposed as a complexity measure of approximate POMDP planning. Existing theoretical results are based on POMDPs with finite and discrete state spaces and measured in the l1- metric space. When considering heuristics, they are assumed to be always admissible. This paper extends the theoretical results on the covering numbers of different search spaces, including the newly defined space reachable under inadmissible heuristics, to the ln-metric spaces. We provide a simple but scalable algorithm for estimating covering numbers. Experimentally, we provide estimated covering numbers of the search spaces reachable by following different policies on several benchmark problems, and analyze their abilities to predict the runtime of POMDP planning algorithms.展开更多
The following results are proved:1. Let X be a screenable p-space and Y a metric space. Then X × Y is screenable.2. Let X be a strongly submetacompact p-space and Y a metacompact Σ-space. ThenN × Y is stron...The following results are proved:1. Let X be a screenable p-space and Y a metric space. Then X × Y is screenable.2. Let X be a strongly submetacompact p-space and Y a metacompact Σ-space. ThenN × Y is strongly submetacompact.3. Suppose that X = ∏ Xn is countably paracompact and for each n < ω, ∏ Xi issubmota-Lindelsf Then X is submeta-Lindelof展开更多
基金This work was supported by 973 Project, the National Natural Science Foundation of China (Grant No. 19871081) and the Natural Science Foundation of Guangdong Province and Anhui Province.
文摘A class of biholomorphic mappings named “quasi-convex mapping” is introduced in the unit ball of a complex Banach space. It is proved that this class of mappings is a proper subset of the class of starlike mappings and contains the class of convex mappings properly, and it has the same growth and covering theorems as the convex mappings. Furthermore, when the Banach space is confined to ?n, the “quasi-convex mapping” is exactly the “quasi-convex mapping of type A” introduced by K. A. Roper and T. J. Suffridge.
基金Supported by the National Natural Science Foundation of China (Grant No. 10471114)
文摘By a ball-covering B of a Banach space X, we mean that it is a collection of open balls off the origin whose union contains the sphere of the unit ball of X. The space X is said to have a ball-covering property, if it admits a ball-covering consisting of countably many balls. This paper, by constructing the equivalent norms on l~∞, shows that ball-covering property is not invariant under isomorphic mappings, though it is preserved under such mappings if X is a Gateaux differentiability space; presents that this property of X is not heritable by its closed subspaces; and the property is also not preserved under quotient mappings.
基金Supported by the Support Program for 1 0 0 Young and Middle-Aged Disciplinary Leaders inGuangxi Higher Education Institution and SF of Guangxi(GSY0 1 3 5 0 2 7)
基金Supported by National Natural Science Foundation of China (Grant No. 10771175)
文摘This paper presents two counterexamples about ball-coverings of Banach spaces and shows a new characterization of uniformly non-square Banach spaces via ball-coverings.
文摘This paper presents learning rates for the least-square regularized regression algorithms with polynomial kernels. The target is the error analysis for the regression problem in learning theory. A regularization scheme is given, which yields sharp learning rates. The rates depend on the dimension of polynomial space and polynomial reproducing kernel Hilbert space measured by covering numbers. Meanwhile, we also establish the direct approximation theorem by Bernstein-Durrmeyer operators in $ L_{\rho _X }^2 $ with Borel probability measure.
基金This project partially supported by National Natural Science Foundation of ChinaThis work was partially supported by NSERC of Canada under grant A-8096
文摘Several theorems on closed (resp. open) covering properties of H-spaces are obtained which improve and generalize the corresponding results of Sperner, Klee, Alexandroff-Pasynkoff, Berge, Ghouila-Houri, Danzer-Grunbaum-Klee, Ky Fan, Shih-Tan, Horvath and Lassonde. As application an almost fixed point theorem for lower semi-continuous map in l.c.-spaces and a generalization of Tychonoffs fixed point theorem are proved in l.c.-spaces which improve those results of Ky Fan and Horvath.
基金Supported by National Natural Science Foundation of China (Grant Nos. 10771175, 10801111 and 11101340)the Natural Science Foundation of Fujian Province (Grant No. 2010J05012) the Fundamental Research Funds for the Central Universities (Grant Nos. 2010121001 and 2011121039)
文摘By a ball-covering B of a Banach space X, we mean that B is a collection of open （or closed） balls off the origin whose union contains the unit sphere of X; and X is said to have the ball-covering property provided it admits a ball-covering of countably many balls. This paper shows that universal finite representability and B-convexity of X can be characterized by properties of ball-coverings of its finite dimensional subspaces.
文摘Partially observable Markov decision processes （POMDPs） provide a rich mathematical framework for planning tasks in partially observable stochastic environments. The notion of the covering number, a metric of capturing the search space size of a POMDP planning problem, has been proposed as a complexity measure of approximate POMDP planning. Existing theoretical results are based on POMDPs with finite and discrete state spaces and measured in the l1- metric space. When considering heuristics, they are assumed to be always admissible. This paper extends the theoretical results on the covering numbers of different search spaces, including the newly defined space reachable under inadmissible heuristics, to the ln-metric spaces. We provide a simple but scalable algorithm for estimating covering numbers. Experimentally, we provide estimated covering numbers of the search spaces reachable by following different policies on several benchmark problems, and analyze their abilities to predict the runtime of POMDP planning algorithms.
文摘The following results are proved:1. Let X be a screenable p-space and Y a metric space. Then X × Y is screenable.2. Let X be a strongly submetacompact p-space and Y a metacompact Σ-space. ThenN × Y is strongly submetacompact.3. Suppose that X = ∏ Xn is countably paracompact and for each n < ω, ∏ Xi issubmota-Lindelsf Then X is submeta-Lindelof
