This article is part exposition of a recent rather technical paper of the last two authors on matrix pavings related to the 1959 Kadison-Singer Extension Problem and part a report on further computational results prov...This article is part exposition of a recent rather technical paper of the last two authors on matrix pavings related to the 1959 Kadison-Singer Extension Problem and part a report on further computational results providing new bounds on the paving parameters for classes of small matrices investigated there and subsequently. A website maintained by the authors provides to all interested the matrices experimentally discovered that yield these bounds along with the proprietary MATLAB software with simple operational directions to load them, pave them, and perform paving searches. The convergence to 1 or not of the infinite sequences of these paving parameters in most cases is equivalent to the Kadison-Singer Extension Problem, and in all cases convergence to 1 negates the problem. The last two sections describe the search process and an interpretation of the data integrated with the results of the precursor to this paper.展开更多
The extension of Minimum Spanning Tree(MST) problem is an NP hard problem which does not exit a polynomial time algorithm. In this paper, a fast optimization method on MST problem——the Gradient Gene Algorithm is int...The extension of Minimum Spanning Tree(MST) problem is an NP hard problem which does not exit a polynomial time algorithm. In this paper, a fast optimization method on MST problem——the Gradient Gene Algorithm is introduced. Compared with other evolutionary algorithms on MST problem, it is more advanced: firstly, very simple and easy to realize; then, efficient and accurate; finally general on other combination optimization problems.展开更多
In this paper,we study Tingley＇s problem on symmetric absolute normalized norms on R^2.We construct new methods for Tingley＇s problem on two-dimensional spaces by using isosceles orthogonality,which does not make us...In this paper,we study Tingley＇s problem on symmetric absolute normalized norms on R^2.We construct new methods for Tingley＇s problem on two-dimensional spaces by using isosceles orthogonality,which does not make use of the notion of natural extension.Furthermore,using our methods,several sufficient conditions for Tingley＇s problem on symmetric absolute normalized norms on R2 are given.As applications,we present various new examples including the two-dimensional Lorentz sequence space d^（2）（ω,q） and its dual d^（2）（ω,q）＊by simple arguments.展开更多
A theorem of Lambrechts and Stanley is used to find the rational cohomology of the complement of an embedding S^(4n-1)→ S^(2n)× S^m as a module and demonstrate that it is not necessarily determined by the map in...A theorem of Lambrechts and Stanley is used to find the rational cohomology of the complement of an embedding S^(4n-1)→ S^(2n)× S^m as a module and demonstrate that it is not necessarily determined by the map induced on cohomology by the embedding, nor is it a trivial extension. This demonstrates that the theorem is an improvement on the classical Lefschetz duality.展开更多
基金supported by Naval Academy Research Council seed grants
文摘This article is part exposition of a recent rather technical paper of the last two authors on matrix pavings related to the 1959 Kadison-Singer Extension Problem and part a report on further computational results providing new bounds on the paving parameters for classes of small matrices investigated there and subsequently. A website maintained by the authors provides to all interested the matrices experimentally discovered that yield these bounds along with the proprietary MATLAB software with simple operational directions to load them, pave them, and perform paving searches. The convergence to 1 or not of the infinite sequences of these paving parameters in most cases is equivalent to the Kadison-Singer Extension Problem, and in all cases convergence to 1 negates the problem. The last two sections describe the search process and an interpretation of the data integrated with the results of the precursor to this paper.
文摘The extension of Minimum Spanning Tree(MST) problem is an NP hard problem which does not exit a polynomial time algorithm. In this paper, a fast optimization method on MST problem——the Gradient Gene Algorithm is introduced. Compared with other evolutionary algorithms on MST problem, it is more advanced: firstly, very simple and easy to realize; then, efficient and accurate; finally general on other combination optimization problems.
文摘In this paper,we study Tingley＇s problem on symmetric absolute normalized norms on R^2.We construct new methods for Tingley＇s problem on two-dimensional spaces by using isosceles orthogonality,which does not make use of the notion of natural extension.Furthermore,using our methods,several sufficient conditions for Tingley＇s problem on symmetric absolute normalized norms on R2 are given.As applications,we present various new examples including the two-dimensional Lorentz sequence space d^（2）（ω,q） and its dual d^（2）（ω,q）＊by simple arguments.
基金supported by the National Science and Engineering Research Council of Canada
文摘A theorem of Lambrechts and Stanley is used to find the rational cohomology of the complement of an embedding S^(4n-1)→ S^(2n)× S^m as a module and demonstrate that it is not necessarily determined by the map induced on cohomology by the embedding, nor is it a trivial extension. This demonstrates that the theorem is an improvement on the classical Lefschetz duality.
