In this paper, we establish a fixed point theorem for set-valued mapping on a topological vector space without ＂local convexity＂. And we also establish some generalized Ky Fan＇s minimax inequalities for set-value v...In this paper, we establish a fixed point theorem for set-valued mapping on a topological vector space without ＂local convexity＂. And we also establish some generalized Ky Fan＇s minimax inequalities for set-value vector mappings, which are the generalization of some previous results.展开更多
It is important to measure wool diameter as the wool quality depends on the fibre diameter D and its deviation CVD. According to IWTO standards, the fibre diameter parameters can be tested with the methods of Airflowm...It is important to measure wool diameter as the wool quality depends on the fibre diameter D and its deviation CVD. According to IWTO standards, the fibre diameter parameters can be tested with the methods of Airflowmeter, DA; Sirolan-Laserscan, DL and CVDL; and OFDA, Do and CVDo. However, these parameters only characterize the average diameter and the variation between the fibres. A single fibre analyzer (SIFAN) can be used to measure fibre profile along the fibre and fibre tensile properties simultaneously. The results obtained from the four methods show that there are i) high relationships between Laserscan values and the results of Airflow, OFDA and SIFAN in the average diameters; ii) correlations between CVDL and CVDo or CVDave; iii) the high correlation between Dave-Dmia but a low correlation between Dave-Dmax; and iv) the relationships between the wool quality and the ratio of Dmln/Dave and Dmin/DL. Based on the results and discussions, the effective measurement of wool diameter should be the SIFAN method. The new parameters of Dmin/DL and Dmin/Dave are the useful value for the evaluation of wool quality in practice.展开更多
The analogy between eigenvalues and singular values has many faces. The current review brings together several examples of this analogy. One example regards the similarity between Symmetric Rayleigh Quotients and Rect...The analogy between eigenvalues and singular values has many faces. The current review brings together several examples of this analogy. One example regards the similarity between Symmetric Rayleigh Quotients and Rectangular Rayleigh Quotients. Many useful properties of eigenvalues stem are from the Courant-Fischer minimax theorem, from Weyl’s theorem, and their corollaries. Another aspect regards “rectangular” versions of these theorems. Comparing the properties of Rayleigh Quotient matrices with those of Orthogonal Quotient matrices illuminates the subject in a new light. The Orthogonal Quotients Equality is a recent result that converts Eckart-Young’s minimum norm problem into an equivalent maximum norm problem. This exposes a surprising link between the Eckart-Young theorem and Ky Fan’s maximum principle. We see that the two theorems reflect two sides of the same coin: there exists a more general maximum principle from which both theorems are easily derived. Ky Fan has used his extremum principle (on traces of matrices) to derive analog results on determinants of positive definite Rayleigh Quotients matrices. The new extremum principle extends these results to Rectangular Quotients matrices. Bringing all these topics under one roof provides new insight into the fascinating relations between eigenvalues and singular values.展开更多
基金The NSF(9452902001003278,10452902001005845) of Guangdong Province
文摘In this paper, we establish a fixed point theorem for set-valued mapping on a topological vector space without ＂local convexity＂. And we also establish some generalized Ky Fan＇s minimax inequalities for set-value vector mappings, which are the generalization of some previous results.
文摘It is important to measure wool diameter as the wool quality depends on the fibre diameter D and its deviation CVD. According to IWTO standards, the fibre diameter parameters can be tested with the methods of Airflowmeter, DA; Sirolan-Laserscan, DL and CVDL; and OFDA, Do and CVDo. However, these parameters only characterize the average diameter and the variation between the fibres. A single fibre analyzer (SIFAN) can be used to measure fibre profile along the fibre and fibre tensile properties simultaneously. The results obtained from the four methods show that there are i) high relationships between Laserscan values and the results of Airflow, OFDA and SIFAN in the average diameters; ii) correlations between CVDL and CVDo or CVDave; iii) the high correlation between Dave-Dmia but a low correlation between Dave-Dmax; and iv) the relationships between the wool quality and the ratio of Dmln/Dave and Dmin/DL. Based on the results and discussions, the effective measurement of wool diameter should be the SIFAN method. The new parameters of Dmin/DL and Dmin/Dave are the useful value for the evaluation of wool quality in practice.
文摘The analogy between eigenvalues and singular values has many faces. The current review brings together several examples of this analogy. One example regards the similarity between Symmetric Rayleigh Quotients and Rectangular Rayleigh Quotients. Many useful properties of eigenvalues stem are from the Courant-Fischer minimax theorem, from Weyl’s theorem, and their corollaries. Another aspect regards “rectangular” versions of these theorems. Comparing the properties of Rayleigh Quotient matrices with those of Orthogonal Quotient matrices illuminates the subject in a new light. The Orthogonal Quotients Equality is a recent result that converts Eckart-Young’s minimum norm problem into an equivalent maximum norm problem. This exposes a surprising link between the Eckart-Young theorem and Ky Fan’s maximum principle. We see that the two theorems reflect two sides of the same coin: there exists a more general maximum principle from which both theorems are easily derived. Ky Fan has used his extremum principle (on traces of matrices) to derive analog results on determinants of positive definite Rayleigh Quotients matrices. The new extremum principle extends these results to Rectangular Quotients matrices. Bringing all these topics under one roof provides new insight into the fascinating relations between eigenvalues and singular values.
