The tridiagonal coefficient matrix for the ＂fixed-fixed＂ spring-mass system was obtained by changing spring length. And then a new algorithm of the inverse problem was designed to construct the masses and the spring...The tridiagonal coefficient matrix for the ＂fixed-fixed＂ spring-mass system was obtained by changing spring length. And then a new algorithm of the inverse problem was designed to construct the masses and the spring constants from the natural frequencies of the ＂fixed-fixed＂ and ＂fixed-fres＂ spring-mass systems. An example was given to illustrate the results.展开更多
Some properties of characteristic polynomials, eigenvalues, and eigenvectors of the Wilkinson matrices W+2n+1 and W-2n+1 are presented. It is proved that the eigenvalues of W+2n+1 just are the eigenvalues of its leadi...Some properties of characteristic polynomials, eigenvalues, and eigenvectors of the Wilkinson matrices W+2n+1 and W-2n+1 are presented. It is proved that the eigenvalues of W+2n+1 just are the eigenvalues of its leading principal submatrix Vn and a bordered matrix of Vn. Recurrence formula are given for the characteristic polynomial of W+2n+1. The eigenvectors of W+2n+1 are proved to be symmetric or skew symmetric. For W-2n+1, it is found that its eigenvalues are zero and the square roots of the eigenvalues of a bordered matrix of V2n. And the eigenvectors of W-2n+1, which the corresponding eigenvalues are opposite in pairs, have close relationship.展开更多
Suppose to toss an independent coin with equal probability of success and failure for each subset of [n] = {1, 2, ..., n}, and form the random hypergraph H(n) by taking as hyperedges the subsets with successful coin t...Suppose to toss an independent coin with equal probability of success and failure for each subset of [n] = {1, 2, ..., n}, and form the random hypergraph H(n) by taking as hyperedges the subsets with successful coin tosses. It is proved that H(n) is almost surely connected. By defining a graph G(S) according to a subset system S, it is shown that the intersecting problem is NP-complete.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant No.10271074)
文摘The tridiagonal coefficient matrix for the ＂fixed-fixed＂ spring-mass system was obtained by changing spring length. And then a new algorithm of the inverse problem was designed to construct the masses and the spring constants from the natural frequencies of the ＂fixed-fixed＂ and ＂fixed-fres＂ spring-mass systems. An example was given to illustrate the results.
基金The Fundamental Research Funds for the Central Universities, China (No.10D10908)
文摘Some properties of characteristic polynomials, eigenvalues, and eigenvectors of the Wilkinson matrices W+2n+1 and W-2n+1 are presented. It is proved that the eigenvalues of W+2n+1 just are the eigenvalues of its leading principal submatrix Vn and a bordered matrix of Vn. Recurrence formula are given for the characteristic polynomial of W+2n+1. The eigenvectors of W+2n+1 are proved to be symmetric or skew symmetric. For W-2n+1, it is found that its eigenvalues are zero and the square roots of the eigenvalues of a bordered matrix of V2n. And the eigenvectors of W-2n+1, which the corresponding eigenvalues are opposite in pairs, have close relationship.
文摘Suppose to toss an independent coin with equal probability of success and failure for each subset of [n] = {1, 2, ..., n}, and form the random hypergraph H(n) by taking as hyperedges the subsets with successful coin tosses. It is proved that H(n) is almost surely connected. By defining a graph G(S) according to a subset system S, it is shown that the intersecting problem is NP-complete.
