2008 NOV 10 - (VerticalNews.com) -- "This paper reviews the equations ax = c and xb = d from a new perspective by studying them in the setting of associative rings with or without involution," investigators in Melbourne, Australia report.

  "Results for rectangular matrices and operators between different Banach and Hilbert spaces are obtained by embedding the 'rectangles' into rings of square matrices or rings of operators acting on the same space. Necessary and sufficient conditions using generalized inverses are given for the existence of the Hermitian, skew-Hermitian, reflexive, antireflexive, positive and real-positive solutions, and the general solutions are described in terms of the original elements or operators," wrote A. Dajic and colleagues, University of Melbourne ...read more

  2008 NOV 10 - (VerticalNews.com) -- According to recent research from Shandong, People's Republic of China, "The study of limit points of eigenvalues of adjacency matrices of graphs was initiated by Hoffman [A.J. Hoffman, On limit points of spectral radii of non-negative symmetric integral matrices, in: Y. Alavi et al. (Eds.), Lecture Notes Math., vol. 303, Springer-Verlag, Berlin, Heidelberg, New York, 1972, pp. 165-172]."

  "There he described all of the limit points of the largest eigenvalue of adjacency matrices of graphs that are no more than root 2 + root 5. In this paper, we investigate limit points of Laplacian spectral radii of graphs. The result is obtained: Let omega = 1/3 ((3)root 19 + 3 root 33 + (3)root 19 - 3 root 33+ 1), beta(0) = 1 and beta(n) (n >= 1) be the largest positive root of P-n(x) = x(n+1) - (1 + x + ... + x(n-1)) (root x + 1)(2). Let alpha(n) = 2 + beta(1/2)(n) + beta(-1/2)(n)," wrote J.M. Guo and colleagues, China University ...read more

  2008 NOV 10 - (VerticalNews.com) -- "In this paper, we give necessary and sufficient conditions for a set of Jordan blocks to correspond to the peripheral spectrum of a nonnegative matrix. For each eigenvalue. lambda, the lambda-level characteristic (with respect to the spectral radius) is defined," scientists writing in the journal Linear Algebra and Its Applications report.

  "The necessary and sufficient conditions include a requirement that the lambda-level characteristic is majorized by the lambda-height characteristic. An algorithm which has been implemented in MATLAB is given to determine when a multiset of Jordan blocks corresponds to the peripheral spectrum of a nonnegative matrix," wrote J.J. Mcdonald and colleagues, Washington State University ...read more