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On block triangular matrices with signed Drazin inverse

Changjiang Bu, Wenzhe Wang, Jiang Zhou, Lizhu Sun (2014)

Czechoslovak Mathematical Journal

The sign pattern of a real matrix A , denoted by sgn A , is the ( + , - , 0 ) -matrix obtained from A by replacing each entry by its sign. Let 𝒬 ( A ) denote the set of all real matrices B such that sgn B = sgn A . For a square real matrix A , the Drazin inverse of A is the unique real matrix X such that A k + 1 X = A k , X A X = X and A X = X A , where k is the Drazin index of A . We say that A has signed Drazin inverse if sgn A ˜ d = sgn A d for any A ˜ ∈ 𝒬 ( A ) , where A d denotes the Drazin inverse of A . In this paper, we give necessary conditions for some block triangular matrices to have signed...

On elliptic curves and random matrix theory

Mark Watkins (2008)

Journal de Théorie des Nombres de Bordeaux

Rubinstein has produced a substantial amount of data about the even parity quadratic twists of various elliptic curves, and compared the results to predictions from random matrix theory. We use the method of Heegner points to obtain a comparable (yet smaller) amount of data for the case of odd parity. We again see that at least one of the principal predictions of random matrix theory is well-evidenced by the data.

On feebly nil-clean rings

Marjan Sheibani Abdolyousefi, Neda Pouyan (2024)

Czechoslovak Mathematical Journal

A ring R is feebly nil-clean if for any a ∈ R there exist two orthogonal idempotents e , f ∈ R and a nilpotent w ∈ R such that a = e - f + w . Let R be a 2-primal feebly nil-clean ring. We prove that every matrix ring over R is feebly nil-clean. The result for rings of bounded index is also obtained. These provide many classes of rings over which every matrix is the sum of orthogonal idempotent and nilpotent matrices.

On graphs with the largest Laplacian index

Bo Lian Liu, Zhibo Chen, Muhuo Liu (2008)

Czechoslovak Mathematical Journal

Let G be a connected simple graph on n vertices. The Laplacian index of G , namely, the greatest Laplacian eigenvalue of G , is well known to be bounded above by n . In this paper, we give structural characterizations for graphs G with the largest Laplacian index n . Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on n and k for the existence of a k -regular graph G of order n with the largest Laplacian...

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