We consider a spectral analysis on the quantum walks graph $$G=(V,E)$$ with local coin operators $$\{C_u\}_{u\in V}$$ and flip flop shift. The have commonly two distinct eigenvalues $$\kappa ,\kappa '$$ $$p=\dim (\ker (\kappa -C_u))$$ for any $$u\in V$$ $$1\le p\le \delta (G)$$ , where $$\delta is minimum degrees of G. show that this walk can be decomposed into cellular automaton $$\ell ^2(V;\m...

For real symmetric and complex Hermitian Gaussian processes whose values are d×d matrices, we characterize the conditions under which probability that at least k eigenvalues collide is positive for 2?k?d, obtain Hausdorff dimension of set collision times.

In this paper we propose a new parallelization of the Davidson algorithm adapted for many eigenvalues. In our parallelization we use a relationship between two consecutive subspaces which allows us to calculate eigenvalues in the subspace through an arrowhead matrix. Theoretical timing estimates for the parallel algorithm are developed and compared against our numerical results on the Paragon. ...

It is well known that the eigenvalues of a real symmetric matrix are not everywhere differentiable. A classical result of Ky Fan states that each eigenvalue of a symmetric matrix is the difference of two convex functions, which implies that the eigenvalues are semismooth functions. Based on a recent result of the authors, it is further proved in this paper that the eigenvalues of a symmetric ma...