Eigenvalues of the Ricci Operator on Four-Dimensional Locally Homogeneous (Pseudo)Riemannian Manifolds with a Four-Dimensional Isotropy Subgroup

The problem of restoring a (pseudo)Riemannian manifold from a given Ricci operator was studied in the papers of many mathematicians. This problem was solved by O. Kowalski and S. Nikcevic for the case of three-dimensional locally homogeneous Riemannian manifolds. The work of G. Calvaruso and O. Kowalski contains the answer to the question above for the case of three –dimensional locally homogeneous Lorentzian manifolds. For the four-dimensional case, similar studies were carried out only in the case of Lie groups with a left-invariant Riemannian metric. The works of A.G. Kremlyov and Yu.G. Nikonorov presented the possible signatures of the eigenvalues of the Ricci operator. However, the question of recovering a four-dimensional Lie group with a left-invariant Riemannian metric from a given Ricci operator remains open. This paper is devoted to the study of the eigenvalues of the Ricci operator on four-dimensional locally homogeneous (pseudo)Riemannian manifolds with a four-dimensional isotropy subgroup. An algorithm for calculating the eigenvalues of the Ricci operator is presented. A theorem on the restoration of such manifolds from a given Ricci operator is proved. It is established that such possibility can happen only in the case when the prescribed operator is diagonalizable and has a unique eigenvalue of multiplicity four.

On similarity of an arbitrary matrix to a block diagonal matrix

Let an n x n -matrix A have m < n (m ? 2) different eigenvalues ?j of the algebraic multiplicity ?j (j = 1,..., m). It is proved that there are ?j x ?j-matrices Aj, each of which has a unique eigenvalue ?j, such that A is similar to the block-diagonal matrix ?D = diag (A1,A2,..., Am). I.e. there is an invertible matrix T, such that T-1AT = ?D. Besides, a sharp bound for the number kT := ||T||||T-1|| is derived. As applications of these results we obtain norm estimates for matrix functions non-regular on the convex hull of the spectra. These estimates generalize and refine the previously published results. In addition, a new bound for the spectral variation of matrices is derived. In the appropriate situations it refines the well known bounds.

THE EFFECTIVE POTENTIAL AND TRANSSHIPMENT IN THERMODYNAMIC FORMALISM AT ZERO TEMPERATURE

For a topologically transitive subshift of finite type defined by a symmetric transition matrix, we introduce a temperature-based problem related to the usual thermodynamic formalism. This problem is described by an operator acting on Hölder continuous observables which is actually superlinear with respect to the max-plus algebra. We thus show that, for each fixed absolute temperature, such an operator admits a unique eigenfunction and a unique eigenvalue. We also study the convergence as the temperature goes to zero and we relate the limit objects to an ergodic version of Kantorovich transshipment problem.

Threshold Effects for the Generalized Friedrichs Model with the Perturbation of Rank One

A familyHμ(p),μ>0,p∈𝕋2of the Friedrichs models with the perturbation of rank one, associated to a system of two particles, moving on the two-dimensional latticeℤ2is considered. The existence or absence of the unique eigenvalue of the operatorHμ(p)lying below threshold depending on the values ofμ>0andp∈Uδ(0)⊂𝕋2is proved. The analyticity of corresponding eigenfunction is shown.

III.—The Eigenvalue Problem for Boolean Matrices

SynopsisIn the case of Boolean matrices a given eigenvector may have a variety of eigenvalues. These eigenvalues form a sublattice of the basic Boolean algebra and the structure of this sublattice is investigated. Likewise a given eigenvalue has a variety of eigenvectors which form a module of the Boolean vector space. The structure of this module is examined. It is also shown that if a vector has a unique eigenvalue λ, then λ satisfies the characteristic equation of the matrix.