Positive definite solutions of some matrix equations

In this paper we investigate some existence questions of positive semi-definite solutions for certain classes of matrix equations known as the generalized Lyapunov equations. We present sufficient and necessary conditions for certain equations and only sufficient for others.     

Recovering differential pencils on compact graphs

We study boundary value problems on compact graphs without circles (i.e. on trees) for second-order ordinary differential equations with nonlinear dependence on the spectral parameter. We establish properties of the spectral characteristics and investigate the inverse spectral problem of recovering the coefficients of the differential equation from the so-called Weyl vector which is a generalization of the Weyl function (m-function) for the classical Sturm-Liouville operator. For this inverse problem we prove the uniqueness theorem and obtain a procedure for constructing the solution by the method of spectral mappings.     

Comparison theorems for self‐adjoint linear Hamiltonian eigenvalue problems

In this work we derive new comparison results for (finite) eigenvalues of two self‐adjoint linear Hamiltonian eigenvalue problems. The coefficient matrices depend on the spectral parameter nonlinearly and the spectral parameter is present also in the boundary conditions. We do not impose any controllability or strict normality assumptions. Our method is based on a generalization of the Sturmian comparison theorem for such systems. The results are new even for the Dirichlet boundary conditions, for linear Hamiltonian systems depending linearly on the spectral parameter, and for Sturm–Liouville eigenvalue problems with nonlinear dependence on the spectral parameter.     

A symmetric generalization of Sturm–Liouville problems in q-difference spaces

Classical Sturm–Liouville problems of q-difference variables are extended for symmetric discrete functions such that the corresponding solutions preserve the orthogonality property. Some illustrative examples are given in this sense.     

Identities for minors of the Laplacian, resistance and distance matrices

It is shown that if L and D are the Laplacian and the distance matrix of a tree respectively, then any minor of the Laplacian equals the sum of the cofactors of the complementary submatrix of D, up to sign and a power of 2. An analogous, more general result is proved for the Laplacian and the resistance matrix of any graph. A similar identity is proved for graphs in which each block is a complete graph on r vertices, and for q-analogues of such matrices of a tree. Our main tool is an identity for the minors of a matrix and its inverse.     

A Prüfer angle approach to semidefinite Sturm–Liouville problems with coupling boundary conditions

It is shown how to reduce the periodic/antiperiodic Sturm–Liouville problems to analysis of the Prüfer angle. This provides an alternative to the more usual approaches via operator theory or the Hill discriminant in the definite case, and leads to new results in the semidefinite case. An extension to more general coupling boundary conditions is also given.     

Chebyshev polynomials on symmetric matrices

In this paper we evaluate Chebyshev polynomials of the second kind on a class of symmetric integer matrices, namely on adjacency matrices of simply laced Dynkin and extended Dynkin diagrams. As an application of these results we explicitly calculate minimal projective resolutions of simple modules of symmetric algebras with radical cube zero that are of finite and tame representation type.     

Convergence rates of some iterative methods for nonsymmetric algebraic Riccati equations arising in transport theory

We determine and compare the convergence rates of various fixed-point iterations for finding the minimal positive solution of a class of nonsymmetric algebraic Riccati equations arising in transport theory.     

Unitary representations of the quantum group SU q (1,1): Structure of the dual space ofU q (sl(2))

Real forms of the quantum universal enveloping algebraU _q (sl(2)) and a topological quantum group associated with this algebra are discussed.