Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings

We solve two inverse spectral problems for star graphs of Stieltjes strings with Dirichlet and Neumann boundary conditions, respectively, at a selected vertex called root. The root is either the central vertex or, in the more challenging problem, a pendant vertex of the star graph. At all other pendant vertices Dirichlet conditions are imposed; at the central vertex, at which a mass may be placed, continuity and Kirchhoff conditions are assumed. We derive conditions on two sets of real numbers to be the spectra of the above Dirichlet and Neumann problems. Our solution for the inverse problems is constructive: we establish algorithms to recover the mass distribution on the star graph (i.e. the point masses and lengths of subintervals between them) from these two spectra and from the lengths of the separate strings. If the root is a pendant vertex, the two spectra uniquely determine the parameters on the main string (i.e. the string incident to the root) if the length of the main string is known. The mass distribution on the other edges need not be unique; the reason for this is the non-uniqueness caused by the non-strict interlacing of the given data in the case when the root is the central vertex. Finally, we relate of our results to tree-patterned matrix inverse problems.

Immunohistochemical demonstration of lumbar intervertebral disc innervation in the dog.

Low back pain is a common ailment in dogs, particularly in specific breeds such as the German shepherd dog. A number of structures such as facet joint capsules, ligaments, dorsal root ganglia, periosteum, vertebral endplates and meninges have been associated with this condition. Yet, in spite of all diagnostic efforts, the origin of pain remains obscure in a substantial proportion of all cases. A further structure often being involved in vertebral column disorders is the intervertebral disc. The presence of nerves, however, is a precondition for pain sensation and, consequently, structures lacking innervation can be left out of consideration as a cause for low back pain. Nerve fibres have been demonstrated at the periphery of the intervertebral disc in man, rabbit and rat. With regard to the dog, however, the extent of intervertebral disc innervation is still being disputed. The goal of the present study, therefore, was to substantiate and expand current knowledge of intervertebral disc innervation. Protein gene product (PGP) 9.5 was used for immunohistochemical examination of serial transversal and sagittal paraffin sections of lumbar discs from adult dogs. This general marker revealed nerve fibres to be confined to the periphery of the intervertebral discs. These results indicate that even limited pathological processes affecting the outer layers of the intervertebral disc are prone to cause low back pain.