Dissipative Operators Generated by the Sturm–Liouville Differential Expression in the Weyl Limit Circle Case

AbstractIn this paper, using Livšic's theorem, we investigate the problem of completeness of the system of eigenfunctions and associated functions of dissipative operators generated by the Sturm–Liouville differential expression on the semi-axis in Weyl's limit-circle case

Non-pharmacological management of osteoporosis: a consensus of the Belgian Bone Club

This consensus article reviews the various aspects of the non-pharmacological management of osteoporosis, including the effects of nutriments, physical exercise, lifestyle, fall prevention, and hip protectors. Vertebroplasty is also briefly reviewed. Non-pharmacological management of osteoporosis is a broad concept. It must be viewed as an essential part of the prevention of fractures from childhood through adulthood and the old age. The topic also includes surgical procedures for the treatment of peripheral and vertebral fractures and the post-fracture rehabilitation. The present document is the result of a consensus, based on a systematic review and a critical appraisal of the literature. Diets deficient in calcium, proteins or vitamin D impair skeletal integrity. The effect of other nutriments is less clear, although an excessive consumption of sodium, caffeine, or fibres exerts negative effects on calcium balance. The deleterious effects of tobacco, excessive alcohol consumption and a low BMI are well accepted. Physical activity is of primary importance to reach optimal peak bone mass but, if numerous studies have shown the beneficial effects of various types of exercise on bone mass, fracture data as an endpoint are scanty. Fall prevention strategies are especially efficient in the community setting, but less evidence is available about their effectiveness in preventing fall-related injuries and fractures. The efficacy of hip protectors remains controversial. This is also true for vertebroplasty and kyphoplasty. Several randomized controlled studies had reported a short-term advantage of vertebroplasty over medical treatment for pain relief, but these findings have been questioned by recent sham-controlled randomized clinical studies

nth order Stieltjes differential boundary operators and Stieltjes differential boundary systems

AbstractThis article discusses linear differential boundary systems, which include nth-order differential boundary relations as a special case, in Lnp[0,1] × Lnp[0,1], 1 ⩽ p < ∞. The adjoint relation in Lnq[0,1] × Lnq[0,1], 1p + 1q = 1, is derived. Green's formula is also found. Self-adjoint relations are found in Ln2[0,1] × Ln2[0,1], and their connection with Coddington's extensions of symmetric operators on subspaces of Lnp[0,1] × Ln2[0,1] is established

Boundary values for an eigenvalue problem with a singular potential

AbstractThe radial Hamiltonian operator H = −d2dx2 − λx2 is considered on [0, ∞). While no boundary condition is needed at ∞ in order to make H self-adjoint, one is indeed needed at 0. This article describes in elementary terms how such conditions may be found. The resulting eigenvalue problem is then discussed and its relation to previous work is explained. Other potentials with singularities at 0 are also mentioned