Root system of singular perturbations of the harmonic oscillator type operators

We analyze perturbations of the harmonic oscillator type operators in a Hilbert space H, i.e. of the self-adjoint operator with simple positive eigenvalues μ k satisfying μ k+1 − μ k ≥ Δ > 0. Perturbations are considered in the sense of quadratic forms. Under a local subordination assumption, the eigenvalues of the perturbed operator become eventually simple and the root system contains a Riesz basis.

Validation of a standardized mapping system of the hip joint for radial MRA sequencing

OBJECTIVE Intraarticular gadolinium-enhanced magnetic resonance arthrography (MRA) is commonly applied to characterize morphological disorders of the hip. However, the reproducibility of retrieving anatomic landmarks on MRA scans and their correlation with intraarticular pathologies is unknown. A precise mapping system for the exact localization of hip pathomorphologies with radial MRA sequences is lacking. Therefore, the purpose of the study was the establishment and validation of a reproducible mapping system for radial sequences of hip MRA. MATERIALS AND METHODS Sixty-nine consecutive intraarticular gadolinium-enhanced hip MRAs were evaluated. Radial sequencing consisted of 14 cuts orientated along the axis of the femoral neck. Three orthopedic surgeons read the radial sequences independently. Each MRI was read twice with a minimum interval of 7 days from the first reading. The intra- and inter-observer reliability of the mapping procedure was determined. RESULTS A clockwise system for hip MRA was established. The teardrop figure served to determine the 6 o'clock position of the acetabulum; the center of the greater trochanter served to determine the 12 o'clock position of the femoral head-neck junction. The intra- and inter-observer ICCs to retrieve the correct 6/12 o'clock positions were 0.906-0.996 and 0.978-0.988, respectively. CONCLUSIONS The established mapping system for radial sequences of hip joint MRA is reproducible and easy to perform.

Cognitive-Behavioural Analysis System of Psychotherapy (CBASP), a drug, or their combination: differential therapeutics for persistent depressive disorder: a study protocol of an individual participant data network meta-analysis.

INTRODUCTION Despite important advances in psychological and pharmacological treatments of persistent depressive disorders in the past decades, their responses remain typically slow and poor, and differential responses among different modalities of treatments or their combinations are not well understood. Cognitive-Behavioural Analysis System of Psychotherapy (CBASP) is the only psychotherapy that has been specifically designed for chronic depression and has been examined in an increasing number of trials against medications, alone or in combination. When several treatment alternatives are available for a certain condition, network meta-analysis (NMA) provides a powerful tool to examine their relative efficacy by combining all direct and indirect comparisons. Individual participant data (IPD) meta-analysis enables exploration of impacts of individual characteristics that lead to a differentiated approach matching treatments to specific subgroups of patients. METHODS AND ANALYSIS We will search for all randomised controlled trials that compared CBASP, pharmacotherapy or their combination, in the treatment of patients with persistent depressive disorder, in Cochrane CENTRAL, PUBMED, SCOPUS and PsycINFO, supplemented by personal contacts. Individual participant data will be sought from the principal investigators of all the identified trials. Our primary outcomes are depression severity as measured on a continuous observer-rated scale for depression, and dropouts for any reason as a proxy measure of overall treatment acceptability. We will conduct a one-step IPD-NMA to compare CBASP, medications and their combinations, and also carry out a meta-regression to identify their prognostic factors and effect moderators. The model will be fitted in OpenBUGS, using vague priors for all location parameters. For the heterogeneity we will use a half-normal prior on the SD. ETHICS AND DISSEMINATION This study requires no ethical approval. We will publish the findings in a peer-reviewed journal. The study results will contribute to more finely differentiated therapeutics for patients suffering from this chronically disabling disorder. TRIAL REGISTRATION NUMBER CRD42016035886.