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The main objective of this paper is to solve the problem of finding a geodesic that best fits a given set of time-labelled points on the Grassmann manifold. To achieve this goal, we first derive a very useful simplified formula for the geodesic arc joining two points on the Grassmannian depending explicitly only on the given points. This allows to simplify the expression for the geodesic distance, which is crucial to generalize the fitting problem, and is also used to obtain a simpler characterization of the geometric mean of a finite set of points lying on the Grassmannian, where the given points enter explicitly. [ABSTRACT FROM AUTHOR]

Objective Major depressive disorder (MDD) is associated with an estimated fourfold risk for premature death, largely attributed to cardiovascular disorders. Pericardial adipose tissue (PAT), a fat compartment surrounding the heart, has been implicated in the development of coronary artery disease. An unanswered question is whether people with chronic MDD are more likely to have elevated PAT volumes versus acute MDD and controls (CTRL). Methods The study group consists of sixteen patients with chronic MDD, thirty-four patients with acute MDD, and twenty-five CTRL. PAT and adrenal gland volume were measured by magnetic resonance tomography. Additional measures comprised factors of the metabolic syndrome, cortisol, relative insulin resistance, and pro-inflammatory cytokines (interleukin-6; IL-6 and tumor necrosis factor-α, TNF-α). Results PAT volumes were significantly increased in patients with chronic MDD > patients with acute MDD > CTRL. Adrenal gland volume was slightly enlarged in patients with chronic MDD > acute MDD > CTRL, although this difference failed to reach significance. The PAT volume was correlated with adrenal gland volume, and cortisol concentrations were correlated with depression severity, measured by BDI-2 and MADRS. Group differences were found concerning the rate of the metabolic syndrome, being most frequent in chronic MDD > acute MDD > CTRL. Further findings comprised increased fasting cortisol, increased TNF-α concentration, and decreased physical activity level in MDD compared to CTRL. Conclusion Our results extend the existing literature in demonstrating that patients with chronic MDD have the highest risk for developing cardiovascular disorders, indicated by the highest PAT volume and prevalence of metabolic syndrome. The correlation of PAT with adrenal gland volume underscores the role of the hypothalamus-pituitary-adrenal system as mediator for body-composition changes. Metabolic monitoring, health advices and motivation for the improvement of physical fitness may be recommended in depressed patients, in particular in chronic depression. [ABSTRACT FROM AUTHOR]