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The main goal of this article is to describe a relation between the asymptotic properties of filtrations on section rings and the geometry at infinity of the space of Kähler potentials. More precisely, for a polarized projective manifold and an ample test configuration, Phong and Sturm associated a geodesic ray of plurisubharmonic metrics on the polarizing line bundle. On the other hand, for the same data, Witt Nyström associated a filtration on the section ring of the polarized manifold. In this article, we establish a folklore conjecture that the pluripotential chordal distance between the geodesic rays associated with two ample test configurations coincides with the spectral distance between the associated filtrations on the section ring. This gives an algebraic description of the boundary at infinity of the space of positive metrics, viewed – as it is usually done for spaces of negative curvature – through geodesic rays. [ABSTRACT FROM AUTHOR]

Determining the form of chronic disorder of consciousness is a challenging task in clinical practice, highlighting the need for objective diagnostic methods to assess the depth of impairment of consciousness. This study contributes to the development of more effective diagnostic and prognostic algorithms and biomarkers for patient management. Resting-state functional MRI data from 15 patients and 12 healthy controls were acquired using a 1.5 T system and processed using the statistical parametric mapping package. A connectivity matrix was constructed to assess brain connectivity, and statistical analyses including permutation tests, network-based statistics, and geodesic distance were used to identify significant differences in network measures between controls and patients. Our results revealed significant differences between the functional networks of the patient and control groups at both local and global levels, with altered metrics of node strength, clustering, and global efficiency. Notably, subcortical structures such as the thalamus, caudate nucleus, and raphe dorsalis nucleus showed disruptions in patients, consistent with the role of these regions in the basal ganglia-thalamo-cortical circuit. Our findings provide insight into the complex problem of how information is processed in the functional brain network in chronic disorders of consciousness, beyond the mere localization of functions. [ABSTRACT FROM AUTHOR]

We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold M when given a sample on a finite point set. We prove that the quality of the sample is given by the L γ (M) -average of the geodesic distance to the point set and determine the value of γ ∈ (0 , ∞ ]. This extends our findings on bounded convex domains [IMA J. Numer. Anal., 2024]. As a byproduct, we prove the optimal rate of convergence of the n th minimal worst case error for L q (M) -approximation for all 1 ≤ q ≤ ∞. Further, a limit theorem for moments of the average distance to a set consisting of i.i.d. uniform points is proven. This yields that a random sample is asymptotically as good as an optimal sample in precisely those cases with γ < ∞. In particular, we obtain that cubature formulas with random nodes are asymptotically as good as optimal cubature formulas if the weights are chosen correctly. This closes a logarithmic gap left open by Ehler, Gräf and Oates [Stat. Comput., 29:1203-1214, 2019]. [ABSTRACT FROM AUTHOR]