In the case that the low-regularity Riemannian metric induces a complete length space, we demonstrate essential self-adjointness of the operator and its higher powers up to the regularity of its coefficients.
In a nonsmooth setting, various metric regularity properties of the subdifferential of a lower semicontinuous convex function acting in a Hilbert space were established by Aragón Artacho and Geoffroy [4] in terms of the quadratic growth condition.
The generalized differentiation in our study is Mordukhovich's coderivative [4], which plays an important role in characterizations of metric regularity and openness properties of set-valued mappings; see [9] and the references therein.
The regularity of the metric at (r_) requires that we must identify (tau rightarrow tau +beta _) and (phi _{i}sim phi _{i}+beta _Omega _{i}), where (beta _) and (Omega _{i}) are the inverse temperature and the angular velocity of the outer horizon [16] begin{aligned} Omega _{i}=frac{a_{i}}{Xi l^{2}}.
Unlike higher-order contingent derivatives based on encounter information, the mth-order variations of a map, based on the different rates of change of the point under consideration in the domain space and the range space of a map, were proposed to obtain the open mapping principle in [26] and consider Hölder metric regularity of set-valued maps in [27].
We studied 61 undergraduates for 30 days using sleep diaries, and quantified sleep regularity using a novel metric, the sleep regularity index (SRI).
The following notion also extends the regularity of an ordered metric space.
We also introduce a notion of regularity for quantum metrics on G, and show how to construct a quantum metric from any ergodic action of G, starting from a regular quantum metric on G. Furthermore, we introduce a quantum Gromov Hausdorff distance between ergodic actions of G when G is separable and show that it induces the above topology.
This study evaluated the external validity of a new metric, the Sleep Regularity Index (SRI), among older adults (n = 1978; mean age 68.7 ± 9.2), as well as relationships between the SRI and cardiometabolic risk using data from the Multi-Ethnic Study of Atherosclerosis (MESA).
