Depth classification based on affine-invariant, weighted and kernel-based spatial depth functions

Several multivariate depth functions have been proposed in the literature, of which some satisfy all the conditions for statistical depth functions while some do not. Spatial depth is known to be invariant to spherical and shift transformations. In this paper, the possibility of using different versions of spatial depth in classification is considered. The covariance-adjusted, weighted, and kernel-based versions of spatial depth functions are presented to classify multivariate outcomes. We extend the maximal depth classification notions for the covariance-adjusted, weighted, and kernel-based spatial depth versions. The classifiers' performance is considered and compared with some existing classification methods using simulated and real datasets.

Analysis of the isopycnal advection in the Lofoten basin (the Norwegian sea)

<p>The Lofoten Basin in the Norwegian Sea is a real reservoir of the Atlantic Waters. The shape of the Basin in the form of a bowl and a great depth with its monotonous increase to the centre results in the Atlantic Water gradually deepen and fill the Basin. The deepening of the Atlantic Waters in the Lofoten Basin determines not only the structure of its waters but also the features of the ocean-atmosphere interaction. Flowing through the transit regions, the Atlantic Waters lose heat to the atmosphere, mix with the surrounding water masses and undergo a transformation, which causes the formation of deep ocean waters. At the same time, the heat input with the Atlantic waters significantly exceeds its loss to the atmosphere in the Lofoten Basin.</p><p>We study isopycnal advection and diapycnal mixing in the Lofoten Basin. We use the GLORYS12V1 oceanic reanalysis data and analyze four isosteric δ-surfaces. We also calculate the depth of their location. We establish that δ-surfaces have the slope eastward with maximal deepening where the quasi-permanent Lofoten Vortex is located. We analyze the temperature distribution on the isosteric δ-surfaces as well as the interannual and seasonal variability of their location depth.</p><p>The maximal depth on the isosteric surfaces is observed in 2010, which is known as the year of the largest mixed layer depths in the Lofoten Basin according to the ARGO buoys. We demonstrate the same correspondence to in 2000, 2010, 2013.</p><p>The maximal depth on the isosteric surfaces is observed is reached in summer. The maximal areas with the greatest depths also are observed in summer in contrast to a minimum in winter. This means certain inertia of changes in the thermohaline characteristics of Atlantic Waters as well as a shift of 1-2 seasons of the influence of deep convection on isosteric surfaces.</p><p>It is shown that isopycnal advection in the Lofoten Basin makes a significant contribution to its importance as the main thermal reservoir of the Nordic Seas.</p>

Generalised superconformal higher-spin multiplets

We propose generalised $$ \mathcal{N} $$ N = 1 superconformal higher-spin (SCHS) gauge multiplets of depth t, $$ {\Upsilon}_{\alpha (n)\overset{\cdot }{\alpha }(m)}^{(t)} $$ ϒ α n α ⋅ m t , with n ≥ m ≥ 1. At the component level, for t > 2 they contain generalised conformal higher-spin (CHS) gauge fields with depths t − 1, t and t + 1. The supermultiplets with t = 1 and t = 2 include both ordinary and generalised CHS gauge fields. Super-Weyl and gauge invariant actions describing the dynamics of $$ {\Upsilon}_{\alpha (n)\overset{\cdot }{\alpha }(m)}^{(t)} $$ ϒ α n α ⋅ m t on conformally-flat superspace backgrounds are then derived. For the case n = m = t = 1, corresponding to the maximal-depth conformal graviton supermultiplet, we extend this action to Bach-flat backgrounds. Models for superconformal non-gauge multiplets, which are expected to play an important role in the Bach-flat completions of the models for $$ {\Upsilon}_{\alpha (n)\overset{\cdot }{\alpha }(m)}^{(t)} $$ ϒ α n α ⋅ m t , are also provided. Finally we show that, on Bach-flat backgrounds, requiring gauge and Weyl invariance does not always determine a model for a CHS field uniquely.

On asymptotic symmetries in higher dimensions for any spin

We investigate asymptotic symmetries in flat backgrounds of dimension higher than or equal to four. For spin two we provide the counterpart of the extended BMS transformations found by Campiglia and Laddha in four-dimensional Minkowski space. We then identify higher-spin supertranslations and generalised superrotations in any dimension. These symmetries are in one-to-one correspondence with spin-s partially-massless representations on the celestial sphere, with supertranslations corresponding in particular to the representations with maximal depth. We discuss the definition of the corresponding asymptotic charges and we exploit the supertranslational ones in order to prove the link with Weinberg’s soft theorem in even dimensions.

KURIL-OKHOTSK REGION

The review of the annual seismicity of the Kurilo-Okhotsk region is submitted. Parameters of 863 earthquakes are determined by records of four Kuril stations. Their distribution of hypocenter depth is given. The maximal depth has made h=620 km in 2013. The distribution of earthquakes on the magnitude and their summarized energy on areas of the region is given. The map of epicenters, together with focal mechanisms for 23 earth-quakes is shown. For each area, seismic conditions are described.

The second Hilbert coefficient of modules with almost maximal depth

Let [Formula: see text] be a good [Formula: see text]-filtration of a finitely generated [Formula: see text]-module [Formula: see text] of dimension [Formula: see text], where [Formula: see text] is a local ring and [Formula: see text] is an [Formula: see text]-primary ideal of [Formula: see text]. In the case of depth [Formula: see text], we give an upper bound for the second Hilbert coefficient [Formula: see text] generalizing the results by Huckaba–Marley, and Rossi–Valla proved that [Formula: see text] is Cohen–Macaulay. We also give a condition for the equality, which relates to the depth of the associated graded module [Formula: see text]. A lower bound on [Formula: see text] is proved generalizing a result by Rees and Narita.

Maximal depth property of finitely generated modules

Let [Formula: see text] be a Noetherian local ring and [Formula: see text] a finitely generated [Formula: see text]-module. We say [Formula: see text] has maximal depth if there is an associated prime [Formula: see text] of [Formula: see text] such that depth [Formula: see text]. In this paper, we study finitely generated modules with maximal depth. It is shown that the maximal depth property is preserved under some important module operations. Generalized Cohen–Macaulay modules with maximal depth are classified. Finally, the attached primes of [Formula: see text] are considered for [Formula: see text].