The infinitesimal symmetry algebra of any Cartan geometry has maximum dimension realized by the flat model, but often this dimension drops significantly when considering non-flat geometries, so a gap phenomenon arises. For general (regular, normal) parabolic geometries of type (G,P), we use Tanaka theory to derive a universal upper bound on the submaximal symmetry dimension. We use Kostant’s version ...

Generalizing polynomials previously studied in the context of linear codes, we define weight polynomials and an enumerator for a matroid M. Our main result is that these polynomials are determined by Betti numbers associated with N0-graded minimal free resolutions of the Stanley-Reisner ideals of M and so-called elongations of M. Generalizing Greene’s the- orem from coding theory, we show that ...

We define generalized Hamming weights for almost affine codes. We show that this definition is natural since we can extend some well known properties of t he generalized Hamming weights for linear codes, to almost affine codes. In addition we discus s duality of almost affine codes, and of the smaller class of multilinear codes.

We find necessary and sufficient conditions for the foliation defined by level sets of a function f(x1, ..., xn) to be totally geodesic in a torsion-free connection and apply them to find the conditions for d-webs of hypersurfaces to be geodesic, and in the case of flat connections, for d-webs (d ≥ n + 1) of hypersurfaces to be hyperplanar webs. These conditions are systems of generalized Euler ...

We prove that any planar 4-web defines a unique projective structure in the plane in such a way that the leaves of the web foliations are geodesics of this projective structure. We also find conditions for the projective structure mentioned above to contain an affine symmetric connection, and conditions for a planar 4-web to be equivalent to a geodesic 4-web on an affine symmetric surface. Similar ...

Polar lows are intense mesoscale cyclones developing in marine polar air masses. This study presents a new global climatology of polar lows based on the ERA5 reanalysis for the years 1979–2020. Criteria for the detection of polar lows are derived based on a comparison of five polarlow archives with cyclones derived by a mesoscale tracking algorithm. The characteristics associated with polar lows are ...

Reproducible climate reconstructions of the Common Era (1 CE to present) are key to placing industrial-era warming into the context of natural climatic variability. Here we present a community-sourced database of temperature-sensitive proxy records from the PAGES2k initiative. The database gathers 692 records from 648 locations, including all continental regions and major ocean basins. The records ...

A simple conceptual model for the global mean surface temperature (GMST) response to CO2 emissions is presented and analysed. It consists of linear long-memory models for the GMST anomaly response ∆T to radiative forcing and atmospheric CO2 - 5 concentration response ∆C to emission rate. The responses are connected by the standard logarithmic relation between CO2 concentration and its radiative ...

To each linear code C over a finite field we associate the matroid M(C) of its parity check matrix. For any matroid M one can define its generalized Hamming weights, and if a matroid is associated to such a parity check matrix, and thus of type M(C) , these weights are the same as those of the code C . In our main result we show how the weights d1,…,dk of a matroid M are determined by the N -graded ...

In graph neural networks (GNNs), pooling operators compute local summaries of input graphs to capture their global properties, and they are fundamental for building deep GNNs that learn hierarchical representations. In this work, we propose the Node Decimation Pooling (NDP), a pooling operator for GNNs that generates coarser graphs while preserving the overall graph topology. During training, the ...