We observe that, up to conjugation, a majority of higher order ODEs (ordinary differential equations) and ODE systems have only fiber-preserving point symmetries. By exploiting Lie's classification of Lie algebras of vector fields, we describe all exceptions to this in the case of scalar ODEs and systems of ODEs on a pair of functions. The scalar ODEs whose symmetry algebras are not fiber preserving ...

We present a complete description of a class of linearizable planar geodesic webs which contain a parallelizable 3-subweb.

Hybrid clustering combines partitional and hierarchical clustering for computational effectiveness and versatility in cluster shape. In such clustering, a dissimilarity measure plays a crucial role in the hierarchical merging. The dissimilarity measure has great impact on the final clustering, and data-independent properties are needed to choose the right dissimilarity measure for the problem ...

We discuss rank 2 sub-Riemannian structures on low-dimensional manifolds and prove that some of these structures in dimensions 6, 7 and 8 have a maximal amount of symmetry but no integrals polynomial in momenta of low degrees, except for those coming from the Killing vector fields and the Hamiltonian, thus indicating nonintegrability of the corresponding geodesic flows.

We introduce some novel adaptations of Life-Like Automata. We focus mainly on some modifications of the original rules where the evolution of the system is depending from the step and the zone of the grid. We study also the properties of some novel non-deterministic adaptations.

We study differential invariants of linear differential operators and use them to find conditions for equivalence of differential operators acting on line bundles over smooth manifolds with respect to groups of automorphisms.

Let R be a real closed field. We prove that for any fixed <i>d</i>, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of R^<i>k</i> defined by polynomials of degrees bounded by <i>d</i> vanishes in dimensions <i>d</i> and larger. This vanishing result is tight. Using a new geometric approach we also prove an upper bound of [mathematical formula] on the ...

We consider symmetric (under the action of products of finite symmetric groups) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of degrees bounded by a fixed constant <i>d</i>. We prove that if a Specht module, S^λ⁠, appears with positive multiplicity in the isotypic decomposition of the ...

Among (regular, normal) parabolic geometries of type (G,P), there is a locally unique maximally symmetric structure and it has symmetry dimension dim(G). The symmetry gap problem concerns the determination of the next realizable (submaximal) symmetry dimension. When G is a complex or split-real simple Lie group of rank at least three or when (G,P)=(G2,P2), we establish a local classification result ...

Arctic Ocean properties and processes are highly relevant to the regional and global coupled climate system, yet still scarcely observed, especially in winter. Team OCEAN conducted a full year of physical oceanography observations as part of the Multidisciplinary drifting Observatory for the Study of the Arctic Climate (MOSAiC), a drift with the Arctic sea ice from October 2019 to September 2020. ...