Divergence operators and odd Poisson brackets

We define the divergence operators on a graded algebra, and we show that, given an odd Poisson bracket on the algebra, the operator that maps an element to the divergence of the hamiltonian derivation that it defines is a generator of the bracket. This is the "odd laplacian", $\Delta$, of Batalin-Vilkovisky quantization. We then study the generators of odd Poisson brackets on supermanifolds, where divergences of graded vector fields can be defined either in terms of berezinian volumes or of graded connections. Examples include generators of the Schouten bracket of multivectors on a manifold (the supermanifold being the cotangent bundle where the coordinates in the fibres are odd) and generators of the Koszul-Schouten bracket of forms on a Poisson manifold (the supermanifold being the tangent bundle, with odd coordinates on the fibres).Comment: 27 pages; new Section 1, introduction and conclusion re-written, typos correcte

Sedentariness in quantum walks

We present a relaxation of the concept of a sedentary family of graphs introduced by Godsil [Linear Algebra Appl. 614:356-375, 2021] and provide sufficient conditions for a given vertex in a graph to exhibit sedentariness. We show that a vertex with at least two twins (vertices that share the same neighbours) is sedentary. We also prove that there are infinitely many graphs containing strongly cospectral vertices that are sedentary, which reveals that, even though strong cospectrality is a necessary condition for pretty good state transfer, there are strongly cospectral vertices which resist high probability state transfer to other vertices. Moreover, we derive results about sedentariness in products of graphs which allow us to construct new sedentary families, such as Cartesian powers of complete graphs and stars.Comment: 26 pages, 3 figure

Curves with constant curvature ratios

Curves in Rn for which the ratios between two consecutive curvatures are constant are characterized by the fact that their tangent indicatrix is a geodesic in a flat torus. For n = 3, 4, spherical curves of this kind are also studied and compared with intrinsic helices in the sphere