Related papers: Liaison Linkages

We describe each multiple curve on the orientable surface of genus-$g$ with $n$ punctures and one boundary component by using this multiple curve's geometric intersection number with the embedded curves in this surface.

Jaeger's directed cycle double cover conjecture can be formulated as a problem of existence of special perfect matchings in a class of graphs that we call hexagon graphs. In this work, we explore the structure of hexagon graphs. We show…

Reductions of higher tangent bundles of Lie groupoids provide natural examples of geometric structures which we would like to call higher algebroids. Such objects can be also constructed abstractly starting from an arbitrary almost Lie…

The linear transports along paths in vector bundles introduced in Ref. [1] are applied to the special case of tensor bundles over a given differentiable manifold. Links with the transports along paths generated by derivations of tensor…

A \emph{$(k,t)$-track layout} of a graph $G$ consists of a (proper) vertex $t$-colouring of $G$, a total order of each vertex colour class, and a (non-proper) edge $k$-colouring such that between each pair of colour classes no two…

In translation surfaces of finite area (corresponding to holomorphic differentials), directions of saddle connections are dense in the unit circle. On the contrary, saddle connections are fewer in translation surfaces with poles…

Given a finite, simple graph $G$, the $k$-component order edge connectivity of $G$ is the minimum number of edges whose removal results in a subgraph for which every component has order at most $k-1$. In general, determining the…

In this paper we present a generalization of the classical configuration model. Like the classical configuration model, the generalized configuration model allows users to specify an arbitrary degree distribution. In our generalized…

The well-known geometric approach to field theory is based on description of classical fields as sections of fibred manifolds, e.g. bundles with a structure group in gauge theory. In this approach, Lagrangian and Hamiltonian formalisms…

It is known that parallel manipulators suffer from singular configurations. Evaluating the distance between a given configuration to the closest singular one is of interest for industrial applications (e.g.\ singularity-free path planning).…

A linkage mechanism consists of rigid bodies assembled by joints which can be used to translate and transfer motion from one form in one place to another. In this paper, we are particularly interested in a family of spacial linkage…

A homothety surface can be assembled from polygons by identifying their edges in pairs via homotheties, which are compositions of translation and scaling. We consider linear trajectories on a 1-parameter family of genus-2 homothety…

The problem of constructing curves with many points over finite fields has received considerable attention in the recent years. Using the class field theory approach, we construct new examples of curves ameliorating some of the known…

We develop a framework that systematically casts the solvability and uniqueness conditions of linearized geometric boundary-value problems into cohomological terms. The theory is designed to be applicable without assumptions on the…

We find a sharp bound for the order of the automorphism group of a stable curve of genus $g$ with $3g-3$ nodes, and a sharp bound for the order of the automorphism group of such a curve with all smooth components. Combined with the results…

Working over imperfect fields, we give a comprehensive classification of genus-one curves that are regular but not geometrically regular, extending the known case of geometrically reduced curves. The description is given intrinsically, in…

In recent years, the question of whether the ranks of elliptic curves defined over $\mathbb{Q}$ are unbounded has garnered much attention. One can create refined versions of this question by restricting one's attention to elliptic curves…

If a smooth, closed, and embedded curve is deformed along its normal vector field at a rate proportional to its curvature, it shrinks to a circular point. This curve evolution is called Euclidean curve shortening and the result is known as…

The gonality of a smooth geometrically connected curve over a field $k$ is the smallest degree of a nonconstant $k$-morphism from the curve to the projective line. In general, the gonality of a curve of genus $g \ge 2$ is at most $2g - 2$.…

This article is expository in nature, outlining some of the many still incompletely understood features of higher spin field theory. We are mainly considering higher spin gauge fields in their own right as free-standing theoretical…