Related papers: Structural rigidity and flexibility using graphs o…
We study time-changes of unipotent flows on finite volume quotients of semisimple linear groups, generalising previous work by Ratner on time-changes of horocycle flows. Any measurable isomorphism between time-changes of unipotent flows…
We investigate the motions of a bar structure consisting of two congruent tetrahedra, whose edges in their basic position form the face diagonals of a rectangular parallelepiped. The constraint of the motion is that the originally…
An important, if relatively less well known aspect of the singularity theorems in Lorentzian Geometry is to understand how their conclusions fare upon weakening or suppression of one or more of their hypotheses. Then, theorems with modified…
Problems of flexible mechanical metamaterials, and highly deformable porous solids in general, are rich and complex due to nonlinear mechanics and nontrivial geometrical effects. While numeric approaches are successful, analytic tools and…
We extend our generic rigidity theory for periodic frameworks in the plane to frameworks with a broader class of crystallographic symmetry. Along the way we introduce a new class of combinatorial matroids and associated linear…
We characterise finite and infinitesimal rigidity for bar-joint frameworks in R^d with respect to polyhedral norms (i.e. norms with closed unit ball P a convex d-dimensional polytope). Infinitesimal and continuous rigidity are shown to be…
This is a survey article on the infinitesimal rigidity of frameworks in Euclidean, hyperbolic, and spherical geometry. We discuss the equivalence of the static and kinematic formulations of the infinitesimal rigidity, the projective…
The famous example of the double-Watt mechanism given by Connelly and Servatius raises some problems concerning the classical definitions of higher-order flexibility and rigidity, respectively. Recently, the author was able to give a proper…
Lagrangian and Hamiltonian neural networks (LNNs and HNNs, respectively) encode strong inductive biases that allow them to outperform other models of physical systems significantly. However, these models have, thus far, mostly been limited…
In this study of the Reidemeister moves within the classical knot theory, we focus on hard diagrams of knots and links, categorizing them as either rigid or shaky based on their adaptability to certain moves. We establish that every link…
A linearly constrained framework in $\mathbb{R}^d$ is a point configuration together with a system of constraints which fixes the distances between some pairs of points and additionally restricts some of the points to lie in given affine…
On the transversals of a subgroup of a group, using the binary operation of the group, structural mappings are defined. Based on these mappings, the notion of the hypergroup over the group is introduced, which generalizes the notion of the…
We propose a novel construction of finite hypergraphs and relational structures that is based on reduced products with Cayley graphs of groupoids. To this end we construct groupoids whose Cayley graphs have large girth not just in the usual…
A Lie groupoid, called \textit{material Lie groupoid}, is associated in a natural way to any elastic material. The corresponding Lie algebroid, called \textit{material algebroid}, is used to characterize the uniformity and the homogeneity…
The monography examines the problem of constructing a group of automorphisms of a graph. A graph automorphism is a mapping of a set of vertices onto itself that preserves adjacency. The set of such automorphisms forms a vertex group of a…
A graph H is strongly immersed in G if H is obtained from G by a sequence of vertex splittings (i.e., lifting some pairs of incident edges and removing the vertex) and edge removals. Equivalently, vertices of H are mapped to distinct…
It is known that families of graphs with a semialgebraic edge relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and that they can be decomposed into very homogeneous semialgebraic pieces up to…
In this paper, we introduce a natural classification of bar and joint frameworks that possess symmetry. This classification establishes the mathematical foundation for extending a variety of results in rigidity, as well as infinitesimal or…
Functional structures from across the engineered and biological world combine rigid elements such as bones and columns with flexible ones such as cables, fibers and membranes. These structures are known loosely as tensegrities, since these…
In 1992, Hendrickson proved that (d+1)-connectivity and redundant rigidity are necessary conditions for a generic (non-complete) bar-joint framework to be globally rigid in $\mathbb{R}^d$. Jackson and Jordan confirmed in 2005 that these…