Related papers: The Flexibility and Rigidity of Leaper Frameworks
It is well-known that the property of a bar-and-joint framework `to be infinitesimally rigid' is invariant under projective transformations of Eucliean $d$-space for every $d\geqslant 2$. It is less known that the property of a…
For a given rational number $x$ and an integer $s\geq 1$, let us consider a generalized polylogarithmic function, often called the Lerch function, defined by $$\Phi_{s}(x,z)= \sum_{k=0}^{\infty}\frac{z^{k+1}}{(k+x+1)^s}\enspace.$$ We prove…
Many problems in robotics require reasoning over a mix of continuous dynamics and discrete events, such as making and breaking contact in manipulation and locomotion. These problems are locally well modeled by linear complementarity…
Dense non-Brownian suspension flows of hard particles display mystifying properties: as the jamming threshold is approached, the viscosity diverges, as well as a length scale that can be identified from velocity correlations. To unravel the…
Motivated by the challenging formation stabilization problem for mobile robotic teams when no distance or relative displacement measurements are available and each robot can only measure some of those angles formed by rays towards its…
We show weighted non-autonomous $L^q(L^p)$ maximal regularity for families of complex second-order systems in divergence form under a mixed regularity condition in space and time. To be more precise, we let $p,q \in (1,\infty)$ and we…
We present a novel parabolic reflector system capable of generating a broader class of shapes beyond canonical parabolas. Using a discretized framework, we construct meshes corresponding to key families of developable surfaces, including…
Ice systems are prototypes of locally constrained dynamics. This is exemplified in Coulomb-liquid phases where a large space of configurations is sampled, each satisfying local ice rules. Dynamics proceeds through `flipping' rings, i.e.,…
The presence of symmetries is one of the central structural features that make some integer programs challenging for state-of-the-art solvers. In this work, we study the efficacy of Linear Programming (LP) hierarchies in the presence of…
In 2005, Bob Connelly showed that a generic framework in $\bR^d$ is globally rigid if it has a stress matrix of maximum possible rank, and that this sufficient condition for generic global rigidity is preserved by the 1-extension operation.…
We give a combinatorial characterization of generic frameworks that are minimally rigid under the additional constraint of maintaining symmetry with respect to a finite order rotation or a reflection. To establish these results we develop a…
We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a flat folding by a sequence of simple folds? There are several models of simple folds;…
Given a continuous function $f:X\to\mathbb{R}$ and a cover $\mathcal{I}$ of its image by intervals, the Mapper is the nerve of a refinement of the pullback cover $f^{-1}(\mathcal{I})$. Despite its success in applications, little is known…
Here it is shown how to combine two generically globally rigid bar frameworks in $d$-space to get another generically globally rigid framework. The construction is to identify $d+1$ vertices from each of the frameworks and erase one of the…
Following a review of related results in rigidity theory, we provide a construction to obtain generically universally rigid frameworks with the minimum number of edges, for any given set of n nodes in two or three dimensions. When a set of…
We use computational method to investigate the number of ways to pack dimers on \emph{odd-by-odd} lattices. In this case, there is always a single vacancy in the lattices. We show that the dimer configuration numbers on $(2k+1) \times…
The rigidity matrix is a fundamental tool for studying the infinitesimal rigidity properties of Euclidean bar-joint frameworks. In this paper we generalize this tool and introduce a rigidity matrix for bar-joint frameworks in arbitrary…
Given a quartic Galois extension $L/\mathbb{Q}$ of number fields and a Hopf-Galois structure $H$ on $L/\mathbb{Q}$, we study the freeness of the ring of integers $\mathcal{O}_L$ as module over the associated order $\mathfrak{A}_H$ in $H$.…
We study uniform stability of discrete groups, Lie groups and Lie algebras in the rank metric, and the connections between uniform stability of these objects. We prove that semisimple Lie algebras are far from being flexibly…
We present a survey of results concerning the use of inductive constructions to study the rigidity of frameworks. By inductive constructions we mean simple graph moves which can be shown to preserve the rigidity of the corresponding…