Related papers: The Flexibility and Rigidity of Leaper Frameworks
The shape of semiflexible polymer rings is studied over their whole range of flexibility. Investigating the joint distribution of asphericity and nature of asphericity as well as their respective averages we find two distinct shape regimes…
We provide a rigorous mathematical analysis of a coupled system consisting of a floating platform in a fluid of finite depth, clamped to a flexible Euler-Bernoulli beam. The superstructure supports a rigid tip mass at its free end,…
A longstanding problem in rigidity theory is to characterize the graphs which are minimally generically rigid in 3-space. The results of Cauchy, Dehn, and Alexandrov give one important class: the triangulated convex spheres, but there is an…
In 2005, Wyart et al. (Europhys. Lett., 72 (2005) 486) showed that the low frequency vibrational properties of jammed amorphous sphere packings can be understood in terms of a length scale, called l*, that diverges as the system becomes…
We study stable commutator length on free $\mathbb{Q}$-groups. We prove that every non-identity element has positive stable commutator length, and that the corresponding free group embeds isometrically. We deduce that a non-abelian free…
We present three results which support the conjecture that a graph is minimally rigid in $d$-dimensional $\ell_p$-space, where $p\in (1,\infty)$ and $p\not=2$, if and only if it is $(d,d)$-tight. Firstly, we introduce a graph bracing…
Fekete, Jord\'an and Kaszanitzky [4] characterised the graphs which can be realised as 2-dimensional, infinitesimally rigid, bar-joint frameworks in which two given vertices are coincident. We formulate a conjecture which would extend their…
Lagrangian multiform theory is a variational framework for integrable systems. In this article we introduce a new formulation which is based on symplectic geometry and which treats position, momentum and time coordinates of a…
An infinite periodic framework in the plane can be represented as a framework on a torus, using a $\mathbb Z^2$-labelled gain graph. We find necessary and sufficient conditions for the generic minimal rigidity of frameworks on the…
A rigidity theory is developed for frameworks in a metric space with two types of distance constraints. Mixed sparsity graph characterisations are obtained for the infinitesimal and continuous rigidity of completely regular bar-joint…
We present a rigorous study of framework rigidity in general finite dimensional normed spaces from the perspective of Lie group actions on smooth manifolds. As an application, we prove an extension of Asimow and Roth's 1978/9 result…
A surface is considered flexible if it allows a continuous deformation that preserves both metric and smoothness. We introduce a novel construction method, called 'base + crinkle,' for generating a broad class of non-self-intersecting…
Motivated by the recent approach of Milman, Shabelman, and Yehudayoff \cite{MilmanShabelmanYehudayoff2025}, we establish, for $p\geq 1$, a complete characterization of the fixed points of the composition of the $L_p$-centroid operator and…
This paper proves a 2017 conjecture of De Loera, La Haye, Oliveros, and Rold\'an-Pensado that the "prime grid" $\big\{(p,q) \in \mathbb{Z}^2 : \text{$p$ and $q$ are prime}\big\} \subseteq \mathbb{R}^2$ contains empty polygons with…
We prove that the space of contractible simple loops of a given fixed area in any compact oriented surface has infinite diameter as a homogeneous space of the group of area-preserving diffeomorphisms endowed with the $L^p$-metric. As a…
We study the structure of strictly singular non-compact operators between $L_p$ spaces. Answering a question raised in [Adv. Math. 316 (2017), 667-690], it is shown that there exist operators $T$, for which the set of points…
Given a connected graph $G$ and a configuration of $t$ pebbles on the vertices of G, a $q$-pebbling step consists of removing $q$ pebbles from a vertex, and adding a single pebble to one of its neighbors. Given a vector…
For a finite point set $E\subset \mathbb{R}^d$ and a connected graph $G$ on $k+1$ vertices, we define a $G$-framework to be a collection of $k + 1$ points in E such that the distance between a pair of points is specified if the…
It is well-known that there exist bar-joint frameworks (without continuous flexions) whose physical models can snap between different realizations due to non-destructive elastic deformations of material. We present a method to measure these…
A two-dimensional grid with dots is called a \emph{configuration with distinct differences} if any two lines which connect two dots are distinct either in their length or in their slope. These configurations are known to have many…