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Register a new userSlider-pinning Rigidity: a Maxwell-Laman-type Theorem
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We define and study slider-pinning rigidity, giving a complete combinatorial characterization. This is done via direction-slider networks, which are a generalization of Whiteleys direction networks.
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A one-to-one correspondence between the infinitesimal motions of bar-joint frameworks in $mathbb{R}^d$ and those in $mathbb{S}^d$ is a classical observation by Pogorelov, and further connections among different rigidity models in various different spaces have been extensively studied. In this paper, we shall extend this line of research to include the infinitesimal rigidity of frameworks consisting of points and hyperplanes. This enables us to understand correspondences between point-hyperplane rigidity, classical bar-joint rigidity, and scene analysis. Among other results, we derive a combinatorial characterization of graphs that can be realized as infinitesimally rigid frameworks in the plane with a given set of points collinear. This extends a result by Jackson and Jord{a}n, which deals with the case when three points are collinear.
We introduce a geometric generalization of Halls marriage theorem. For any family $F = {X_1, dots, X_m}$ of finite sets in $mathbb{R}^d$, we give conditions under which it is possible to choose a point $x_iin X_i$ for every $1leq i leq m$ in such a way that the points ${x_1,...,x_m}subset mathbb{R}^d$ are in general position. We give two proofs, one elementary proof requiring slightly stronger conditions, and one proof using topological techniques in the spirit of Aharoni and Haxells celebrated generalization of Halls theorem.
Following an argument proposed by Mason, we prove that there are no algebraically special asymptotically simple vacuum space-times with a smooth, shear-free, geodesic congruence of principal null directions extending transversally to a cross-section of Scri. Our analysis leaves the door open for escaping this conclusion if the congruence is not smooth, or not transverse to Scri. One of the elements of the proof is a new rigidity theorem for the Trautman-Bondi mass.
We consider surfaces with boundary satisfying a sixth order nonlinear elliptic partial differential equation corresponding to extremising the $L^2$-norm of the gradient of the mean curvature. We show that such surfaces with small $L^2$-norm of the second fundamental form and satisfying so-called `flat boundary conditions are necessarily planar.
We consider the problem of finding an inductive construction, based on vertex splitting, of triangulated spheres with a fixed number of additional edges (braces). We show that for any positive integer $b$ there is such an inductive construction of triangulations with $b$ braces, having finitely many base graphs. In particular we establish a bound for the maximum size of a base graph with $b$ braces that is linear in $b$. In the case that $b=1$ or $2$ we determine the list of base graphs explicitly. Using these results we show that doubly braced triangulations are (generically) minimally rigid in two distinct geometric contexts arising from a hypercylinder in $mathbb{R}^4$ and a class of mixed norms on $mathbb{R}^3$.
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