Related papers: Universal rigidity on the line, point order
In various places in the literature it is stated that every separable linear order embeds into the real line. This is, however, not the case, at least not with respect to the usual definition of separability. We correct this misconception.
In this paper we study the rank of planar rigidity matrix of 4-valent graphs, both in case of generic realizations and configurations in general position, under various connectivity assumptions on the graphs. For each case considered, we…
This paper describes the construction of a canonical compactification of the space of trajectories and of the unstable/stable sets of a generic gradient like vector field on a closed manifold as well as a canonical structure of a smooth…
The uniform one-dimensional fragment U1 is a recently introduced extension of the two-variable fragment FO2. The logic U1 enables the use of relation symbols of all arities and thereby extends the scope of applications of FO2. In this…
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…
We investigate a novel setting for polytope rigidity, where a flex must preserve edge lengths and the planarity of faces, but is allowed to change the shapes of faces. For instance, the regular cube is flexible in this notion. We present…
We formalize the ``metric bundle'' viewpoint by defining, for any smooth $n$--manifold $M$, the open fiberwise cones $\mathcal{G}^{p,q}\subset S^2\Tstar M$ of nondegenerate symmetric bilinear forms with fixed signature $(p,q)$, and we…
In this paper we formulate and prove necessary and sufficient geometric conditions for existence of generic tensegrities in the plane for arbitrary graphs. The conditions are written in terms of "meet-join" relations for the configuration…
We give a linear upper bound on the number of distinct volume-equivalent frameworks of bipyramids, up to rigid motions. As a corollary, we show that global volume rigidity is not a generic property of simplicial complexes.
The threshold network model is a type of finite random graphs. In this paper, we introduce a generalized threshold network model. A pair of vertices with random weights is connected by an edge when real-valued functions of the pair of…
A significant range of geometric structures whose rigidity is explored for both practical and theoretical purposes are formed by modifying generically isostatic triangulated spheres. In the block and hole structures (P, p), some edges are…
We study the generalized roundness of finite metric spaces whose distance matrix $D$ has the property that every row of $D$ is a permutation of the first row. The analysis provides a way to characterize subsets of the Hamming cube $\{ 0, 1…
We prove a topological rigidity theorem for closed hypersurfaces of the Euclidean sphere and of an elliptic space form. It asserts that, under a lower bound hypothesis on the absolute value of the principal curvatures, the hypersurface is…
The main result of this paper is a probabilistic construction of finite rigid structures. It yields a finitely axiomatizable class of finite rigid structures where no L^omega_{infty, omega} formula with counting quantifiers defines a linear…
We study a number of local and global classification problems in generalized complex geometry. In the first topic, we characterize the local structure of generalized complex manifolds by proving that a generalized complex structure near a…
We investigate the problem of assembling general shapes and patterns in a model in which particles move based on uniform external forces until they encounter an obstacle. While previous work within this model of assembly has focused on…
Many mechanical structures, both engineered and biological, combine heavy rigid elements such as bones and beams with lightweight flexible ones such as cables and membranes. These are referred to as tensegrities, reflecting that cables can…
We compare three notions of genericity of separable metric structures. Our analysis provides a general model theoretic technique of showing that structures are generic in descriptive set theoretic (topological) sense and in measure…
We prove that for every $d\geq 3$ the homomorphism order of the class of line graphs of finite graphs with maximal degree $d$ is universal. This means that every finite or countably infinite partially ordered set may be represented by line…
Starting from a Unified Field Theory (UFT) proposed previously by the author, the possible fermionic representations arising from the same spacetime are considered from the algebraic and geometrical viewpoint. We specifically demonstrate in…