Related papers: Zero-sum cycles in flexible polyhedra
The present paper considers volume formulae, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation…
A projective mirror polyhedron is a projective polyhedron endowed with reflections across its faces. We construct an explicit diffeomorphism between the moduli space of a mirror projective polyhedron with fixed dihedral angles in…
Motivated by the electroweak hierarchy problem, we consider theories with two extra dimensions in which the four-dimensional scalar fields are components of gauge boson in full space, namely the Gauge-Higgs unification framework. We briefly…
Associated to every complete affine 3-manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface S. We classify such complete affine structures when Sigma is homeomorphic to a three-holed sphere. In particular, for…
A polyhedron is flexible if it can be continuously deformed preserving the shape and dimensions of every its face. In the late 1970's Klaus Steffen constructed a sphere-homeomorphic embedded flexible polyhedron with triangular faces and…
We consider the question of which zero-dimensional schemes deform to a collection of distinct points; equivalently, we ask which Artinian k-algebras deform to a product of fields. We introduce a syzygetic invariant which sheds light on this…
We consider algebraic manifolds $Y$ of dimension 3 over $\Bbb{C}$ with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$ and $i>0$. Let $X$ be a smooth completion of $Y$ with $D=X-Y$, an effective divisor on $X$ with normal crossings. If the…
To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible…
In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth…
We consider F-theory compactifications on a mirror pair of elliptic Calabi-Yau threefolds. This yields two different six-dimensional theories, each of them being nonperturbatively equivalent to some compactification of heterotic strings on…
In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also…
This paper investigates the rigidity of bordered polyhedral surfaces. Using the variational principle, we show that bordered polyhedral surfaces are determined by boundary value and discrete curvatures on the interior edges. As a corollary,…
This research provides algorithms and numerical methods to geometrically control the magnitude of the internal and external forces in the reciprocal diagrams of 3D/Polyhedral Graphic statics (3DGS). In 3DGS, the form of the structure and…
We consider the problem of decomposing a regular non-negative function as a sum of squares of functions which preserve some form of regularity. In the same way as decomposing non-negative polynomials as sum of squares of polynomials allows…
For a nonempty polyhedral set $P\subset \mathbb R^d$, let $\mathcal F(P)$ denote the set of faces of $P$, and let $N(P,F)$ be the normal cone of $P$ at the nonempty face $F\in\mathcal F(P)$. We prove that the function $\sum_{F\in\mathcal…
We examine three-dimensional metric deformations based on a tetrad transformation through the action the matrices of scalar fields. We describe by this approach to deformation the results obtained by Coll et al. in [1], where it is stated…
A fundamental theorem of Laman characterises when a bar-joint framework realised generically in the Euclidean plane admits a non-trivial continuous deformation of its vertices. This has recently been extended in two ways. Firstly to…
A certain real number, depending on two neighbouring sides of a quadrilateral and the diagonal meeting these two sides at their common point, is shown to be invariant under affinity. As an application we demonstrate a nice formula for the…
In this paper, we are interested in flat metric structures with conical singularities on surfaces which are obtained by deforming translation surface structures. The moduli space of such flat metric structures can be viewed as some…
In heterotic flux compactification with supersymmetry, three different connections with torsion appear naturally, all in the form $\omega+a H$. Supersymmetry condition carries $a=-1$, the Dirac operator has $a=-1/3$, and higher order term…