Related papers: Maximal Angle Flow on the Shape Sphere of Triangle…

In this paper we develop an approach to conformal geometry of piecewise flat metrics on manifolds. In particular, we formulate the combinatorial Yamabe problem for piecewise flat metrics. In the case of surfaces, we define the combinatorial…

Optimal geometrical arrangements, such as the stacking of atoms, are of relevance in diverse disciplines. A classic problem is the determination of the optimal arrangement of spheres in three dimensions in order to achieve the highest…

We show that every graph of maximum degree three can be drawn in three dimensions with at most two bends per edge, and with 120-degree angles between any two edge segments meeting at a vertex or a bend. We show that every graph of maximum…

Consider a graph with a rotation system, namely, for every vertex, a circular ordering of the incident edges. Given such a graph, an angle cover maps every vertex to a pair of consecutive edges in the ordering -- an angle -- such that each…

We investigate two-dimensional liquid bridges trapped between pairs of identical horizontal cylinders. The cylinders support forces due to surface tension and hydrostatic pressure which balance the weight of the liquid. The shape of the…

Consider a polygon P and all neighboring circles (circles going through three consecutive vertices of P). We say that a neighboring circle is extremal if it is empty (no vertices of P inside) or full (no vertices of P outside). It is well…

In part 1, we proposed a model of dynamics of wetting for slow movements near a contact line formed at the interface of two immiscible fluids and a solid when viscous dissipation remains bounded. The contact line is not a material line and…

We present a detailed analysis of the flow in a 180 sharp bend of square cross-section. Besides numerous applications where this generic configuration is found, its main fundamental interest resides in the co-existence of a recirculation…

We study flip graphs of triangulations whose maximum vertex degree is bounded by a constant $k$. In particular, we consider triangulations of sets of $n$ points in convex position in the plane and prove that their flip graph is connected if…

Orbits in triaxial ellipsoidal potentials precess about either the major or minor axis of the ellipsoid. In standard perturbation theory it can be shown that a circular orbit will precess about the minor axis if its angular momentum vector…

We present a conjecture on the exact location of the multicritical point in the phase diagram of spin glass models in finite dimensions. By generalizing our previous work, we combine duality and gauge symmetry for replicated random systems…

The conventional no-slip boundary condition leads to a non-integrable stress singularity at a moving contact line. This makes numerical simulations challenging, especially when capillary effects are essential for the dynamics of the flow.…

I investigate the caustics produced by the fall of collisionless dark matter in and out of a galaxy in the limit of negligible velocity dispersion. The outer caustics are spherical shells enveloping the galaxy. The inner caustics are rings.…

We investigate the convergence of the mean curvature flow of arbitrary codimension in Riemannian manifolds with bounded geometry. We prove that if the initial submanifold satisfies a pinching condition, then along the mean curvature flow…

A rotating system, such as a star, liquid drop, or atomic nucleus, may rotate as an oblate spheroid about its symmetry axis or, if the angular velocity is greater than a critical value, as a triaxial ellipsoid about a principal axis. The…

We investigate triangulations of the two-dimensional sphere and torus with the faces properly colored white and black. We focus on matchings between white triangles and incident vertices. On the torus our objects are perfect pairings,…

It is conjectured that every cusped hyperbolic 3-manifold has a decomposition into positive volume ideal hyperbolic tetrahedra (a "geometric" triangulation of the manifold). Under a mild homology assumption on the manifold we construct…

The optimal one-sided parametric polynomial approximants of a circular arc are considered. More precisely, the approximant must be entirely in or out of the underlying circle of an arc. The natural restriction to an arc's approximants…

The onset of convection in a rotating cylindrical annulus with parallel ends filled with a compressible fluid is studied in the anelastic approximation. Thermal Rossby waves propagating in the azimuthal direction are found as solutions. The…

This paper explores the electrical and thermal conductivity of complex contact spots on the surface of a half-space. Employing an in-house Fast Boundary Element Method implementation, various complex geometries were studied. Our…