Related papers: On discrete constant principal curvature surfaces
In this paper, we study a natural discretization of the smooth Gaussian curvature on surfaces. A discrete uniformization theorem is established for this discrete Gaussian curvature. We further investigate the prescribing combinatorial…
The atomic structure and elastic properties of silicon carbide nanowires of different shapes and effective sizes were studied using density functional theory and classical molecular dynamics. The surface relaxation led to surface…
This article begins by reviewing the causal set approach in discrete quantum gravity. In our version of this approach a special role is played by covariant causal sets which we call $c$-causets. The importance of $c$-causets is that they…
In this survey, we study three different notions of curvature that are defined on graphs, namely, combinatorial curvature, Bakry-\'Emery curvature, and Ollivier's Ricci curvature. For each curvature notion, the definition and its motivation…
We study a generalization of constant Gauss curvature -1 surfaces in Euclidean 3-space, based on Lorentzian harmonic maps, that we call pseudospherical frontals. We analyze the singularities of these surfaces, dividing them into those of…
We consider constant mean curvature surfaces of finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors in…
Vector fields and line fields, their counterparts without orientations on tangent lines, are familiar objects in the theory of dynamical systems. Among the techniques used in their study, the Morse--Smale decomposition of a (generic) field…
We consider surfaces with parallel mean curvature vector field and finite total curvature in product spaces of type $\mathbb{M}^n(c)\times\mathbb{R}$, where $\mathbb{M}^n(c)$ is a space form, and characterize certain of these surfaces. When…
This work develops optimal preconditioners for the discrete H(curl) and H(div) problems on two-dimensional surfaces by nodal auxiliary space preconditioning [R. Hiptmair, J. Xu: SIAM J. Numer. Anal. \textbf{45}, 2483-2509 (2007)]. In…
We make a systematic study of the focal surface of a congruence of lines in the projective space. Using differential techniques together with techniques from intersection theory, we reobtain in particular all the invariants of the focal…
The discrete tori are graph analogues of the real tori, which are defined by the Cayley graphs of a finite product of finite cyclic groups. In this paper, using the theory of the heat kernel on the discrete tori established by Chinta,…
Carbon nanotube (CNT)-polymer composites are widely engineered using surface coatings and chemical treatments to improve interfacial bonding and load transfer. It has been suggested in the nanocomposite literature that nanotube curvature,…
The Separatrix Theorem of C. Camacho and P. Sad guarantees the existence of invariant curve (separatrix) passing through the singularity of germ of holomorphic foliation on complex surface, when the surface underlying the foliation is…
Carrier injection into carbon nanotubes and graphene nanoribbons, contacted by a metal coating over an arbitrary length, is studied by various means: Minimal models allow for exact analytic solutions which can be transferred to the original…
Allotropes of carbon, including one-dimensional carbon nanotubes and two-dimensional graphene sheets, continue to draw attention as promising platforms for probing the physics of electrons in lower dimensions. Recent research has shown that…
The structure of simplicial manifolds in a model of Causal Dynamical Triangulations in 3+1 dimensions with the spatial topology of a 3-torus is analyzed with the help of topological observables, such as loops with nonzero winding numbers…
An investigation concerning the tautness of suspended carbon nanotubes (CNTs), grown using the chemical vapor deposition (CVD) method is presented. The suspended nanotubes were analyzed with both a transmission electron microscope (TEM),…
Carbon fiber and graphene-based nanostructures such as carbon nanotubes (CNTs) and defective structures have extraordinary potential as strong and lightweight materials. A longstanding bottleneck has been lack of understanding and…
Smooth and curved microstructural topologies found in nature - from soap films to trabecular bone - have inspired several mimetic design spaces for architected metamaterials and bio-scaffolds. However, the design approaches so far have been…
We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global…