Discrete Triangles Research Articles

Control of the Nucleation Density of Molybdenum Disulfide in Large-Scale Synthesis Using Chemical Vapor Deposition.

Atmospheric pressure chemical vapor deposition (CVD) is presently a promising approach for preparing two-dimensional (2D) MoS2 crystals at high temperatures on SiO2/Si substrates. In this work, we propose an improved CVD method without hydrogen, which can increase formula flexibility by controlling the heating temperature of MoO3 powder and sulfur powder. The results show that the size and coverage of MoS2 domains vary largely, from discrete triangles to continuous film, on substrate. We find that the formation of MoS2 domains is dependent on the nucleation density of MoS2. Laminar flow theory is employed to elucidate the cause of the different shapes of MoS2 domains. The distribution of carrier gas speeds at the substrate surface leads to a change of nucleation density and a variation of domain morphology. Thus, nucleation density and domain morphology can be actively controlled by adjusting the carrier gas flow rate in the experimental system. These results are of significance for understanding the growth regulation of 2D MoS2 crystals.

On discrete triangles characterization

For a given real triangle T its discretization on a discrete point set S consists of points from S which fall into T. If the number of such points is finite the obtained discretization of T will be called discrete triangle. In this paper we show that all discrete triangles from a fixed discretizing set are determined uniquely by their 10 discrete moments which have the order up to 3. Of a particular interest is the case when S is the integer grid, i.e., S= Z 2 . The discretization of a real triangle on Z 2 is called digital triangle. It turns out that the proposed characterization preserves a coding of digital triangles from an integer grid of a given size, say m× m, within an O( logm) amount of memory space per coded digital triangle. That is the theoretical minimum. A possible extension of the proposed coding scheme for digital triangles to the coding digital convex k-gons and arbitrary digital convex shapes is discussed, as well.

Graceful planes and lines

In the framework of the theory of arithmetic geometry (Reveillès, Géométrie Discrète, Calcul en Nombres Entiers et Algorithmique, Thèse d’état, Université Louis Pasteur, Strasbourg, 1991), we propose an approach to discretize polyhedra by meshes of discrete triangles. We propose a general discretization scheme based on reducing the 3D problem to a 2D problem. We introduce new classes of discrete planes and lines called graceful planes and graceful lines. Naive planes and graceful lines are used to construct as thin as possible triangular mesh discretization admitting an analytical description. The interiors of the triangles are portions of naive planes, while the sides are graceful lines. These primitives serve as an optimal ground for obtaining thin tunnel-free discretizations, within the adopted generation scheme. We also extend our considerations to arbitrary surfaces and curves.

Thin discrete triangular meshes

In this paper we present an approach to describe polyhedra by meshes of discrete triangles. The study is based on the theory of arithmetic discrete geometry, (J.-P. Reveillès, Géométrie discrète, calcul en nombres entiers et algorithmique, Thèse d’état, Université Louis Pasteur, Strasbourg, December 1991). As distinct from the previous investigations on this topic, the triangles we introduce are parts of the thinnest possible discrete 6-tunnel-free planes, i.e., those that are usually used in practice. Given a plane P in the space, we define a 6-tunnel-free discrete plane, called a regular plane, which appears to be the best approximation to P . Given a mesh of triangles, we propose a method to approximate any triangle by a discrete triangular patch – a portion of a regular plane, and we prove that the resulting triangular mesh is 6-tunnel-free. The properties of the approximation obtained make the suggested approach convenient for practical applications.