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Topological Analysis of Local Structure in Atomic Systems
Published in Jeffrey P. Simmons, Lawrence F. Drummy, Charles A. Bouman, Marc De Graef, Statistical Methods for Materials Science, 2019
Emanuel A. Lazar, David J. Srolovitz
The language and tools of graph theory can be used to record complete topological information of a Voronoi cell by looking at it as a planar graph. Briefly, a graph is a set of points called vertices, and a set of connections between those vertices are called edges. A planar graph is one whose vertices and edges can be drawn in the plane without any edges crossing. Two graphs are isomorphic if there is a correspondence between their vertices so that two vertices are connected by an edge in one graph if and only if corresponding vertices are connected by an edge in the other graph [1059]. Mathematical theorems from the early twentieth century [965, 1142] guarantee that every Voronoi cell can be uniquely represented as a planar graph, thus allowing us to make precise statements about Voronoi cells using the language of graph theory. Figure 15.8 illustrates planar graphs corresponding to the three Voronoi cells of Figure 15.7.
Introduction
Published in Vlad P. Shmerko, Svetlana N. Yanushkevich, Sergey Edward Lyshevski, Computer Arithmetics for Nanoelectronics, 2018
Vlad P. Shmerko, Svetlana N. Yanushkevich, Sergey Edward Lyshevski
A graph G is planar if it is isomorphic to a graph G' such that (a) the vertices and edges of G' are contained in the same plane, and (b) at most one vertex occupies, or one edge passes through any point on the plane. In other words, a graph is planar if it can be drawn on a plane with no two edges intersecting. Example 5.10 In Figure 5.11a, a planar graph is shown. There are no crossings between edges in this graph. The nonplanar graph G2 in Figure 5.11b can be planarized, that is, converted to an isomorphic planar graph G3 as depicted in Figure 5.11c.
The Emergence of Order in Space
Published in Pier Luigi Gentili, Untangling Complex Systems, 2018
The mechanochemical patterning is responsible for cell polarity (Goehring and Grill 2013). Cell polarity is the asymmetric organization of a cell. Cell polarity is crucial for certain cellular functions, such as cell migration, directional cell growth, and asymmetric cell division. It is also relevant in the formation of tissues. For example, epithelial cells are examples of polarized cells that feature distinct apical and basal plasma membrane domains (see Figure 9.14). The apical-basal polarity drives the opposing surfaces of the cell to acquire distinct functions and chemical components. There is also planar cell polarity that aligns cells and cellular structures such as hairs and bristles within the epithelial plane. Cell polarity is so fundamental that it is ubiquitous among living beings: it is present not only in animals and plants, but also in fungi, prokaryotes, protozoa, and even archaebacteria.
Crossing edge minimization in radial outerplanar layered graphs using segment paths
Published in Optimization Methods and Software, 2023
Francisco Madera-Ramírez, Joel Antonio Trejo-Sánchez, José López-Martínez, Jorge Ríos-Martínez
A graph is an ordered pair consisting of a finite set V of vertices and a finite set E of edges, that is, pairs of vertices. If edge ∈E, vertices u and v are said to be adjacent and is said to be incident to u and v. A drawing Γ of a graph G maps each vertex v to a distinct point of the plane and each edge to a simple open Jordan curve [19] with endpoints and . A drawing is planar if no two distinct edges intersect except, possibly, at common points.
Performance Analysis of Dolph-Tschebyscheff Array for Different SLL and Array Length
Published in IETE Technical Review, 2023
Maloth Gopal, S. S. Patil, K. P. Ray
Antennas with high directional characteristics are essential in diverse applications to cater to the requirements of long-distance wireless communication. An array antenna, which is a collection of radiating elements organized in a linear/planar geometrical configuration, is used to achieve this. The relative displacement between the elements, the excitation amplitude and phase of the individual elements, and their relative radiation pattern determines the overall radiation pattern of an array antenna [1]. The total field of an array is calculated by multiplying the field of a single element by an array factor. The array factor is determined by the number of elements, the inter-element distance and phase, as well as the element excitation amplitudes. Dipoles, loops, slots, microstrips, horns, reflectors, and other elements are employed in a variety of antenna arrays used for personal, commercial, and military applications. Generally, array elements are considered identical to reduce the complexity.
Fabrication of ridge waveguide on the ion-implanted TGG crystal by femtosecond laser ablation
Published in Journal of Modern Optics, 2020
Jing-Yi Chen, Jie Zhang, Liao-Lin Zhang, Chun-Xiao Liu
The optical waveguide structure confines the light to a guiding area with a width of several micrometers, which can not only enhance the density of the light intensities but also significantly reduce the dimensions of optical devices. Therefore, researches on the arts of optical waveguide fabrication are particularly important. Ion implantation/irradiation is an effective material surface modification technology, which can precisely select the ion energy and ion dose to change the refractive index within a specific area of the material, and hence the optical waveguide is fabricated [10–12]. Planar or slab waveguides, the most common optical waveguide structures, restrict light in one dimension. However, in the tremendous demand for fabricating miniaturized and integrated optical devices, two-dimensional (2D) waveguides such as channel and ridge waveguides are more widely applied, owing to their higher intensity density in the waveguide region confined by horizontal and vertical directions [13]. On the basis of ion-implanted planar waveguide wafer, surface patterning technology needs to be incorporated for preparing the bi-directional confinement ridge waveguide. Recently, the femtosecond laser ablation reported in some literatures has proved to be a simple and precise means in micromachining ridge waveguide structures [14–17]. Controlling the trajectory of the ablating laser, the ridge region could be prepared without a mask in the whole process [16,18].