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Representation of Stochastic Microstructures
Published in Jeffrey P. Simmons, Lawrence F. Drummy, Charles A. Bouman, Marc De Graef, Statistical Methods for Materials Science, 2019
Topological invariants refer to proper properties of an object that are preserved under homeomorphisms (the continuous stretching or bending of an object into a new shape). Here we will limit the discussion to metrics related to homology or the mathematical classification of objects based on the number of “holes” and connected/disconnected components in the structure. The strength of this approach to characterizing stochastic materials is that it provides a description of the long-range connectivity in a microstructure. The drawback of homology as a metric is the lack of scale in the topological relationships. It is intuitive to expect that the size and shape of clusters or percolating paths are important [390], however here we are only consider metrics that are invariant to stretching and bending. Size and anisotropy are neglected from a topological point of view.
Onion-Like Inorganic Fullerenes from a Polyhedral Perspective
Published in Klaus D. Sattler, st Century Nanoscience – A Handbook, 2020
Ch. Chang, A. B. C. Patzer, D. Sülzle, H. Bauer
where × is called the Euler characteristic and g the genus (i.e. the number of holes in or tunnels through the polyhedral framework). The Euler characteristic is a topological invariant that describes the shape of a structure regardless of how it is bent. Positive/negative × values indicate positive/negative curvature of the polyhedral structure. For every convex polyhedron, i.e. a polyhedron having no holes, tunnels, kinks etc. the genus g is equal to 0 and, therefore, χ is always equal to 2 and the Euler relation1 becomes: υ−e+f=2.
Two-Dimensional Topological Insulators
Published in Grigory Tkachov, Topological Quantum Materials, 2015
In the preceding chapter, we used simple intuitive arguments to introduce the two-dimensional topological insulators as 2D systems with an insulating interior and a Kramers pair of conducting edge states. Such systems are also known as quantum spin Hall insulators. We now intend to construct theoretical models that would elucidate the origin of quantum spin Hall insulators. We have already mentioned the original theoretical works on the quantum spin Hall effect in the context of graphene [7, 8] and semiconductor materials [11–13]. These models are closely related to the notion of Chern insulators. It goes back to Haldane’s paper [6] on a condensed matter realization of the quantum electrodynamical “parity anomaly” in 2 + 1 dimensions. Chern insulators have a nontrivial band structure characterized by a topological invariant, the Chern number, hence the name. These systems exhibit broken time-reversal symmetry, chiral edge states and a quantum Hall effect in the absence of any external magnetic field and Landau levels. Albeit unusual, this was envisioned by the theory of Thouless, Kohmoto, Nightingale and den Nijs [3]. It establishes a general relation between an observable—the Hall conductivity—and the topological Chern number. We shall explore different models of Chern insulators and use them as building blocks for quantum spin Hall insulators. In particular, a nontrivial band topology will be explained in terms of an intrinsic band inversion in reciprocal space [12, 13]. This key attribute of topological insulators reflects their striking quantum mechanics combining the elements of both Dirac and Schroedinger equations.
Topological perception on attention to product shape
Published in International Journal of Design Creativity and Innovation, 2020
Topology is a major branch of mathematics concerned with spatial properties that are preserved under one-to-one, continuous transformations, such as stretching and bending, but not breaking or fusing. Important topological properties include connectedness and compactness. Hence, solid figures such a cube and a tetrahedron are topologically equivalent, because one can be transformed into the other through continuous transformations. An attribute of an object is called topologically invariant if it does not change under a continuous transformation. If two objects have different topological invariants, they must be different homeomorphisms. One of the most common topological invariants is the Euler characteristic, which is commonly denoted by χ and conventionally defined for the surfaces of a polyhedron as χ= V – E + F, where V, E, and F are the numbers of vertices (corners), edges, and faces in the polyhedron, respectively. For example, the surface of a convex polyhedron has the Euler characteristic χ = V – E + F = 2. On the other hand, several human-made goods are homeomorphic to either spheres or rings and can be molded forming tiles. The topological transformations can be pictured as distorting a rubber sheet without creating holes or fusing edges. The Euler characteristics of a sphere and a ring (i.e. a torus) are 2 and 0, respectively.
Bandgaps and topological interfaces of metabeams with periodic acoustic black holes
Published in Mechanics of Advanced Materials and Structures, 2022
On the other hand, in recent years, the researches of topological phenomena in quantum Hall and topological insulators have been extensively studied in condensed matter physics [29,30]. Topological insulators, based on a great deal of qualitative yet robust topological invariants instead of specific geometric features or constitutive parameters, can conduct localized waves along edges and interfaces with negligible scattering and losses [31]. Recently, the interface state in ABH has been examined in beams [32] and plates [33], which extend the design of topological properties in vibration and noise control. Lyu et al. [32] explored the topological interface in a periodic ABH beam from both numerical and experimental aspects. Multiple topological inversion points are found and it opened a new possibility for manipulating the topological inversion points in a certain range. Semperlotti et al. [33] studied the phononic plates with the periodic lattice of ABH which can support topological edge states. They found that the ABH can achieve Dirac dispersion in very low frequency and extend the design of topological properties in frequency ranges. Besides, the artificial structures in acoustic systems are explored to achieve acoustic topological interface states in the Bragg bandgaps [34,35] and the local resonant bandgaps [36,37]. Meanwhile, the topological interface states of the flexural waves [38,39] and longitudinal waves [40,41] have been studied in one-dimensional elastic systems. Motivated by this, the topological phenomenon is illustrated based on the mass-spring model, and a compound ABH metabeam is proposed to achieve the topological interface state in this paper.
Topological indices of line graph of transition metal tetra cyano benzene organic network
Published in Molecular Physics, 2023
Muhammad Faisal Nadeem, Muhammad Talha Farooq, Eman Alzahrani, Hala M. Abo-Dief, Zeinhom M. El-Bahy
Cheminformatics is a relatively young field that combines chemistry, mathematics and information science. Topological descriptors are essential in molecular chemistry, specially in analyses of QSPR and quantitative structure–activity relationship (QSAR) . The topological invariant is a real number linked with molecular constitutions which suggests links among molecular structures and several physical properties, chemical reactions and processes, which describes the graph's topology, and under graph isomorphism, they are invariant. Topological descriptors are usually classified into three groups degree [10], distance [11,12] and spectrum [13–16]. Topological descriptors based on degree and distances are also studied [17–22].