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Hypothetical Zeolite
Hypothetical zeolites are combinatorial models of potential structures of the minerals known as zeolites. They are four-regular periodic graphs, with vertices representing the tetrahedral atom (usually Si or Al) and the edges representing the bridging oxygen atoms. Alternatively, by ignoring the tetrahedral atoms, zeolites can be represented by a six-valent periodic graph of oxygen atoms, with each O-O edge defining an edge of a tetrahedron. At present there are millions of hypothetical zeolite frameworks known. For any fixed number of vertices, these are a small fraction of possible 4-regular periodic graphs, because the geometric constraints imposed by the chemistry of zeolites rules out most possibilities. Enumeration methods There are several methods for enumerating zeolite frameworks. The method of Earl et al. uses simulated annealing to arrange a fixed number of tetrahedral atoms under periodic symmetry constraints. The method of Treacy and Rivin uses combinatorial enume ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeolite
Zeolites are microporous, crystalline aluminosilicate materials commonly used as commercial adsorbents and catalysts. They mainly consist of silicon, aluminium, oxygen, and have the general formula ･y where is either a metal ion or H+. These positive ions can be exchanged for others in a contacting electrolyte solution. exchanged zeolites are particularly useful as solid acid catalysts. The term ''zeolite'' was originally coined in 1756 by Swedish mineralogist Axel Fredrik Cronstedt, who observed that rapidly heating a material, believed to have been stilbite, produced large amounts of steam from water that had been adsorbed by the material. Based on this, he called the material ''zeolite'', from the Greek , meaning "to boil" and , meaning "stone". Zeolites occur naturally but are also produced industrially on a large scale. , 253 unique zeolite frameworks have been identified, and over 40 naturally occurring zeolite frameworks are known. Every new zeolite structure that is ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Graph (crystallography)
In crystallography, a periodic graph or crystal net is a three-dimensional periodic graph, i.e., a three-dimensional Euclidean graph whose vertices or nodes are points in three-dimensional Euclidean space, and whose edges (or bonds or spacers) are line segments connecting pairs of vertices, periodic in three linearly independent axial directions. There is usually an implicit assumption that the set of vertices are uniformly discrete, i.e., that there is a fixed minimum distance between any two vertices. The vertices may represent positions of atoms or complexes or clusters of atoms such as single-metal ions, molecular building blocks, or secondary building units, while each edge represents a chemical bond or a polymeric ligand. Although the notion of a periodic graph or crystal net is ultimately mathematical (actually a crystal net is nothing but a periodic realization of an abelian covering graph over a finite graph ), and is closely related to that of a Tessellation of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enumeration
An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration (for example, whether the set must be finite, or whether the list is allowed to contain repetitions) depend on the discipline of study and the context of a given problem. Some sets can be enumerated by means of a natural ordering (such as 1, 2, 3, 4, ... for the set of positive integers), but in other cases it may be necessary to impose a (perhaps arbitrary) ordering. In some contexts, such as enumerative combinatorics, the term ''enumeration'' is used more in the sense of ''counting'' – with emphasis on determination of the number of elements that a set contains, rather than the production of an explicit listing of those elements. Combinatorics In combinatorics, enumeration means counting, i.e., determining the exact number of elemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simulated Annealing
Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem. It is often used when the search space is discrete (for example the traveling salesman problem, the boolean satisfiability problem, protein structure prediction, and job-shop scheduling). For problems where finding an approximate global optimum is more important than finding a precise local optimum in a fixed amount of time, simulated annealing may be preferable to exact algorithms such as gradient descent or branch and bound. The name of the algorithm comes from annealing in metallurgy, a technique involving heating and controlled cooling of a material to alter its physical properties. Both are attributes of the material that depend on their thermodynamic free energy. Heating and cooling the material affects both the temperature and the the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Embedding
In topological graph theory, an embedding (also spelled imbedding) of a Graph (discrete mathematics), graph G on a surface (mathematics), surface \Sigma is a representation of G on \Sigma in which points of \Sigma are associated with graph theory, vertices and simple arcs (Homeomorphism, homeomorphic images of [0,1]) are associated with graph theory, edges in such a way that: * the endpoints of the arc associated with an edge e are the points associated with the end vertices of e, * no arcs include points associated with other vertices, * two arcs never intersect at a point which is interior to either of the arcs. Here a surface is a compact space, compact, connected space, connected 2-manifold. Informally, an embedding of a graph into a surface is a drawing of the graph on the surface in such a way that its edges may intersect only at their endpoints. It is well known that any finite graph can be embedded in 3-dimensional Euclidean space \mathbb^3.. A planar graph is one that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a spindle torus. If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is called a ''toroid'', as in a square toroid. Real-world objects that approximate a torus of revolution include swim rings, inner tubes and ringette rings. Eyeglass lenses that combine spherical and cylindrical correction are toric lenses. A torus should not be confused with a '' solid torus'', which is formed by r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedra
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of a polytope, a more general concept in any number of dimensions. Definition Convex polyhedra are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts,. some more rigorous than others, and there is not universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra) or include shapes that are often not considered as valid polyhed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
