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Geometric lattice - Wikipedia

Every interval of a geometric lattice (the subset of the lattice between given lower and upper bound elements) is itself geometric; taking an interval of a geometric lattice corresponds to forming a minor of the associated matroid. Geometric lattices are complemented, and because of the interval property they are also relatively complemented.

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We design the lattices to extremize effective properties. As customary in topology optimization, we design a periodic unit cell of the lattice and obtain the effective properties via numerical homogenization. Each bar is represented as a cylindrical offset surface of a medial axis parameterized by the positions of the endpoints of the medial axis.

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11. Geometric Lattices - University of Hawaii

Accordingly, we define a geometric lattice to be an algebraic semimodular lattice in which every element is a join of atoms. As we have already described, the points, lines, planes, etc. (and the empty set) of a finite dimensional Euclidean geometry (ℜn) form a geometric lattice.

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researchers new tools to carefully design lattice geometry. These materials are generally classified as architected mate-rials, and have demonstrated near-ideal performance [4–6]. There are various typical fabrication methods, including two-photon lithography direct laser writing of hollow tube alu-mina nano-lattices [7]. Zheng et al. has shown the fabrica- tion of a record ultra-light and ...

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Fig. 1shows some common types of two-dimensional lattice materials, including square, diamond and triangle. According to Fig. 1, the unit cell, which represents a microstructure, is geometrically characterized by a lattice distance a, a thickness of cell wall w/2, and a translation angle γ. For square lattices, the angle γtakes the value of 90o.

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LECTURE Matroids and geometric lattices

The lattice of flats and geometric lattices For a matroid M define L(M) to be the poset of flats of M, ordered by inclusion. Since the intersection of flats is a flat, L(M) is a meet-semilattice; and since L(M) has a top element S, it follows from Lemma 2.3 that L(M) is a lattice, which we call the lattice of flats of M.

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Quantum spin liquids by geometric lattice design

ful platform for materials design and discovery 9,10. Within this framework, in close synergy to interface and heteroepitaxial strain, geometrical lattice engineering (GLE) can be applied to forge unusual artificial lat-tice geometries by stacking a specific number of atomic planes along high-index crystallographic directions 11–13. Here ...

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Geometric lattice | Semantic Scholar

In the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumptions of finiteness. Geometric lattices and matroid lattices, respectively, form the lattices of flats of finite and infinite matroids, and every geometric or matroid lattice comes from a matroid in this way.

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