Geometry optimization for tunable band gap and wave guiding in periodic grid structures

A proper lattice structure consisting of homogeneous material is designed in this paper to investigate the maximum bandwidth of perfect lattice structures and tunable property of waveguide with linear geometric defect by means of selecting optimal geometric lattice cell. A simulation model based on finite element method is used to calculate dispersion curves and transmission spectrums of lattice structures with different geometric parameters. Meanwhile, a simplified theoretical model of unit cell, which considers the mass of grid bar and stiffness of node area, is applied to validate the accuracy of simulation result and may provide an effective approach for prediction of band gap lower boundary. Then, the validated numerical results show different orders of widest band gap that can be realized by different optimal geometric structures. Moreover, waveguide property can be effectively controlled and manipulated by changing defect parameters. The present study may establish theoretical and simulation foundation to control and manipulate band structures and other acoustic propagation characteristics of waveguide devices.

Convexity in Ordered Matroids and the Generalized External Order

In 1980, Las Vergnas defined a notion of discrete convexity for oriented matroids, which Edelman subsequently related to the theory of anti-exchange closure functions and convex geometries. In this paper, we use generalized matroid activity to construct a convex geometry associated with an ordered, unoriented matroid. The construction in particular yields a new type of representability for an ordered matroid defined by the affine representability of its corresponding convex geometry. The lattice of convex sets of this convex geometry induces an ordering on the matroid independent sets which extends the external active order on matroid bases. We show that this generalized external order forms a supersolvable meet-distributive lattice refining the geometric lattice of flats, and we uniquely characterize the lattices isomorphic to the external order of a matroid. Finally, we introduce a new trivariate generating function generalizing the matroid Tutte polynomial.

Lattice AdS geometry and continuum limit

We construct the lattice AdS geometry. The lattice AdS2 geometry and AdS3 geometry can be extended from the lattice AdS2 induced metric, which provided the lattice Schwarzian theory at the classical limit. Then we use the lattice embedding coordinates to rewrite the lattice AdS2 geometry and AdS3 geometry with the manifest isometry. The lattice AdS2 geometry can be obtained from the lattice AdS3 geometry through the compactification without the lattice artifact. The lattice embedding coordinates can also be used in the higher-dimensional AdS geometry. Because the lattice Schwarzian theory does not suffer from the issue of the continuum limit, the lattice AdS2 geometry can be obtained from the higher-dimensional AdS geometry through the compactification, and the lattice AdS metric does not depend on the angular coordinates, we expect that the continuum limit should exist in the lattice Einstein gravity theory from this geometric lattice AdS geometry. Finally, we apply this lattice construction to construct the holographic tensor network without the issue of a continuum limit.

A microtubule RELION-based pipeline for cryo-EM image processing

Microtubules are polar filaments built from αβ-tubulin heterodimers that exhibit a range of architectures in vitro and in vivo. Tubulin heterodimers are arranged helically in the microtubule wall but many physiologically relevant architectures exhibit a break in helical symmetry known as the seam. Noisy 2D cryo-electron microscopy projection images of pseudo-helical microtubules therefore depict distinct but highly similar views owing to the high structural similarity of α- and β-tubulin. The determination of the αβ-tubulin register and seam location during image processing is essential for alignment accuracy that enables determination of biologically relevant structures. Here we present a pipeline designed for image processing and high-resolution reconstruction of cryo-electron microscopy microtubule datasets, based in the popular and user-friendly RELION image-processing package, Microtubule RELION-based Pipeline (MiRP). The pipeline uses a combination of supervised classification and prior knowledge about geometric lattice constraints in microtubules to accurately determine microtubule architecture and seam location. The presented method is fast and semi-automated, producing near-atomic resolution reconstructions with test datasets that contain a range of microtubule architectures and binding proteins.AbbreviationsMiRP, Microtubule RELION-based Pipeline; cryo-EM, cryo-electron microscopy; MT, microtubule; CTF, contrast transfer function; PF, protofilament.

Some problems about geometric lattice

In this paper, the survey about some results of the convex lattice set are given and the invariance of projection problem of convex lattice set is also obtained. And combining a famous result in the graph theory, several conjectures about the convex lattice set are presented.

Characterization and interpretation of twin related row-matching orientation relationships between Mg2Sn precipitates and the Mg matrix

Two irrational orientation relationships (ORs) are newly observed between β-Mg2Sn precipitates and the α-Mg matrix in an aged Mg–Sn-based alloy, namely OR9: [011]β-9//[0 1 {\bar 1} 0]α, (100)β-9about −15.6° from (0001)αand OR10: [0 {\bar 1} {\bar 1}]β-10//[0 1{\bar 1} 0]α, (100)β-10about 4.0° from ({\bar 2} 1 1 0)α. The ORs have a common reciprocal row-matching condition, which shows that rows of diffraction spots parallel tog(1 1{\bar 1})β-9[org({\bar 1} {\bar 1}1)β-10] from lattice β match those parallel tog(2 {\bar 1} {\bar 1} 4)αfrom lattice α. A twin relationship exists between OR9-type and OR10-type precipitates, of which the major side facets, parallel to the twinning plane, are invariably normal to the matching rows in reciprocal space. The reciprocal row matching of diffraction spots is proven to correspond to a real row matching of lattice sites in direct space. The real matching rows of lattice sites lie in the major side facets, resulting in periodically distributed good matching lattice bands in the facets,i.e.a singular interfacial structure. This explains why the irrational ORs are preferred. This study demonstrates an explicit association of geometric lattice matching with the ORs and the morphologies of precipitates developed during a phase transformation.

DNA Codeword Design: Theory and Applications

This is a survey of the origin, current progress and applications of a major roadblock to the development of analytic models for DNA computing (a massively parallel programming methodology) and DNA self-assembly (a nanofabrication methodology), namely the so-called CODEWORD DESIGN problem. The problem calls for finding large sets of single DNA strands that do not crosshybridize to themselves or to their complements and has been recognized as an important problem in DNA computing, self-assembly, DNA memories and phylogenetic analyses because of their error correction and prevention properties. Major recent advances include the development of experimental techniques to search for such codes, as well as a theoretical framework to analyze this problem, despite the fact that it has been proven to be NP-complete using any single concrete metric space to model the Gibbs energy. In this framework, codeword design is reduced to finding large sets of strands maximally separated in DNA spaces and, therefore, the key to finding such sets would lie in knowledge of the geometry of these spaces. A new general technique has been recently found to embed them in Euclidean spaces in a hybridization-affinity-preserving manner, i.e., in such a way that oligos with high/low hybridization affinity are mapped to neighboring/remote points in a geometric lattice, respectively. This isometric embedding materializes long-held metaphors about codeword design in terms of sphere packing and error-correcting codes and leads to designs that are in some cases known to be provably nearly optimal for some oligo sizes. It also leads to upper and lower bounds on estimates of the size of optimal codes of size up to 32–mers, as well as to infinite families of solutions to CODEWORD DESIGN, based on estimates of the kissing (or contact) number for sphere packings in Euclidean spaces. Conversely, this reduction suggests interesting new algorithms to find dense sphere packing solutions in high dimensional spheres using results for CODEWORD DESIGN previously obtained by experimental or theoretical molecular means, as well as a proof that finding these bounds exactly is NP-complete in general. Finally, some research problems and applications arising from these results are described that might be of interest for further research.

Geometric Lattice Structure of Covering and Its Application to Attribute Reduction through Matroids

The reduction of covering decision systems is an important problem in data mining, and covering-based rough sets serve as an efficient technique to process the problem. Geometric lattices have been widely used in many fields, especially greedy algorithm design which plays an important role in the reduction problems. Therefore, it is meaningful to combine coverings with geometric lattices to solve the optimization problems. In this paper, we obtain geometric lattices from coverings through matroids and then apply them to the issue of attribute reduction. First, a geometric lattice structure of a covering is constructed through transversal matroids. Then its atoms are studied and used to describe the lattice. Second, considering that all the closed sets of a finite matroid form a geometric lattice, we propose a dependence space through matroids and study the attribute reduction issues of the space, which realizes the application of geometric lattices to attribute reduction. Furthermore, a special type of information system is taken as an example to illustrate the application. In a word, this work points out an interesting view, namely, geometric lattice, to study the attribute reduction issues of information systems.