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In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A fairly complete picture of the representation theory for CW left regular bands is obtained.

Modern computational protocols based on the density functional theory (DFT) infer that polyhedral closo ten-vertex carboranes are key starting stationary states in obtaining ten-vertex cationic carboranes. The rearrangement of the bicapped square polyhedra into decaborane-like shapes with open hexagons in boat conformations is caused by attacks of N-heterocyclic carbenes (NHCs) on the closo motifs. Single-point computations on the stationary points found during computational examinations of the reaction pathways have clearly shown that taking the “experimental” NHCs into account requires the use of dispersion correction. Further examination has revealed that for the purposes of the description of reaction pathways in their entirety, i.e., together with all transition states and intermediates, a simplified model of NHCs is sufficient. Many of such transition states resemble in their shapes those that dictate Z-rearrangement among various isomers of closo ten-vertex carboranes. Computational results are in very good agreement with the experimental findings obtained earlier.

Significance This work demonstrates a structure–property correlation in metallic glasses for the community of amorphous solids. It associates geometrically unfavored motifs, i.e., those most disordered local polyhedral packing structures in a metallic glass, with the soft spots defined from the vibrational modes and correlates them with shear transformation zones composed of atoms with large nonaffine displacements. The statistical correlation established thus ties together the heterogeneity inherent in the amorphous structure with the spatial heterogeneity in the mechanical (elastic and plastic) properties of a metallic glass. In a 3D model mimicking realistic Cu ₆₄Zr ₃₆ metallic glass, we uncovered a direct link between the quasi-localized low-frequency vibrational modes and the local atomic packing structure. We also demonstrate that quasi-localized soft modes correlate strongly with fertile sites for shear transformations: geometrically unfavored motifs constitute the most flexible local environments that encourage soft modes and high propensity for shear transformations, whereas local configurations preferred in this alloy, i.e., the full icosahedra (around Cu) and Z16 Kasper polyhedra (around Zr), contribute the least.

With the advantage of fast charge transfer, heterojunction engineering is identified as a viable method to reinforce the anodes' sodium storage performance. Also, vacancies can effectively strengthen the Na+ adsorption ability and provide extra active sites for Na+ adsorption. However, their synchronous engineering is rarely reported. Herein, a hybrid of Co0.85Se/WSe2 heterostructure with Se vacancies and N‐doped carbon polyhedron (CoWSe/NCP) has been fabricated for the first time via a hydrothermal and subsequent selenization strategy. Spherical aberration‐corrected transmission electron microscopy confirms the phase interface of the Co0.85Se/WSe2 heterostructure and the existence of Se vacancies. Density functional theory simulations reveal the accelerated charge transfer and enhanced Na+ adsorption ability, which are contributed by the Co0.85Se/WSe2 heterostructure and Se vacancies, respectively. As expected, the CoWSe/NCP anode in sodium‐ion battery achieves outstanding rate capability (339.6 mAh g−1 at 20 A g−1), outperforming almost all Co/W‐based selenides. A hybrid of Co0.85Se/WSe2 heterostructure with Se vacancies and N‐doped carbon polyhedron (CoWSe/NCP) has been fabricated for the first time. The Co0.85Se/WSe2 heterostructure can accelerate charge transfer and the Se vacancies can strengthen the Na+ adsorption ability. As an anode for sodium‐ion batteries, the CoWSe/NCP hybrid achieves outstanding rate performance, outperforming almost all Co/W‐based selenides, together with competitive cycling stability.

The polyhedra with A 3 , B 3 / C 3 , H 3 reflection symmetry group G in the real 3 D space are considered. The recursive rules for finding orbits with smaller radii, which provide the structures of nested polytopes, are demonstrated.

This paper surveys the construction, properties, and applications of generalized barycentric coordinates on polygons and polyhedra. Applications include: surface mesh parametrization in geometric modelling; image, curve, and surface deformation in computer graphics; and polygonal and polyhedral finite element methods.

Conventional embeddings of the edge-graphs of Platonic polyhedra, {f, z}, where f, z denote the number of edges in each face and the edge-valence at each vertex, respectively, are untangled in that they can be placed on a sphere (𝕊²) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of 𝕊¹ on the sphere. The most symmetric (flag-transitive) realizations of those polyhedral graphs are those of the classical Platonic polyhedra, whose symmetries are *2fz, according to Conway’s two-dimensional (2D) orbifold notation (equivalent to Schönflies symbols Ih, Oh , and Td ). Tangled Platonic {f , z} polyhedra—which cannot lie on the sphere without edge-crossings—are constructed as windings of helices with three, five, seven,... strands on multigenus surfaces formed by tubifying the edges of conventional Platonic polyhedra, have (chiral) symmetries 2fz (I, O, andT), whose vertices, edges, and faces are symmetrically identical, realized with two flags. The analysis extends to the “θz ” polyhedra, {2, z}. The vertices of these symmetric tangled polyhedra overlap with those of the Platonic polyhedra; however, their helicity requires curvilinear (or kinked) edges in all but one case. We show that these 2fz polyhedral tangles are maximally symmetric; more symmetric embeddings are necessarily untangled. On one hand, their topologies are very constrained: They are either self-entangled graphs (analogous to knots) or mutually catenated entangled compound polyhedra (analogous to links). On the other hand, an endless variety of entanglements can be realized for each topology. Simpler examples resemble patterns observed in synthetic organometallic materials and clathrin coats in vivo.

We develop a general approach to valid inference after model selection. At the core of our framework is a result that characterizes the distribution of a post-selection estimator conditioned on the selection event. We specialize the approach to model selection by the lasso to form valid confidence intervals for the selected coefficients and test whether all relevant variables have been included in the model.