Let I be a 3D digital image, and let Q(I) be the associated cubical complex. In this paper we show how to simplify the combinatorial structure of Q(I) and obtain a homeomorphic cellular complex P (I) with fewer cells. We introduce formulas for a diagonal approximation on a general polygon and use it to compute cup products on the cohomology H∗(P (I)). The cup product encodes important geometric...

Crystal engineering has recently emerged as a method of choice for the design and construction of organic as well as metal-organic functional materials. Crystal engineering attempts to establish packing trends in whole families of compounds and seeks to establish connections between structure and function. The utility of crystal engineering has also been expanded to the nanoscience and the deve...

the copper(ii) complex [cu(pic)2]·2h2o was synthesized with 2-picolinic acid (hpic) and copper acetate as reactants by room temperature solid-solid reaction. the composition and structure of the complex was characterized by elemental analyses, single crystal x-ray diffraction, x-ray powder diffraction, ft-ir spectroscopy and thermogravimetry-differential scanning calorimetry. the crystal struct...

A certain inequality is shown to hold for the values of the Möbius function of the poset obtained by attaching a maximum element to a lower Eulerian Cohen– Macaulay poset. In two important special cases, this inequality provides partial results supporting Stanley’s nonnegativity conjecture for the toric h-vector of a lower Eulerian Cohen–Macaulay meet-semilattice and Adin’s nonnegativity conjec...

Almtract--We present a comprehensive survey of the theory of hypercube graphs. Basic properties related to distance, coloring, domination and genus are reviewed. The properties of the n-cube defined by its subgraphs are considered next, including thickness, coarseness, Hamiltonian cycles and induced paths and cycles. Finally, various embedding and packing problems are discussed, including the d...

Cohomology and cohomology ring of three-dimensional (3D) objects are topological invariants that characterize holes and their relations. Cohomology ring has been traditionally computed on simplicial complexes. Nevertheless, cubical complexes deal directly with the voxels in 3D images, no additional triangulation is necessary, facilitating efficient algorithms for the computation of topological ...

We prove a Matrix-Tree Theorem enumerating the spanning trees of a cell complex in terms of the eigenvalues of its cellular Laplacian operators, generalizing a previous result for simplicial complexes. As an application, we obtain explicit formulas for spanning tree enumerators and Laplacian eigenvalues of cubes; the latter are integers. We prove a weighted version of the eigenvalue formula, pr...

Abstract This article proposes a way of doing type theory informally, assuming cubical style reasoning. It can thus be viewed as first step toward alternative to the program informalization carried out in homotopy book for dependent augmented with axioms univalence and higher inductive types. We adopt cartesian proposed by Angiuli, Brunerie, Coquand, Favonia, Harper, Licata implicit foundation,...