Open porous lattice structures have found a wide range of applications as lightweight load‐bearing and energy absorbent in various fields. A common requirement for any application is the preliminary assessment mechanical response proposed architecture. Herein, uniform continuous functionally graded (FGLSs) made titanium alloy, Ti–6Al–4V, are designed using cubic pillar octahedron unit cells at ...

Density functional theory (DFT) at the hybrid B3LYP level has been applied to the germanium clusters Ge(9)(z) clusters (z = -6, -4, -3, -2, 0, +2, and +4) starting from three different initial configurations. Double-zeta quality LANL2DZ basis functions extended by adding one set of polarization (d) and one set of diffuse (p) functions were used. The global minimum for Ge(9)(2)(-) is the tricapp...

Mr. C. Stephanos posed the following question in the “Intermédiaire des Mathématiciens ” [3]: “Do there exist polyhedra with invariant facets that are susceptible to an infinite family of transformations that only alter solid angles and dihedrals?” I announced, in the same Journal [4], a special concave octahedron possessing the required property. Cauchy, on the other hand, has proved [5] that ...

We have used all-electron local ~LDA! and nonlocal ~GGA! approximations to the density-functional theory to determine binding energies, equilibrium geometries, vibrational frequencies, and magnetic properties of RhN clusters (N<6). We present careful tests on the Rh2 dimer that compare results as calculated with a large ~18-single Gaussian! and a very large ~23-single Gaussian! basis sets. Whil...

The structure of the rhombohedral (R-) AlPdCo phase was determined by single-crystal X-ray diffraction: space group R 3 (No. 148), a = 2.91019(8) nm and c = 1.31854(4) nm, V = 9.6709(5) nm3, atoms/cell = 660, F(000) = 12820, R = 0.0481 for the observed 3354 reflections measured by Mo K¡ radiation (­ = 0.071069 nm). R-AlPdCo has two types of pseudo-Mackay clusters (pMCs) around Co(12) at 18f and...

We herein present the results of a series of calculations performed on some representative cluster models of hybrid organic-inorganic halide perovskites, (MA)jPbkXl (l = 2j + k; MA = methylammonium, +CH3NH3; X = halide). In particular, aimed at finding possible analogies with the bulk, we focused our initial attention on neutral clusters of iodides (X = I) constituted by an increasing number of...

We define and study a recurrence relation in Z, called the hexahedron recurrence, which is similar to the octahedron recurrence (Hirota bilinear difference equation) and cube recurrence (Miwa equation). Like these examples, solutions to the hexahedron recurrence are partition functions for configurations on a certain graph, and have a natural interpretation in terms of cluster algebras. We give...

We prove that alternating links with two totally geodesic checkerboard surfaces are three projection the 1-skeleton of octahedron, cuboctahedron and icosidodecahedron. also characterize these as right-angled completely realisable show all hyperbolic weaving knots exceptions have both not geodesic.