Nature utilizes self-assembly to fabricate structures on length scales ranging from the atomic to the macro scale. Self-assembly has emerged as a paradigm in engineering that enables the highly parallel fabrication of complex, and often three-dimensional, structures from basic building blocks. Although there have been several demonstrations of this self-assembly fabrication process, rules that ...

With using the carbon nano-tube (CNT) of high chemical activity, nano-TiCx particles with different growth shapes were synthesized through the self-propagating high temperature in the 80 wt.% metal (Cu, Al, and Fe)-Ti-CNT systems. The growth shapes of the TiCx particles are mainly octahedron in the Cu- and Al-Ti-CNT systems, while mainly cube- and sphere-like in the Fe-Ti-CNT system.

An earlier paper describes a program to prove the Kepler conjecture on sphere packings. This paper carries out the second step of that program. A sphere packing leads to a decomposition ofR3 into polyhedra. The polyhedra are divided into two classes. The first class of polyhedra, called quasi-regular tetrahedra, have density at most that of a regular tetrahedron. The polyhedra in the remaining ...

We propose a new heuristic for pure 0–1 programs, which finds feasible integer points by enumerating extended facets of the octahedron, the outer polar of the unit hypercube. We give efficient algorithms to carry out the enumeration, and we explain how our heuristic can be embedded in a branch-and-cut framework. Finally, we present computational results on a set of pure 0–1 programs taken from ...

Between tradition (Hilbert’s 12-th Problem) and actual challenges (coding theory) we attack infinite two-dimensional Galois theory. From a number theoretic point of view we work over Q(x). Geometrically, one has to do with towers of Shimura surfaces and Shimura curves on them. We construct and investigate a tower of rational Picard modular surfaces along a Galois group isomorphic to the (double...

We prove that the 82 link complement is the minimal volume orientable hyperbolic manifold with 4 cusps. Its volume is twice of the volume V8 of the ideal regular octahedron, i.e. 7.32... = 2V8. The proof relies on Agol’s argument used to determine the minimal volume hyperbolic 3-manifolds with 2 cusps. We also need to estimate the volume of a hyperbolic 3-manifold with totally geodesic boundary...

Woodall, D.R., A zero-free interval for chromatic polynomials, Discrete Mathematics 101 (1992) 333-341. It is proved that, for a wide class of near-triangulations of the plane, the chromatic polynomial has no zeros between 2 and 2.5. Together with a previously known result, this shows that the zero of the chromatic polynomial of the octahedron at 2.546602. . . is the smallest non-integer real z...

The electron paramagnetic resonance spectra of Cu doped Na 2 HAs0 4 -7H 2 0 single crystals were studied at room temperature. The results indicate the substitutional entrance of Cu in two magnetically inequivalent Na sites. Charge compensation is supposed to be fulfilled by proton vacancies. The spin Hamiltonian parameters were determined. The ground state for Cu seems to indicate the dominance...