Intrinsic universality and the computational power of self-assembly

This short survey of recent work in tile self-assembly discusses the use of simulation to classify and separate the computational and expressive power of self-assembly models. The journey begins with the result that there is a single universal tile set that, with proper initialization and scaling, simulates any tile assembly system. This universal tile set exhibits something stronger than Turing universality: it captures the geometry and dynamics of any simulated system. From there we find that there is no such tile set in the noncooperative, or temperature 1, model, proving it weaker than the full tile assembly model. In the two-handed or hierarchal model, where large assemblies can bind together on one step, we encounter an infinite set, of infinite hierarchies, each with strictly increasing simulation power. Towards the end of our trip, we find one tile to rule them all: a single rotatable flipable polygonal tile that can simulate any tile assembly system. It seems this could be the beginning of a much longer journey, so directions for future work are suggested.Comment: In Proceedings MCU 2013, arXiv:1309.104

Effect of pentagons in sp3 systems on electronic, elastic, and vibrational properties: Case of chiral structures

We present first-principles calculations of carbon and silicon chiral framework structures (CFSs). In this system, proposed recently by Pickard and Needs [Phys. Rev. B 81, 014106 (2010)], atoms form only pentagonal cycles. This configuration enables unambiguous analysis of the effects of pentagons on electronic, vibrational, and thermodynamic properties. The local density approximation electronic band gaps in CFSs were found to be equal to or greater than those of clathrates using the same formalism, as confirmed by GW calculations: 1.8 and 5.5 eV for Si and C-CFS, respectively. We show that, as in clathrates, an increasing electronic band gap is correlated with the contraction of the valence bands, resulting from the frustration of the p shells. The electron localized function and Wannier analysis confirm the sp3 nature of the bonds. Finally, we discuss vibrational and related properties. We show that CFSs present singularities, in particular, that the higher frequencies are not located at the Γ point

Spectrum of the exponents of best rational approximation

Using the new theory of W. M. Schmidt and L. Summerer called parametric geometry of numbers, we show that the going-up and going-down transference inequalities of W. M. Schmidt and M. Laurent describe the full spectrum of the $n$ exponents of best rational approximation to points in $\mathbb{R}^{n+1}$.Comment: 13 pages, 4 figures, minor corrections since version

Construction of points realizing the regular systems of Wolfgang Schmidt and Leonard Summerer

In a series of recent papers, W. M. Schmidt and L. Summerer developed a new theory by which they recover all major generic inequalities relating exponents of Diophantine approximation to a point in $\mathbb{R}^n$, and find new ones. Given a point in $\mathbb{R}^n$, they first show how most of its exponents of Diophantine approximation can be computed in terms of the successive minima of a parametric family of convex bodies attached to that point. Then they prove that these successive minima can in turn be approximated by a certain class of functions which they call $(n,\gamma)$-systems. In this way, they bring the whole problem to the study of these functions. To complete the theory, one would like to know if, conversely, given an $(n,\gamma)$-system, there exists a point in $\mathbb{R}^n$ whose associated family of convex bodies has successive minima which approximate that function. In the present paper, we show that this is true for a class of functions which they call regular systems.Comment: 11 pages, 1 figure, to appear in Journal de th\'eorie des nombres de Bordeau