Optimal shapes of compact strings

Optimal geometrical arrangements, such as the stacking of atoms, are of relevance in diverse disciplines. A classic problem is the determination of the optimal arrangement of spheres in three dimensions in order to achieve the highest packing fraction; only recently has it been proved that the answer for infinite systems is a face-centred-cubic lattice. This simply stated problem has had a profound impact in many areas, ranging from the crystallization and melting of atomic systems, to optimal packing of objects and subdivision of space. Here we study an analogous problem--that of determining the optimal shapes of closely packed compact strings. This problem is a mathematical idealization of situations commonly encountered in biology, chemistry and physics, involving the optimal structure of folded polymeric chains. We find that, in cases where boundary effects are not dominant, helices with a particular pitch-radius ratio are selected. Interestingly, the same geometry is observed in helices in naturally-occurring proteins.Comment: 8 pages, 3 composite ps figure

An Overview of Schema Theory

The purpose of this paper is to give an introduction to the field of Schema Theory written by a mathematician and for mathematicians. In particular, we endeavor to to highlight areas of the field which might be of interest to a mathematician, to point out some related open problems, and to suggest some large-scale projects. Schema theory seeks to give a theoretical justification for the efficacy of the field of genetic algorithms, so readers who have studied genetic algorithms stand to gain the most from this paper. However, nothing beyond basic probability theory is assumed of the reader, and for this reason we write in a fairly informal style. Because the mathematics behind the theorems in schema theory is relatively elementary, we focus more on the motivation and philosophy. Many of these results have been proven elsewhere, so this paper is designed to serve a primarily expository role. We attempt to cast known results in a new light, which makes the suggested future directions natural. This involves devoting a substantial amount of time to the history of the field. We hope that this exposition will entice some mathematicians to do research in this area, that it will serve as a road map for researchers new to the field, and that it will help explain how schema theory developed. Furthermore, we hope that the results collected in this document will serve as a useful reference. Finally, as far as the author knows, the questions raised in the final section are new.Comment: 27 pages. Originally written in 2009 and hosted on my website, I've decided to put it on the arXiv as a more permanent home. The paper is primarily expository, so I don't really know where to submit it, but perhaps one day I will find an appropriate journa

Rotating mesons in the presence of higher derivative corrections from gauge-string duality

We consider a rotating quark-antiquark $(q\bar{q})$ pair in $\mathcal{N}=4$ thermal plasma. By using AdS/CFT correspondence, the properties of this system have been investigated. We study variation of rotating string radius at the boundary as a function of the tip of U-shape string and angular velocity of rotating meson. We also extend the results to the higher derivative corrections i.e. ${\cal{R}}^2$ and ${\cal{R}}^4$ which correspond to finite coupling corrections on the rotating quark-antiquark system in the hot plasma. In ${\cal{R}}^4$ case and for fixed angular velocity as $\lambda^{-1}$ decreases the string endpoints get more and more separated. To study ${\cal{R}}^2$ corrections, rotating quark-antiquark system in Gauss-Bonnet background has been considered. We summarize the effects of these corrections in the conclusion section.Comment: 21 pages, 11 figures, NPB version, corrections to the effects of higher derivative term

The geometrical origin of the strain-twist coupling in double helices

The geometrical coupling between strain and twist in double helices is investigated. Overwinding, where strain leads to further winding, is shown to be a universal property for helices, which are stretched along their longitudinal axis when the initial pitch angle is below the zero-twist angle (39.4 deg). Unwinding occurs at larger pitch angles. The zero-twist angle is the unique pitch angle at the point between overwinding and unwinding, and it is independent of the mechanical properties of the double helix. This suggests the existence of zero-twist structures, i.e. structures that display neither overwinding, nor unwinding under strain. Estimates of the overwinding of DNA, chromatin, and RNA are given.Comment: 8 pages, 4 figures; typos fixed; added ref. and acknowledgemen