| Recent experimental advances have opened up the possibility of equilibrium self-assembly of functionalized nanoblocks with a high degree of controllable specific interactions. We propose design principles for selecting short-range interactions between self-assembling components to maximize yield. We illustrate the approach with an example from colloidal engineering. In general, it might be necessary to use more component than strictly required for enforcing the ground state configuration.;Next, we generalize our theory to an arbitrary set of interactions and apply it to the well-studied problem of protein folding. We compute the similarity of all pairs of amino acids from their interaction energies and occurrence frequencies, by estimating the ground state energy of proteins comprised of subsets of amino acids. We show that the natural frequencies of amino acids effectively enhance the similarity between residues that are most frequently interchanged due to mutations/errors. Protein composition in synergy with the genetic code is potentially a result of evolutionary driving force minimizing impact of amino acid substitutions on protein structure.;Lastly, self-assembly using DNA origami is considered. Using an extension of our simple model, a catalogue of synthetic DNA sequences is proposed that could potentially improve yield in these systems.;Part II is on the search for finite-time singularities in the incompressible Euler equations. A promising mechanism for generating singularities is stretching of vortex filaments. An exhaustive search of all possible filament initial conditions, however, is not practically feasible. Here, we show that two interacting vortex filaments can not generate a singularity for any initial conditions, by analyzing the asymptotic self-similar limit of their collapse. Essentially, our approach entails a separation of the dynamics of the filament shape, from the shrinking of its core. We solve for the dynamics using a self-similar ansatz and show that the core does not shrink fast enough for a self-consistent collapse. The results are generalized to an arbitrary number of filaments. |
