Approximate Euclidean Steiner trees

An approximate Steiner tree is a Steiner tree on a given set of terminals in Euclidean space such that the angles at the Steiner points are within a specified error e from 120 degrees. This notion arises in numerical approximations of minimum Steiner trees (W. D. Smith, Algorithmica, 7 (1992), 137–177). We investigate the worst-case relative error of the length of an approximate Steiner tree compared to the shortest tree with the same topology. Rubinstein, Weng and Wormald (J. Global Optim. 35 (2006), 573–592) conjectured that this relative error is at most linear in e, independent of the number of terminals. We verify their conjecture for the two-dimensional case as long as the error e is sufficiently small in terms of the number of terminals. We derive a lower bound linear in e for the relative error in the two-dimensional case when e is sufficiently small in terms of the number of terminals. We find improved estimates of the relative error for larger values of e, and calculate exact values in the plane for three and four terminals

Generalised k-Steiner tree problems in normed planes

The 1-Steiner tree problem, the problem of constructing a Steiner minimum tree containing at most one Steiner point, has been solved in the Euclidean plane by Georgakopoulos and Papadimitriou using plane subdivisions called oriented Dirichlet cell partitions. Their algorithm produces an optimal solution within $\mathcal{O}(n^2)$ time. In this paper we generalise their approach in order to solve the $k$-Steiner tree problem, in which the Steiner minimum tree may contain up to $k$ Steiner points for a given constant $k$. We also extend their approach further to encompass arbitrary normed planes, and to solve a much wider class of problems, including the $k$-bottleneck Steiner tree problem and other generalised $k$-Steiner tree problems. We show that, for any fixed $k$, such problems can be solved in $\mathcal{O}(n^{2k})$ time

Solid state nanofibers based on self-assemblies: from cleaving from self-assemblies to multilevel hierarchical constructs

Self-assemblies and their hierarchies are useful to construct soft materials with structures at different length scales and to tune the materials properties for various functions. Here we address routes for solid nanofibers based on different forms of self-assemblies. On the other hand, we discuss rational "bottom-up" routes for multi-level hierarchical self-assembled constructs, with the aim of learning more about design principles for competing interactions and packing frustrations. Here we use the triblock copolypeptide poly(l-lysine)-b-poly(γ-benzyl-l-glutamate)-b-poly(l-lysine) complexed with 2′-deoxyguanosine 5′-monophosphate. Supramolecular disks (G-quartets) stabilized by metal cations are formed and their columnar assembly leads to a packing frustration with the cylindrical packing of helical poly(γ-benzyl-l-glutamate), which we suggest is important in controlling the lateral dimensions of the nanofibers. We foresee routes for functionalities by selecting different metal cations within the G-quartets. On the other hand, we discuss nanofibers that are cleaved from bulk self-assemblies in a "top-down" manner. After a short introduction based on cleaving nanofibers from diblock copolymeric self-assemblies, we focus on native cellulose nanofibers, as cleaved from plant cell wall fibers, which are expected to have feasible mechanical properties and to be templates for functional nanomaterials. Long nanofibers with 5-20 nm lateral dimensions can be cleaved within an aqueous medium to allow hydrogels and water can be removed to allow highly porous, lightweight, and flexible aerogels. We further describe inorganic/organic hybrids as prepared by chemical vapour deposition and atomic layer deposition of the different nanofibers. We foresee functional materials by selecting inorganic coatings. Finally we briefly discuss how the organic template can be removed e.g., by thermal treatments to allow completely inorganic hollow nanofibrillar structures. © 2009 The Royal Society of Chemistry

Solid state nanofibers based on self-assemblies: from cleaving from self-assemblies to multilevel hierarchical constructs

Self-assemblies and their hierarchies are useful to construct soft materials with structures at different length scales and to tune the materials properties for various functions. Here we address routes for solid nanofibers based on different forms of self-assemblies. On the other hand, we discuss rational "bottom-up'' routes for multi-level hierarchical self-assembled constructs, with the aim of learning more about design principles for competing interactions and packing frustrations. Here we use the triblock copolypeptide poly(L-lysine)-b-poly(gamma-benzyl-L-glutamate)-b-poly(L-lysine) complexed with 2'-deoxyguanosine 5'-monophosphate. Supramolecular disks (G-quartets) stabilized by metal cations are formed and their columnar assembly leads to a packing frustration with the cylindrical packing of helical poly(gamma-benzyl-L-glutamate), which we suggest is important in controlling the lateral dimensions of the nanofibers. We foresee routes for functionalities by selecting different metal cations within the G-quartets. On the other hand, we discuss nanofibers that are cleaved from bulk self-assemblies in a "top-down'' manner. After a short introduction based on cleaving nanofibers from diblock copolymeric self-assemblies, we focus on native cellulose nanofibers, as cleaved from plant cell wall fibers, which are expected to have feasible mechanical properties and to be templates for functional nanomaterials. Long nanofibers with 5-20 nm lateral dimensions can be cleaved within an aqueous medium to allow hydrogels and water can be removed to allow highly porous, lightweight, and flexible aerogels. We further describe inorganic/organic hybrids as prepared by chemical vapour deposition and atomic layer deposition of the different nanofibers. We foresee functional materials by selecting inorganic coatings. Finally we briefly discuss how the organic template can be removed e. g., by thermal treatments to allow completely inorganic hollow nanofibrillar structures