Long Proteins with Unique Optimal Foldings in the H-P Model

It is widely accepted that (1) the natural or folded state of proteins is a global energy minimum, and (2) in most cases proteins fold to a unique state determined by their amino acid sequence. The H-P (hydrophobic-hydrophilic) model is a simple combinatorial model designed to answer qualitative questions about the protein folding process. In this paper we consider a problem suggested by Brian Hayes in 1998: what proteins in the two-dimensional H-P model have unique optimal (minimum energy) foldings? In particular, we prove that there are closed chains of monomers (amino acids) with this property for all (even) lengths; and that there are open monomer chains with this property for all lengths divisible by four.Comment: 22 pages, 18 figure

An O(n log n)-Time Algorithm for the Restricted Scaffold Assignment

The assignment problem takes as input two finite point sets S and T and establishes a correspondence between points in S and points in T, such that each point in S maps to exactly one point in T, and each point in T maps to at least one point in S. In this paper we show that this problem has an O(n log n)-time solution, provided that the points in S and T are restricted to lie on a line (linear time, if S and T are presorted).Comment: 13 pages, 8 figure

Problems in enforcing Dutch building regulations

Purpose - The paper seeks to define the nature of the policy problems in Dutch building control. Design/methodology/approach - The authors use Dunn's four-phase methodology for public policy analysis, consisting of problem sensing, problem search, problem definition, and problem specification. Both a literature review and a field study into the operation of local building control authorities were undertaken. The field study incorporates characteristics of a survey, with methodology developed by Fowler. Findings - Dutch building control legislation has been subject to many changes over the 100 years or so that it has been in force as it has responded to society's changing priorities. Throughout this period building regulation has become more detailed and more uniform across the country. Nevertheless, almost no legal changes have been made to the enforcement system. Responsibility for building control still lies with the municipalities and implementation is still not established by national legislation or policy document. Ongoing attempts to deregulate and standardise the legislative framework should therefore not stop at changing the regulations. Changes in the supervision system might offer an alternative route to improving the quality of the (technical) building control and clarifying the tasks and responsibilities of building control staff. Research limitations/implications - The analysis focuses on problems in building control and does not consider design and construction problems. Practical implications - The field study contains important lessons forbuilding control practitioners and policymakers regarding current deficiencies in the implementation of building control legislation. Originality/value - The paper provides a model for the analysis, and comparative study, of building control systems in other jurisdictions

Coordinated Motion Planning: Reconfiguring a Swarm of Labeled Robots with Bounded Stretch

We present a number of breakthroughs for coordinated motion planning, in which the objective is to reconfigure a swarm of labeled convex objects by a combination of parallel, continuous, collision-free translations into a given target arrangement. Problems of this type can be traced back to the classic work of Schwartz and Sharir (1983), who gave a method for deciding the existence of a coordinated motion for a set of disks between obstacles; their approach is polynomial in the complexity of the obstacles, but exponential in the number of disks. Despite a broad range of other non-trivial results for multi-object motion planning, previous work has largely focused on sequential schedules, in which one robot moves at a time, with objectives such as the number of moves; attempts to minimize the overall makespan of a coordinated parallel motion schedule (with many robots moving simultaneously) have defied all attempts at establishing the complexity in the absence of obstacles, as well as the existence of efficient approximation methods. We resolve these open problems by developing a framework that provides constant-factor approximation algorithms for minimizing the execution time of a coordinated, parallel motion plan for a swarm of robots in the absence of obstacles, provided their arrangement entails some amount of separability. In fact, our algorithm achieves constant stretch factor: If all robots want to move at most d units from their respective starting positions, then the total duration of the overall schedule (and hence the distance traveled by each robot) is O(d). Various extensions include unlabeled robots and different classes of robots. We also resolve the complexity of finding a reconfiguration plan with minimal execution time by proving that this is NP-hard, even for a grid arrangement without any stationary obstacles. On the other hand, we show that for densely packed disks that cannot be well separated, a stretch factor Omega(N^{1/4}) may be required. On the positive side, we establish a stretch factor of O(N^{1/2}) even in this case. The intricate difficulties of computing precise optimal solutions are demonstrated by the seemingly simple case of just two disks, which is shown to be excruciatingly difficult to solve to optimality

The Distance Geometry of Music

We demonstrate relationships between the classic Euclidean algorithm and many other fields of study, particularly in the context of music and distance geometry. Specifically, we show how the structure of the Euclidean algorithm defines a family of rhythms which encompass over forty timelines (\emph{ostinatos}) from traditional world music. We prove that these \emph{Euclidean rhythms} have the mathematical property that their onset patterns are distributed as evenly as possible: they maximize the sum of the Euclidean distances between all pairs of onsets, viewing onsets as points on a circle. Indeed, Euclidean rhythms are the unique rhythms that maximize this notion of \emph{evenness}. We also show that essentially all Euclidean rhythms are \emph{deep}: each distinct distance between onsets occurs with a unique multiplicity, and these multiplicies form an interval $1,2,...,k-1$. Finally, we characterize all deep rhythms, showing that they form a subclass of generated rhythms, which in turn proves a useful property called shelling. All of our results for musical rhythms apply equally well to musical scales. In addition, many of the problems we explore are interesting in their own right as distance geometry problems on the circle; some of the same problems were explored by Erd\H{o}s in the plane.Comment: This is the full version of the paper: "The distance geometry of deep rhythms and scales." 17th Canadian Conference on Computational Geometry (CCCG '05), University of Windsor, Canada, 200

An asymptotic result for Laguerre-Sobolev orthogonal polynomials

AbstractLet {Sn} denote the sequence of polynomials orthogonal with respect to the Sobolev inner product (f,g)s = ∫0+∞ f(x)g(x)xαe−xdx+λ∫0+∞ f′(x)g′(x)xαe−xdx where α > − 1, λ > 0 and the leading coefficient of the Sn is equal to the leading coefficient of the Laguerre polynomial Ln(α). Then, if x∈Cß[0,+∞), limn→∞Sn(x)Ln(α−1)(x) is a constant depending on λ