On the groundstate energy of tight knots

New results on the groundstate energy of tight, magnetic knots are presented. Magnetic knots are defined as tubular embeddings of the magnetic field in an ideal, perfectly conducting, incompressible fluid. An orthogonal, curvilinear coordinate system is introduced and the magnetic energy is determined by the poloidal and toroidal components of the magnetic field. Standard minimization of the magnetic energy is carried out under the usual assumptions of volume- and flux-preserving flow, with the additional constraints that the tube cross-section remains circular and that the knot length (ropelength) is independent from internal field twist (framing). Under these constraints the minimum energy is determined analytically by a new, exact expression, function of ropelength and framing. Groundstate energy levels of tight knots are determined from ropelength data obtained by the SONO tightening algorithm developed by Pieranski (Pieranski, 1998) and collaborators. Results for torus knots are compared with previous work done by Chui & Moffatt (1995), and the groundstate energy spectrum of the first prime knots (up to 10 crossings) is presented and analyzed in detail. These results demonstrate that ropelength and framing determine the spectrum of magnetic knots in tight configuration.Comment: 26 pages, 9 figure

Mapping the Evolution of "Clusters": A Meta-analysis

This paper presents a meta-analysis of the “cluster literature” contained in scientific journals from 1969 to 2007. Thanks to an original database we study the evolution of a stream of literature which focuses on a research object which is both a theoretical puzzle and an empirical widespread evidence. We identify different growth stages, from take-off to development and maturity. We test the existence of a life-cycle within the authorships and we discover the existence of a substitutability relation between different collaborative behaviours. We study the relationships between a “spatial” and an “industrial” approach within the textual corpus of cluster literature and we show the existence of a “predatory” interaction. We detect the relevance of clustering behaviours in the location of authors working on clusters and in measuring the influence of geographical distance in co-authorship. We measure the extent of a convergence process of the vocabulary of scientists working on clusters.Cluster, Life-Cycle, Cluster Literature, Textual Analysis, Agglomeration, Co-Authorship

The Multi-path Traveling Salesman Problem with Stochastic Travel Costs: Building Realistic Instances for City Logistics Applications

One of the main issues related to routing problems applied in an urban context with uncertainty related to the transportation costs is how to define realistic instances. In this paper, we overcome this issue, providing a standard methodology to extend routing instances from the literature incorporating real data provided by sensors networks. In order to test the methodology, we consider a routing problem specifically designed for City Logistics and Smart City applications, the multi-path Traveling Salesman Problem with stochastic travel costs, where several paths connect each pair of nodes and each path shows a stochastic travel cost with unknown distribution

Stochastic programming framework for Lithuanian pension payout modelling

The paper provides a scientific approach to the problem of selecting a pension fund by taking into account some specific characteristics of the Lithuanian Republic (LR) pension accumulation system. The decision making model, which can be used to plan a long-term pension accrual of the Lithuanian Republic (LR) citizens, in an optimal way is presented. This model focuses on factors that influence the sustainability of the pension system selection under macroeconomic, social and demographic uncertainty. The model is formalized as a single stage stochastic optimization problem where the long-term optimal strategy can be obtained based on the possible scenarios generated for a particular participant. Stochastic programming methods allow including the pension fund rebalancing moment and direction of investment, and taking into account possible changes of personal income, changes of society and the global financial market. The collection of methods used to generate scenario trees was found useful to solve strategic planning problems

Sensitivity Analysis in Stochastic Second Order Cone Programming for Mobile Ad Hoc Networks

AbstractIn this paper sensitivity analysis is adopted in order to understand the randomness of a stochastic second order cone program for mobile ad hoc networks [3]. The algorithm looks for a destination node and sets up a route by means of the expected zone, the region where the sender node expects to find the destination node and the requested zone defined by the sender node for spreading the route request to the destination node. Sensitivity analysis is performed by considering different costs of flooding and latency penalty. Evaluation of EVPI and VSS [2]-[4] allows us to find the range of values in which it is safe to save time by using a deterministic approach instead of a stochastic one

Reduced cost-based variable fixing in two-stage stochastic programming

The explicit consideration of uncertainty is essential in addressing most planning and operation issues encountered in the management of complex systems. Unfortunately, the resulting stochastic programming formulations, integer ones in particular, are generally hard to solve when applied to realistically-sized instances. A common approach is to consider the simpler deterministic version of the formulation, even if it is well known that the solution quality could be arbitrarily bad. In this paper, we aim to identify meaningful information, which can be extracted from the solution of the deterministic problem, in order to reduce the size of the stochastic one. Focusing on two-stage formulations, we show how and under which conditions the reduced costs associated to the variables in the deterministic formulation can be used as an indicator for excluding/retaining decision variables in the stochastic model. We introduce a new measure, the Loss of Reduced Costs-based Variable Fixing (LRCVF), computed as the difference between the optimal values of the stochastic problem and its reduced version obtained by fixing a certain number of variables. We relate the LRCVF with existing measures and show how to select the set of variables to fix. We then illustrate the interest of the proposed LRCVF and related heuristic procedure, in terms of computational time reduction and accuracy in finding the optimal solution, by applying them to a wide range of problems from the literature

Kinetic energy of vortex knots and unknots

New results on the kinetic energy of ideal vortex filaments in the shape of torus knots and unknots are presented. These knots are given by small-amplitude torus knot solutions (Ricca, 1993) to the Localized Induction Approximation (LIA) law. The kinetic energy of different knot and unknot types is calculated and presented for comparison. These results provide new information on relationships between geometry, topology and dynamics of complex vortex systems and help to establish possible connections between aspects of structural complexity of dynamical systems and vortical flows.Comment: 14 pages, 5 figure