The Traveling Salesman Problem Under Squared Euclidean Distances

Let $P$ be a set of points in $\mathbb{R}^d$, and let $\alpha \ge 1$ be a real number. We define the distance between two points $p,q\in P$ as $|pq|^{\alpha}$, where $|pq|$ denotes the standard Euclidean distance between $p$ and $q$. We denote the traveling salesman problem under this distance function by TSP($d,\alpha$). We design a 5-approximation algorithm for TSP(2,2) and generalize this result to obtain an approximation factor of $3^{\alpha-1}+\sqrt{6}^{\alpha}/3$ for $d=2$ and all $\alpha\ge2$. We also study the variant Rev-TSP of the problem where the traveling salesman is allowed to revisit points. We present a polynomial-time approximation scheme for Rev-TSP$(2,\alpha)$ with $\alpha\ge2$, and we show that Rev-TSP$(d, \alpha)$ is APX-hard if $d\ge3$ and $\alpha>1$. The APX-hardness proof carries over to TSP$(d, \alpha)$ for the same parameter ranges.Comment: 12 pages, 4 figures. (v2) Minor linguistic change

Tinbergen's work: Change and continuity

This article provides an overview of Tinbergen's economic writings, comprising well over 900 publications. Six broad areas have been distinguished to allow for the classification and discussion of Tinbergen's economic work. These six areas embody the change in Tinbergen's areas of interest apart from the shift from physics to economics early in his career. Tinbergen's work, however, is not only characterised by change of areas but also by continuity in approach. To mark this continuity four main characteristics of Tinbergen's work have been elaborated. The article concludes with Tinbergen's observations on the achievements to be expected from scientific research. J. Kol was Associate Professor of International Economics and Economic Integration at the Faculty of Economics of the Erasmus University in Rotterdam; P. de Wolff is Emeritus Professor at the University of Amsterdam and was Director of the Central Planning Bureau in The Hague as successor of J. Tinbergen. The authors are grateful to Mrs. I.M. Lageweg and Mr C.J. van Opijnen for their assistance with references

A Simplified Model of the Causation of Technological Unemployment

Also published in: ‘Jan Tinbergen Selected Papers’, edited by L.H. Klaassen, L.M. Koyck and H.J. Witteveen, Nort- Holland Publishing Company, Amsterdam, 1959, pp. 165-181The purpose of the present paper is to find the influence on employment of some of the outstanding "data" (extra-economic determining factor-). For this purpose, a simplified model has been constructed in which these data and the chief economic variables find their places. Since it is not intended to picture cyclic variations and causations, the model may be called a "long-run model." It excludes some of the most typical cyclical phenomena such as stock-exchange speculation and the existence of small lags of all kinds which are of importance to the explar~ationo f cyclcs but do not seen1 to be so for long-run developments. Since the investigators were interested chiefly in studying the consequences of technological development for employnlent and the consequences of some of the best-known devices to improve employment, spccial attention was given to the corresponding sections of economic life. Thc calculations have been made for the United States prewar structure (using figures for 1910) and for the postwar pre-Roosevelt structure (using averages for 1919-1932

Finite Size Scaling of the 2D Six-Clock model

We investigate the isotropic-anisotropic phase transition of the two-dimensional XY model with six-fold anisotropy, using Monte Carlo renormalization group method. The result indicates difficulty of observing asymptotic critical behavior in Monte Carlo simulations, owing to the marginal flow at the fixed point.Comment: Short note. revtex, 6 pages, 3 figures. To appear in J. Phys. Soc. Jpn. Vol.70 No. 2 (Feb 2001

Polymer brush collapse under shear flow

Shear responsive surfaces offer potential advances in a number of applications. Surface functionalisation using polymer brushes is one route to such properties, particularly in the case of entangled polymers. We report on neutron reflectometry measurements of polymer brushes in entangled polymer solutions performed under controlled shear, as well as coarse-grained computer simulations corresponding to these interfaces. Here we show a reversible and reproducible collapse of the brushes, increasing with the shear rate. Using two brushes of greatly different chain lengths and grafting densities, we demonstrate that the dynamics responsible for the structural change of the brush are governed by the free chains in solution rather than the brush itself, within the range of parameters examined. The phenomenon of the brush collapse could find applications in the tailoring of nanosensors, and as a way to dynamically control surface friction and adhesion

Spin-orbit coupled particle in a spin bath

We consider a spin-orbit coupled particle confined in a quantum dot in a bath of impurity spins. We investigate the consequences of spin-orbit coupling on the interactions that the particle mediates in the spin bath. We show that in the presence of spin-orbit coupling, the impurity-impurity interactions are no longer spin-conserving. We quantify the degree of this symmetry breaking and show how it relates to the spin-orbit coupling strength. We identify several ways how the impurity ensemble can in this way relax its spin by coupling to phonons. A typical resulting relaxation rate for a self-assembled Mn-doped ZnTe quantum dot populated by a hole is 1 $\mu$s. We also show that decoherence arising from nuclear spins in lateral quantum dots is still removable by a spin echo protocol, even if the confined electron is spin-orbit coupled.Comment: 18 pages, 1 figur