A Generalization of the Convex Kakeya Problem

Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal \Theta(n log n)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G.Comment: 14 pages, 9 figure

Low temperature thermodynamics of charged bosons in a random potential and the specific heat of La_{2-x}Sr_{x}CuO_{4} below Tc

We propose a simple analytical form of the partition function for charged bosons localised in a random potential and derive the consequent thermodynamics below the superfluid transition temperature. In the low temperature limit, the specific heat, C, depends on the localisation length exponent nu: C is linear for nu1 we find C proportional to T^{1/nu}. This unusual sub-linear temperature dependence of the specific heat has recently been observed in La_{2-x}Sr_{x}CuO_{4} below Tc.Comment: Revtex, 6 pages, 4 postscript figure

First order quantum phase transitions

Quantum phase transitions have been the subject of intense investigations in the last two decades [1]. Among other problems, these phase transitions are relevant in the study of heavy fermion systems, high temperature superconductors and Bose-Einstein condensates. More recently there is increasing evidence that in many systems which are close to a quantum critical point (QCP) different phases are in competition. In this paper we show that the main effect of this competition is to give rise to inhomogeneous behavior associated with quantum first order transitions. These effects are described theoretically using an action that takes into account the competition between different order parameters. The method of the effective potential is used to calculate the quantum corrections to the classical functional. These corrections generally change the nature of the QCP and give rise to interesting effects even in the presence of non-critical fluctuations. An unexpected result is the appearance of an inhomogeneous phase with two values of the order parameter separated by a first order transition. Finally, we discuss the universal behavior of systems with a weak first order zero temperature transition in particular as the transition point is approached from finite temperatures. The thermodynamic behavior along this line is obtained and shown to present universal features.Comment: 7 pages, 5 figures. Invited talk at ICM2006, Kyoto. To appear in JMM

Water sensor feedback control system for surface irrigation

A water sensor feedback control system was developed to control semiautomatic irrigation of basins and borders. When water reaches a sensor at a downfield irrigation cutoff point, a signal to terminate irrigation is sent via wire or infrared (IR) telemetry to a station controller or receiver at the upper end of the field. The sensor uses a monostable interface to strengthen the signal for wire transmission, and prevents continuous IR transmission during the time the sensor is immersed or remains wet. The water sensor controller, powered by a 12-V battery, uses a silicon controlled rectifier (SCR) triggered by the feedback signal to discharge a capacitor through an electric solenoid. The energized solenoid actuates a gate release or valve to terminate irrigation of one field segment and begin irrigation of another. The water sensor system was tested in a level basin irrigation system. A sensor costs approximately U.S. $30, while single and dual station controllers cost about$65 to 80. Both can be portable to minimize the number of units required

Electronic Single Station Irrigation Timer/Controller

Automation is being used increasingly to improve farm water application efficiency and to reduce labor requirements. For many years, farmers have wanted and needed time-controlled devices and structures to change their irrigation sets automatically, particularly when it is inconvenient for them to make the change, such as at night. Such equipment has not been available commercially

D-wave Bose-Einstein condensation and the London penetration depth in superconducting cuprates

We show that bipolaron formation leads to a d-wave Bose-Einstein condensate in cuprates. It is the bipolaron energy dispersion rather than a particular pairing interaction which is responsible for the d-wave symmetry. The unusual low-temperature dependence of the magnetic field penetration depth in cuprates is explained by the localisation of bosons in the random potential. The temperature dependence of the penetration depth is linear with positive or negative slope depending on the random field profile.Comment: 4 pages (RevTeX), 4 figure

Automated Telemetric Irrigation Controller

An electronic, microprocessor-based controller was developed and tested for automating surface irrigation systems. Communication between the central controller and individual satellite field stations is by tone telemetry over a single 3-conductor wire. The reliable Dual Tone Multiple Frequency or Touch Tone system is the same as that used in telephone communications. The system is designed to actuate momentarily energized pilot valves commonly used in automated surface irrigation systems. Because of its low power requirement, the control system can be battery-powered. It is being field tested in three different, automated surface systems

Supernova pointing with low- and high-energy neutrino detectors

A future galactic SN can be located several hours before the optical explosion through the MeV-neutrino burst, exploiting the directionality of $\nu$-$e$-scattering in a water Cherenkov detector such as Super-Kamiokande. We study the statistical efficiency of different methods for extracting the SN direction and identify a simple approach that is nearly optimal, yet independent of the exact SN neutrino spectra. We use this method to quantify the increase in the pointing accuracy by the addition of gadolinium to water, which tags neutrons from the inverse beta decay background. We also study the dependence of the pointing accuracy on neutrino mixing scenarios and initial spectra. We find that in the ``worst case'' scenario the pointing accuracy is $8^\circ$ at 95% C.L. in the absence of tagging, which improves to $3^\circ$ with a tagging efficiency of 95%. At a megaton detector, this accuracy can be as good as $0.6^\circ$. A TeV-neutrino burst is also expected to be emitted contemporaneously with the SN optical explosion, which may locate the SN to within a few tenths of a degree at a future km$^2$ high-energy neutrino telescope. If the SN is not seen in the electromagnetic spectrum, locating it in the sky through neutrinos is crucial for identifying the Earth matter effects on SN neutrino oscillations.Comment: 13 pages, 7 figures, Revtex4 format. The final version to be published in Phys. Rev. D. A few points in the original text are clarifie

Kang-Redner Anomaly in Cluster-Cluster Aggregation

The large time, small mass, asymptotic behavior of the average mass distribution \pb is studied in a $d$-dimensional system of diffusing aggregating particles for $1\leq d \leq 2$. By means of both a renormalization group computation as well as a direct re-summation of leading terms in the small reaction-rate expansion of the average mass distribution, it is shown that \pb \sim \frac{1}{t^d} (\frac{m^{1/d}}{\sqrt{t}})^{e_{KR}} for $m \ll t^{d/2}$, where $e_{KR}=\epsilon +O(\epsilon ^2)$ and $\epsilon =2-d$. In two dimensions, it is shown that \pb \sim \frac{\ln(m) \ln(t)}{t^2} for $m \ll t/ \ln(t)$. Numerical simulations in two dimensions supporting the analytical results are also presented.Comment: 11 pages, 6 figures, Revtex

Quasicondensate and superfluid fraction in the 2D charged-boson gas at finite temperature

The Bogoliubov - de Gennes equations are solved for the Coulomb Bose gas describing a fluid of charged bosons at finite temperature. The approach is applicable in the weak coupling regime and the extent of its quantitative usefulness is tested in the three-dimensional fluid, for which diffusion Monte Carlo data are available on the condensate fraction at zero temperature. The one-body density matrix is then evaluated by the same approach for the two-dimensional fluid with e^2/r interactions, to demonstrate the presence of a quasi-condensate from its power-law decay with increasing distance and to evaluate the superfluid fraction as a function of temperature at weak coupling.Comment: 9 pages, 2 figure