Hyperelliptic curves, continued fractions, and Somos sequences

We detail the continued fraction expansion of the square root of a monic polynomials of even degree. We note that each step of the expansion corresponds to addition of the divisor at infinity, and interpret the data yielded by the general expansion. In the quartic and sextic cases we observe explicitly that the parameters appearing in the continued fraction expansion yield integer sequences defined by bilinear relations instancing sequences of Somos type.Comment: Published at http://dx.doi.org/10.1214/074921706000000239 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

The response of a floating ice sheet to an accelerating line load

The two-dimensional response of a thin, floating sheet of ice to a line load that accelerates from rest at $t = 0$ to a uniform velocity V for $t \geq T$ is determined through an integral-transform solution of the linearized equations of motion. If $T = 0$ – i.e. if the load is impulsively started with velocity V – the solution exhibits singularities at $V = c_0$, the shallow-water-gravity-wave speed, and $V = c_{\min}$, the minimum speed for transverse motion of the ice, but these singularities are avoided by the acceleration of the load through the critical speeds

Change in fluoride content of Maltese tapwaters : implications for oral health

Dietary intake of fluoride is believed to be largely derived from drinking water. In Malta, tap water presently contains less fluoride than is found in naturally occurring ground water obtained from aquifers. Over 55% of water production comes from reverse osmosis and such water is practically fluoride-free. Before the introduction of reverse osmosis water into the water budget, fluoride levels measured from 10 different tap waters averaged 0.51 ppm F-. Now, fluoride content measured from 19 tap water sources averaged 0.21 ppm F- and more than a third of these sources are practically fluoride-free. The incidence of dental caries in Malta has so far been low according to the international standards. We suggest that the lowering of the fluoride level in the water supplies may favour an escalation in the incidence of tooth decay in the future.peer-reviewe

Quadratic irrational integers with partly prescribed continued fraction expansion

We generalise remarks of Euler and of Perron by explaining how to detail all quadratic irrational integers for which the symmetric part of the period of their continued fraction expansion commences with prescribed partial quotients. The function field case is particularly striking.Comment: 10 pages; dedicated to the memory of Bela Brindz

Densely Integrated Photonic Devices based on Microring Resonators in Access Networks

The application of optical fibers has led to virtually lossless point to point data links in the core network with practically unlimited bandwidth. In response to increasing bandwidth demands of consumers the optical techniques employed in these networks are now gradually extended towards the consumers premises. At the access network level, where equipment is shared by a few users at the most, cost is the major issue. In addition the demand of optical transparency at the nodes and hubs, excluding conversion between the optical and electrical domains, results in a high degree of complexity of the devices. Fortunately, through the use of flexible WDM bandwidth allocation schemes that provide bandwidth where needed, and cheap mass-produced densely integrated optical components, the cost can be reduced significantly. Promising building blocks in these components are Microring Resonators (MRs) built using high refractive index waveguides which can be used to create densely integrated wavelength filters as well as more complex functionality.\ud In this work we present a 1x4 reconfigurable optical add-drop multiplexer (rOADM) as well as a 1x4x4 reconfigurable wavelength router in variants operating in the second and third telecom window. The devices, which can be used at bit rates up to 40 Gbit/s, have a footprint less than 2 mm2 and are based on thermally tunable vertically coupled microring resonators fabricated in the high contrast (Δn≈0.55) Si3N4/SiO2 materials system. In addition to the presented devices a new polarization diversity scheme is proposed that solves the issue of polarization dependence behavior of microring resonators without the need of doubling the optical circuit

Specialisation and reduction of continued fractions of formal power series

We discuss and illustrate the behaviour of the continued fraction expansion of a formal power series under specialisation of parameters or their reduction modulo $p$ and sketch some applications of the reduction theorem here proved.Comment: 7 page

Environmental forensic evidence from perchlorate in dust fall

Summer in Malta is accompanied by fireworks as part of the numerous (about 85) religious festivals (festa) that occur throughout the period. We attempted to establish whether firework activity is truly following official trade statistics, which imply a decreasing trend, or otherwise. Firework manufacture critically depends on the availability of oxidising agents, two of which (KNO3, KClO3) being controlled by permits but not potassium perchlorate (KClO4) which is freely available. Recent changes in legislation have slightly decreased the quantity of KClO3 and increased that of KNO3 and according to official trade statistics, consumption of KClO4 has decreased from 17 t/a in 2010 to ≈ 2.5 t/a during 2011--2014. However, from levels of perchlorate in dust fall and rate of total deposition we calculated the quantity of KClO4 used during 2012 and obtained ≥ 90 t/a, 36 times the official figure. This situation is serious since the danger of accidental explosion during manufacture escalates as production intensifies. Also, contamination of the environment by firework-waste appears destined to remain high with possible consequent effects on human health.peer-reviewe

Elliptic curves and continued fractions

We detail the continued fraction expansion of the square root of the general monic quartic polynomial, noting that each line of the expansion corresponds to addition of the divisor at infinity. We analyse the data yielded by the general expansion. In that way we obtain `elliptic sequences' satisfying Somos relations. I mention several new results on such sequences. The paper includes a detailed `reminder exposition' on continued fractions of quadratic irrationals in function fields.Comment: v1; final -- I hop

Noteworthy : Guns and butter

Defense industries ; Consumption (Economics)