Iterated harmonic numbers

The harmonic numbers are the sequence 1, 1+1/2, 1+1/2+1/3, ... Their asymptotic difference from the sequence of the natural logarithm of the positive integers is Euler's constant gamma. We define a family of natural generalizations of the harmonic numbers. The jth iterated harmonic numbers are a sequence of rational numbers that nests the previous sequences and relates in a similar way to the sequence of the jth iterate of the natural logarithm of positive integers. The analogues of several well-known properties of the harmonic numbers also hold for the iterated harmonic numbers, including a generalization of Euler's constant. We reproduce the proof that only the first harmonic number is an integer and, providing some numeric evidence for the cases j = 2 and j = 3, conjecture that the same result holds for all iterated harmonic numbers. We also review another proposed generalization of harmonic numbers.Comment: 13 pages, 2 figure

It Takes More Than an Apple a Day

http://dx.doi.org/10.1126/science.337.6101.146

Radar Altimetry as a Robust Tool for Monitoring the Active Lava Lake at Erebus Volcano, Antarctica

The level of lava within a volcanic conduit reflects the overpressure within a connected magma reservoir. Continuous monitoring of lava level can therefore provide critical insights into volcanic processes, and aid hazard assessment. However, accurate measurements of lava level are not easy to make, partly owing to the often dense fumes that hinder optical techniques. Here, we present the first radar instrument designed for the purpose of monitoring lava level, and report on its successful operation at Erebus volcano, Antarctica. We describe the hardware and data processing steps followed to extract a time series of lava lake level, demonstrating that we can readily resolve ~1 m cyclic variations in lake level that have previously been recognised at Erebus volcano. The performance of the radar (continuous, automated data collection in temperatures of around -30°C) indicates the suitability of this approach for sustained automated measurements at Erebus and other volcanoes with lava lakes

Information Tradeoff Relations for Finite-Strength Quantum Measurements

In this paper we give a new way to quantify the folklore notion that quantum measurements bring a disturbance to the system being measured. We consider two observers who initially assign identical mixed-state density operators to a two-state quantum system. The question we address is to what extent one observer can, by measurement, increase the purity of his density operator without affecting the purity of the other observer's. If there were no restrictions on the first observer's measurements, then he could carry this out trivially by measuring the initial density operator's eigenbasis. If, however, the allowed measurements are those of finite strength---i.e., those measurements strictly within the interior of the convex set of all measurements---then the issue becomes significantly more complex. We find that for a large class of such measurements the first observer's purity increases the most precisely when there is some loss of purity for the second observer. More generally the tradeoff between the two purities, when it exists, forms a monotonic relation. This tradeoff has potential application to quantum state control and feedback.Comment: 15 pages, revtex3, 3 eps figure

Mapping threatened Thai bovids provides opportunities for improved conservation outcomes in Asia

Wild bovids provide important ecosystem functions as seed dispersers and vegetation modifiers. Five wild bovids remain in Thailand: gaur (Bos gaurus), banteng (Bos javanicus), wild water buffalo (Bubalus arnee), mainland serow (Capricornis sumatraensis) and Chinese goral (Naemorhedus griseus). Their populations and habitats have declined substantially and become fragmented by land-use change. We use ecological niche models to quantify how much potential suitable habitat for these species remains within protected areas in Asia and then specifically Thailand. We combined species occurrence data from several sources (e.g. mainly camera traps and direct observation) with environmental variables and species-specific and single, large accessible areas in ensemble models to generate suitability maps, using out-of-sample predictions to validate model performance against new independent data. Gaur, banteng and buffalo models showed reasonable model accuracy throughout the entire distribution (greater than or equal to 62%) and in Thailand (greater than or equal to 80%), whereas serow and goral models performed poorly for the entire distribution and in Thailand, though 5 km movement buffers markedly improved the performance for serow. Large suitable areas were identified in Thailand and India for gaur, Cambodia and Thailand for banteng and India for buffalo. Over 50% of suitable habitat is located outside protected areas, highlighting the need for habitat management and conflict mitigation outside protected areas