On the minimal travel time needed to collect n items on a circle

Consider n items located randomly on a circle of length 1. The locations of the items are assumed to be independent and uniformly distributed on [0,1). A picker starts at point 0 and has to collect all n items by moving along the circle at unit speed in either direction. In this paper we study the minimal travel time of the picker. We obtain upper bounds and analyze the exact travel time distribution. Further, we derive closed-form limiting results when n tends to infinity. We determine the behavior of the limiting distribution in a positive neighborhood of zero. The limiting random variable is closely related to exponential functionals associated with a Poisson process. These functionals occur in many areas and have been intensively studied in recent literature

In-Degree and PageRank of Web pages: Why do they follow similar power laws?

The PageRank is a popularity measure designed by Google to rank Web pages. Experiments confirm that the PageRank obeys a `power law' with the same exponent as the In-Degree. This paper presents a novel mathematical model that explains this phenomenon. The relation between the PageRank and In-Degree is modelled through a stochastic equation, which is inspired by the original definition of the PageRank, and is analogous to the well-known distributional identity for the busy period in the M/G/1 queue. Further, we employ the theory of regular variation and Tauberian theorems to analytically prove that the tail behavior of the PageRank and the In-Degree differ only by a multiplicative factor, for which we derive a closed-form expression. Our analytical results are in good agreement with experimental data.Comment: 20 pages, 3 figures; typos added; reference adde

Heterodyne detection of the 752.033-GHz H2O rotational absorption line

A tunable high resolution two stage heterodyne radiometer was developed for the purpose of investigating the intensity and lineshape of the 752.033 GHz rotational transition of water vapor. Single-sideband system noise temperatures of approximately 45,000 K were obtained using a sensitive GaAs Schottky diode as the first stage mixer. First local oscillator power was supplied by a CO2 laser pumped formic acid laser (761.61 GHz), generating an X-band IF signal with theoretical line center at 9.5744 GHz. Second local oscillator power was provided by means of a 3 GHz waveguide cavity filter with only 9 dB insertion loss. In absorption measurements of the H2O taken from a laboratory simulation of a high altitude rocket plume, the center frequency of the 752 GHz line was determined to within 1 MHz of the reported value. A rotational temperature 75 K, a linewidth 5 MHz and a Doppler shift 3 MHz were measured with the line-of-sight intersecting the simulated-plume axis at a distance downstream of 30 nozzle diameters. These absorption data were obtained against continuum background radiation sources at temperatures of 1175 and 300 K

Polarization of Astronomical Maser Radiation. IV. Circular Polarization Profiles

Profile comparison of the Stokes parameters $V$ and $I$ is a powerful tool for maser data analysis, providing the first direct methods for unambiguous determination of (1) the maser saturation stage, (2) the amplification optical depth and intrinsic Doppler width of unsaturated masers, and (3) the comparative magnitudes of Zeeman splitting and Doppler linewidth. Circular polarization recently detected in OH 1720 MHz emission from the Galactic center appears to provide the first direct evidence for maser saturation.Comment: 14 pages, 1 Postscript figures (included), uses aaspp4.sty. To appear in Astrophysical Journa

A unified view on beamformers for M/EEG source reconstruction

Beamforming is a popular method for functional source reconstruction using magnetoencephalography (MEG) and electroencephalography (EEG) data. Beamformers, which were first proposed for MEG more than two decades ago, have since been applied in hundreds of studies, demonstrating that they are a versatile and robust tool for neuroscience. However, certain characteristics of beamformers remain somewhat elusive and there currently does not exist a unified documentation of the mathematical underpinnings and computational subtleties of beamformers as implemented in the most widely used academic open source software packages for MEG analysis (Brainstorm, FieldTrip, MNE, and SPM). Here, we provide such documentation that aims at providing the mathematical background of beamforming and unifying the terminology. Beamformer implementations are compared across toolboxes and pitfalls of beamforming analyses are discussed. Specifically, we provide details on handling rank deficient covariance matrices, prewhitening, the rank reduction of forward fields, and on the combination of heterogeneous sensor types, such as magnetometers and gradiometers. The overall aim of this paper is to contribute to contemporary efforts towards higher levels of computational transparency in functional neuroimaging

What are the determinants of childhood infections in India’s peri-urban slums? A case study of eight cities

BACKGROUND: Respiratory Tract Infections (RTIs) and Gastro-Intestinal (GI) infections are the leading causes of child mortality and morbidity. This study investigates the associations between the individual, household and slum-level determinants of children’s health and vulnerability to RTIs and GI infections in peri-urban slums in India; an area of research interest at the Childhood Infections and Pollution Consortium. METHODS: The 2015–16 Indian National Family Health Survey was used for data analysis on children aged 0–5 years. NFHS-4 includes data on slums in eight Indian cities, including Delhi, Meerut, Kolkata, Indore, Mumbai, Nagpur, Hyderabad, Chennai. The outcome variables, having fever and cough (FeCo) and diarrhoea in the last two weeks, were used to define the phenotype of infections; for this analysis fever and cough were measures of RTIs and diarrhoea was used to measure GI infections. Exposures considered in this study include variables at the individual, household and slum level and were all informed by existing literature. Multilevel models were used to estimate the association between exposures and outcomes variables; a prior of Cauchy distribution with a scale of 2.5 was selected when building the multilevel logistic models. RESULTS: The total sample size of the number of children included in the analysis was n = 1,424. Data was imputed to account for missingness, and the original and imputed sample showing similar distributions. Results showed that diarrhoea and FeCo were both found to be more present in younger children than older children by a few months. In fixed effects, the odds of developing FeCo were higher if the mother perceives the child was born smaller than average (AOR 4.41, 1.13–17.17, P<0.05) at individual level. On the other hand, the odds of the diarrhoea outcome were lower if the child was older (AOR 0.97, 0.96–0.98, P<0.05) at individual level, and household’s water source was public tap or standpipe (AOR 0.54, 0.31–0.96, P<0.05) at household level. CONCLUSION: The determinants of health, both social and related to health care, at all levels demonstrated linkages to child morbidity in RTIs and GI infections. The empirical evidence highlights the need for contextualised ideas at each level, including one health approach when designing interventions to improve child health

Stability of dark solitons in three dimensional dipolar Bose-Einstein condensates

The dynamical stability of dark solitons in dipolar Bose-Einstein condensates is studied. For standard short-range interacting condensates dark solitons are unstable against transverse excitations in two and three dimensions. On the contrary, due to its non local character, the dipolar interaction allows for stable 3D stationary dark solitons, opening a qualitatively novel scenario in nonlinear atom optics. We discuss in detail the conditions to achieve this stability, which demand the use of an additional optical lattice, and the stability regimes.Comment: 4 pages, 3 eps figure