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We consider a generalized form of the coupon collection problem in which a random number, S, of balls is drawn at each stage from an urn initially containing n white balls (coupons). Each white ball drawn is colored red and returned to the urn; red balls drawn are simply returned to the urn. The question considered is then: how many white balls (uncollected coupons) remain in the urn after the kn draws? Our analysis is asymptotic as n → ∞. We concentrate on the case when kn draws are made, where kn / n → ∞ (the superlinear case), although we sketch known results for other ranges of kn. A Gaussian limit is obtained via a martingale representation for the lower superlinear range, and a Poisson limit is derived for the upper boundary of this range via the Chen-Stein approximation.
The UK was one of few European countries to document a substantial wave of pandemic (H1N1) 2009 influenza in summer 2009. The First Few Hundred (FF100) project ran from April–June 2009 gathering information on early laboratory-confirmed cases across the UK. In total, 392 confirmed cases were followed up. Children were predominantly affected (median age 15 years, IQR 10–27). Symptoms were mild and similar to seasonal influenza, with the exception of diarrhoea, which was reported by 27%. Eleven per cent of all cases had an underlying medical condition, similar to the general population. The majority (92%) were treated with antiviral drugs with 12% reporting adverse effects, mainly nausea and other gastrointestinal complaints. Duration of illness was significantly shorter when antivirals were given within 48 h of onset (median 5 vs. 9 days, P=0·01). No patients died, although 14 were hospitalized, of whom three required mechanical ventilation. The FF100 identified key clinical and epidemiological characteristics of infection with this novel virus in near real-time.
The Keck Interferometer Nuller (KIN) is one of the major scientific and technical precursors to the Terrestrial Planet Finder Interferometer (TPF-I) mission. KIN's primary objective is to measure the level of exo-zodiacal mid-infrared emission around nearby main sequence stars, which requires deep broad-band nulling of astronomical sources of a few Janskys at 10 microns. A number of new capabilitites are needed in order to reach that goal with the Keck telescopes: mid-infrared coherent recombination, interferometric operation in “split pupil” mode, N-band optical path stabilization using K-band fringe tracking and internal metrology, and eventually, active atmospheric dispersion correction. We report here on the progress made implementing these new functionalities, and discuss the initial levels of extinction achieved on the sky.
The extended Korteweg-de Vries equation which includes nonlinear and dispersive terms cubic in the wave amplitude is derived from the water-wave equations and the Lagrangian for the water-wave equations. For the special case in which only the higher-order nonlinear term is retained, the extended Korteweg-de Vries equation is transformed into the Korteweg-de Vries equation. Modulation equations for this equation are then derived from the modulation equations for the Korteweg-de Vries equation and the undular bore solution of the extended Korteweg-de Vries equation is found as a simple wave solution of these modulation equations. The modulation equations are also used to extend the solution for the resonant flow of a fluid over topography. This resonant flow occurs when, in the weakly nonlinear, long-wave limit, the basic flow speed is close to a linear long-wave phase speed for one of the long-wave modes. In addition to the effect of higher-order terms, the effect of boundary-layer viscosity is also considered. These solutions (with and without viscosity) are compared with recent experimental and numerical results.
In completely or partially oriented percolation models, a conceptually simple method, using barriers to enclose all open paths from the origin, improves the best previous lower bounds for the critical percolation probabilities.
The principal results of this paper concern the asymptotic behavior of the number of arcs in the optimal routes of first-passage percolation processes on the square lattice. Assuming that the underlying distribution has an atom at zero less than λ–1, where λ is the connectivity constant, Lp and (in some cases) almost sure convergence theorems are proved for the normalized route length processes. The proofs involve the extension of much of the existing theory of first-passage percolation to the case where negative time coordinates are permitted.
We consider several problems in the theory of first-passage percolation on the two-dimensional integer lattice. Our results include: (i) a mean ergodic theorem for the first-passage time from (0,0) to the line x = n; (ii) a proof that the time constant is zero when the atom at zero of the underlying distribution exceeds C, the critical percolation probability for the square lattice; (iii) a proof of the a.s. existence of routes for the unrestricted first-passage processes; (iv) a.s. and mean ergodic theorems for a class of reach processes; (v) continuity results for the time constant as a functional of the underlying distribution.
We extend some results of Hammersley and Welsh concerning first-passage percolation on the two-dimensional integer lattice. Our results include: (i) weak renewal theorems for the unrestricted reach processes; (ii) an L1-ergodic theorem for the unrestricted first-passage time from (0, 0) to the line X = n; and (iii) weakening of the boundedness restrictions on the underlying distribution in Hammersley and Welsh's weak renewal theorems for the cylinder reach processes.