21,315 research outputs found

    On Lerch's transcendent and the Gaussian random walk

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    Let X1,X2,...X_1,X_2,... be independent variables, each having a normal distribution with negative mean β<0-\beta<0 and variance 1. We consider the partial sums Sn=X1+...+XnS_n=X_1+...+X_n, with S0=0S_0=0, and refer to the process {Sn:n0}\{S_n:n\geq0\} as the Gaussian random walk. We present explicit expressions for the mean and variance of the maximum M=max{Sn:n0}.M=\max\{S_n:n\geq0\}. These expressions are in terms of Taylor series about β=0\beta=0 with coefficients that involve the Riemann zeta function. Our results extend Kingman's first-order approximation [Proc. Symp. on Congestion Theory (1965) 137--169] of the mean for β0\beta\downarrow0. We build upon the work of Chang and Peres [Ann. Probab. 25 (1997) 787--802], and use Bateman's formulas on Lerch's transcendent and Euler--Maclaurin summation as key ingredients.Comment: Published at http://dx.doi.org/10.1214/105051606000000781 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Finite-size scaling of directed percolation above the upper critical dimension

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    We consider analytically as well as numerically the finite-size scaling behavior in the stationary state near the non-equilibrium phase transition of directed percolation within the mean field regime, i.e., above the upper critical dimension. Analogous to equilibrium, usual finite-size scaling is valid below the upper critical dimension, whereas it fails above. Performing a momentum analysis of associated path integrals we derive modified finite-size scaling forms of the order parameter and its higher moments. The results are confirmed by numerical simulations of corresponding high-dimensional lattice models.Comment: 4 pages, one figur

    Optimal Tradeoff Between Exposed and Hidden Nodes in Large Wireless Networks

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    Wireless networks equipped with the CSMA protocol are subject to collisions due to interference. For a given interference range we investigate the tradeoff between collisions (hidden nodes) and unused capacity (exposed nodes). We show that the sensing range that maximizes throughput critically depends on the activation rate of nodes. For infinite line networks, we prove the existence of a threshold: When the activation rate is below this threshold the optimal sensing range is small (to maximize spatial reuse). When the activation rate is above the threshold the optimal sensing range is just large enough to preclude all collisions. Simulations suggest that this threshold policy extends to more complex linear and non-linear topologies

    Exploring prospects of novel drugs for tuberculosis

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    Tuberculosis remains a disease with an enormous impact on public health worldwide. With the continuously increasing epidemic of drug-resistant tuberculosis, new drugs are desperately needed. However, even for the treatment of drug-sensitive tuberculosis, new drugs are required to shorten the treatment duration and thereby prevent development of drug resistance. Within the past ten years, major advances in tuberculosis drug research have been made, leading to a considerable number of antimycobacterial compounds which are now in the pipeline. Here we discuss a number of these novel promising tuberculosis drugs, as well as the discovery of two new potential drug targets for the development of novel effective drugs to curb the tuberculosis pandemic, ie, the coronin 1 and protein kinase G pathways. Protein kinase G is secreted by mycobacteria and is responsible for blocking lysosomal delivery within the macrophage. Coronin 1 is responsible for activating the phosphatase, calcineurin, and thereby preventing phagosome-lysosome fusion within the macrophage. Blocking these two pathways may lead to rapid killing of mycobacteri
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