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    Convex Hulls of L\'evy Processes

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    Let X(t)X(t), t≥0t\geq0, be a L\'evy process in Rd\mathbb{R}^d starting at the origin. We study the closed convex hull ZsZ_s of {X(t):0≤t≤s}\{X(t): 0\leq t\leq s\}. In particular, we provide conditions for the integrability of the intrinsic volumes of the random set ZsZ_s and find explicit expressions for their means in the case of symmetric α\alpha-stable L\'evy processes. If the process is symmetric and each its one-dimensional projection is non-atomic, we establish that the origin a.s. belongs to the interior of ZsZ_s for all s>0s>0. Limit theorems for the convex hull of L\'evy processes with normal and stable limits are also obtained.Comment: 11 page

    Model of ionic currents through microtubule nanopores and the lumen

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    It has been suggested that microtubules and other cytoskeletal filaments may act as electrical transmission lines. An electrical circuit model of the microtubule is constructed incorporating features of its cylindrical structure with nanopores in its walls. This model is used to study how ionic conductance along the lumen is affected by flux through the nanopores when an external potential is applied across its two ends. Based on the results of Brownian dynamics simulations, the nanopores were found to have asymmetric inner and outer conductances, manifested as nonlinear IV curves. Our simulations indicate that a combination of this asymmetry and an internal voltage source arising from the motion of the C-terminal tails causes a net current to be pumped across the microtubule wall and propagate down the microtubule through the lumen. This effect is demonstrated to enhance and add directly to the longitudinal current through the lumen resulting from an external voltage source, and could be significant in amplifying low-intensity endogenous currents within the cellular environment or as a nano-bioelectronic device.Comment: 43 pages, 6 figures, revised versio
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