Optimization in First-Passage Resetting

We investigate classic diffusion with the added feature that a diffusing particle is reset to its starting point each time the particle reaches a specified threshold. In an infinite domain, this process is non-stationary and its probability distribution exhibits rich features. In a finite domain, we define a non-trivial optimization in which a cost is incurred whenever the particle is reset and a reward is obtained while the particle stays near the reset point. We derive the condition to optimize the net gain in this system, namely, the reward minus the cost.Comment: 4 pages, 3 figures, revtex 4-1 format. Version 1 contains changes in response to referee comments. Version 2: A missing factor of 2 in an inline formula has been correcte

Chemical control of root deflection and tap root elongation in containerized nursery stock

These studies were designed 1. to test the effectiveness of a 7% cupric hydroxide [Cu(OH)2]/latex paint formulation (Spin Out™) to control root deflection in a wide assortment of containerized nursery stock, and 2. to control tap root elongation of selected coarsely rooted species by inserting six different types of materials painted with Spin Out™ or impregnated with Spin Out™ WP (wetable powder) at the bottom of the container. Seedlings or rooted cuttings of 54 taxa of ornamental trees, shrubs, perennials and grasses were grown in plastic containers, half of which were painted inside with Spin Out™. Root deflection was measured subjectively by a panel of four judges using a scale from 1 to 5, with 1 indicating root deflection of less than 1.3 cm, (excellent control) and 5 indicating severe root deflection (no control). While excellent control of root deflection was not always achieved in treated containers, root deflection was consistently reduced compared to untreated containers. This eliminated the need for corrective root pruning. Treatment means ranged from 1.0 to 2.5 with 83% ≤ 1.5. Control means ranged from 1.8 to 5.0 with 85% ≥ to 3.0. No visual signs of copper toxicity were observed. Cupric hydroxide did not inhibit or restrict the growth of stem structures such as rhizomes, stolons or basal suckers. Tap roots of three coarse rooted species, Nyssa sylvatica Marshall (black gum), Quercus acutissma Carruth. (sawtooth oak) and Castanea mollissima Bl. (Chinese chestnut) were subjected to six treatment materials which were either cut to fit or placed on the bottom of a 7.6 1 container. Each treatment material (paint only, Styrofoam plug tray, 3M floor buffer mat, peat fiber sheet, stone and weed barrier fabric) was either painted with Spin Out™ or impregnated with Spin Out™ WP. Treatments that allowed the tap root to penetrate the material, i.e. weed barrier fabric, stone and 3M floor buffing mat, were more effective in controlling tap root elongation compared to controls. Weed barrier fabric significantly reduced tap root length of Quercus acutissima and Nyssa sylvatica by 80% and 67%, respectively, compared to controls and by 65% and 53%, respectively, compared to the paint only treatment. In some cases the 3M floor buffing mat and stone treatments were more effective than the weed barrier fabric but were impractical because of weight or expense. The interior walls of all treatment containers were painted with Spin Out™ which significantly inhibited lateral root deflection down the side of the container compared to controls

Maxima of Two Random Walks: Universal Statistics of Lead Changes

We investigate statistics of lead changes of the maxima of two discrete-time random walks in one dimension. We show that the average number of lead changes grows as $\pi^{-1}\ln(t)$ in the long-time limit. We present theoretical and numerical evidence that this asymptotic behavior is universal. Specifically, this behavior is independent of the jump distribution: the same asymptotic underlies standard Brownian motion and symmetric Levy flights. We also show that the probability to have at most n lead changes behaves as $t^{-1/4}[\ln t]^n$ for Brownian motion and as $t^{-\beta(\mu)}[\ln t]^n$ for symmetric Levy flights with index $\mu$. The decay exponent $\beta(\mu)$ varies continuously with the Levy index when $02$.Comment: 7 pages, 6 figure

On the time to reach maximum for a variety of constrained Brownian motions

Published: J. Phys. A: Math. Theor. 41, 365005 (2008).International audienceWe derive P(M,t_m), the joint probability density of the maximum M and the time t_m at which this maximum is achieved for a class of constrained Brownian motions. In particular, we provide explicit results for excursions, meanders and reflected bridges associated with Brownian motion. By subsequently integrating over M, the marginal density P(t_m) is obtained in each case in the form of a doubly infinite series. For the excursion and meander, we analyse the moments and asymptotic limits of P(t_m) in some detail and show that the theoretical results are in excellent accord with numerical simulations. Our primary method of derivation is based on a path integral technique; however, an alternative approach is also outlined which is founded on certain "agreement formulae" that are encountered more generally in probabilistic studies of Brownian motion processes