Diffusive search with spatially dependent resetting

We consider a stochastic search model with resetting for an unknown stationary target $a\in \mathbb{R}$ with known distribution $\mu$. The searcher begins at the origin and performs Brownian motion with diffusion constant $D$. The searcher is also armed with an exponential clock with spatially dependent rate $r=r\left(\cdot \right)$, so that if it has failed to locate the target by the time the clock rings, then its position is reset to the origin and it continues its search anew from there. Denote the position of the searcher at time $t$ by $X\left(t\right)$. Let $E_0^{\left(r\right)}$ denote expectations for the process $X\left(\cdot \right)$. The search ends at time $T_a=inf\{t\ge 0:X\left(t\right)=a\}$. The expected time of the search is then ${\int }_{\mathbb{R}}\left(E_0^{\left(r\right)}{T}_a\right)\mu \left(da\right)$. Ideally, one would like to minimize this over all resetting rates $r$. We obtain quantitative growth rates for $E_0^{\left(r\right)}{T}_a$ as a function of $a$ in terms of the asymptotic behavior of the rate function $r$, and also a rather precise dichotomy on the asymptotic behavior of the resetting function $r$ to determine whether $E_0^{\left(r\right)}{T}_a$ is finite or infinite. We show generically that if $r\left(x\right)$ is of the order ${\left|x\right|}^{2l}$, with $l>-1$, then $log{E}_0^{\left(r\right)}{T}_a$ is of the order ${\left|a\right|}^{l+1}$; in particular, the smaller the asymptotic size of $r$, the smaller the asymptotic growth rate of $E_0^{\left(r\right)}{T}_a$. The asymptotic growth rate of $E_0^{\left(r\right)}{T}_a$ continues to decrease when $r\left(x\right)\sim \frac{D\lambda }{x^2}$ with $\lambda >1$; now the growth rate of $E_0^{\left(r\right)}{T}_a$ is more or less of the order ${\left|a\right|}^{\frac{1+\sqrt{1+8\lambda }}{2}}$. Note that this exponent increases to $\infty$ when $\lambda$ increases to $\infty$ and decreases to 2 when $\lambda$ decreases to 1. However, if $\lambda =1$, then $E_0^{\left(r\right)}{T}_a=\infty$, for $a\ne 0$. Our results suggest that for many distributions $\mu$ supported on all of $\mathbb{R}$, a near optimal (or optimal) choice of resetting function $r$ in order to minimize ${\int }_{{\mathbb{R}}^d}\left(E_0^{\left(r\right)}{T}_a\right)\mu \left(da\right)$ will be one which decays quadratically as $\frac{D\lambda }{x^2}$ for some $\lambda >1$. We also give explicit, albeit rather complicated, variational formulas for ${inf}_{r\gneqq 0}{\int }_{{\mathbb{R}}^d}\left(E_0^{\left(r\right)}{T}_a\right)\mu \left(da\right)$. For distributions $\mu$ with compact support, one should set $r=\infty$ off of the support. We also discuss this case.     

Fluvio-mechanical resetting of the Al and Ti centres in quartz

ESR dating of fluvial terraces is usually based on the assumption of sunlight resetting of the Al centres in quartz. Very long bleaching times (months of sunlight exposure) are required to reset this centre to a stable, non-zero ESR intensity. Considering that highly light sensitive optically stimulated luminescence (OSL) analyses of fluvial samples often indicate partial bleaching, ESR dating should be extremely difficult. Two samples were analysed, a fluvial sediment from a Chinese river terrace and an Australian granite. Using a Höhnle sunlight simulator, the Ti centres of both samples were completely bleached within 1 to 65 h, while the Al centre required extreme bleaching times to reach stable residual intensities (∼1000 h). As an alternative to light resetting, tumbler experiments were carried out to simulate fluvial action. The samples were treated under OSL sample preparation conditions. After relatively short tumbling times, corresponding to a few km of fluvial transport, both Al- and Ti-centres were partially reset. In a granite sample, the tumbling reduced the intensity of the Al centre more effectively than light exposure.     

A bundle type approach to the unconstrained minimization of convex nonsmooth functions

A numerical method for the unconstrained minimization of a convex nonsmooth function of several variables is presented. It is closely related to the bundle type approach and to the conjugate subgradient method. A way is suggested to reduce the amount of information to be stored during the computational procedure. Global convergence of the method to the minimum is proved.     

Paper II: The interpretation of measurement-time-dependent single-aliquot equivalent-dose estimates using predictions from a simple empirical model

Narrow spectrum (blue-green) laboratory partial bleaching of aliquots of natural sedimentary quartz has been shown to have a significant effect on equivalent dose (measured using a single aliquot procedure) calculated as a function of measurement time (D_e(t)). A simple model, based on empirical data, is presented in which the effects of partial bleaching on D_e(t) are predicted, taking into account the influences of pre- and post-burial doses. The model is applied to the case of heterogeneous populations of partially bleached single grains with various ranges of residual doses. Modelling shows that under realistic conditions, D_e histogram plots are unable to distinguish between samples having age overestimates and those with correct ages, whereas the proposed D_e–Z plot is able to make this distinction. Furthermore, modelling shows that D_e–Z plots can identify sub-populations of grains/aliquots that show most evidence of full bleaching, which can then be used to estimate a correct mean D_e value.