Contract Adjustment under Uncertainty

Consider a contract over trade in continuous time between two players, according to which one player makes a payment to the other, in exchange for an exogenous service. At each point in time, either player may unilaterally require an adjustment of the contract payment, involving adjustment costs for both players. Players’ payoffs from trade under the contract, as well as from trade under an adjusted contract, are exogenous and stochastic. We consider players’ choice of whether and when to adjust the contract payment. It is argued that the optimal strategy for each player is to adjust the contract whenever the contract payment relative to the outcome of an adjustment passes a certain threshold, depending among other things of the adjustment costs. There is strategic substitutability in the choice of thresholds, so that if one player becomes more aggressive by choosing a threshold closer to unity, the other player becomes more passive. If players may invest in order to reduce the adjustment costs, there will be over-investment compared to the welfare maximizing levels.

Adaptive independent Metropolis--Hastings

We propose an adaptive independent Metropolis--Hastings algorithm with the ability to learn from all previous proposals in the chain except the current location. It is an extension of the independent Metropolis--Hastings algorithm. Convergence is proved provided a strong Doeblin condition is satisfied, which essentially requires that all the proposal functions have uniformly heavier tails than the stationary distribution. The proof also holds if proposals depending on the current state are used intermittently, provided the information from these iterations is not used for adaption. The algorithm gives samples from the exact distribution within a finite number of iterations with probability arbitrarily close to 1. The algorithm is particularly useful when a large number of samples from the same distribution is necessary, like in Bayesian estimation, and in CPU intensive applications like, for example, in inverse problems and optimization.Comment: Published in at http://dx.doi.org/10.1214/08-AAP545 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Hydrological connectivity of soil pipes determined by ground-penetrating radar tracer detection

Soil pipes are common and important features of many catchments, particularly in semi-arid and humid areas, and can contribute a large proportion of runoff to river systems. They may also significantly influence catchment sediment and solute yield. However, there are often problems in finding and defining soil pipe networks which are located deep below the surface. Ground penetrating radar (GPR) has been used for non-destructive identification and mapping of soil pipes in blanket peat catchments. While GPR can identify subsurface cavities, it cannot alone determine hydrological connectivity between one cavity and another. This paper presents results from an experiment to test the ability of GPR to establish hydrological connectivity between pipes through use of a tracer solution. Sodium chloride was injected into pipe cavities previously detected by the radar. The GPR was placed downslope of the injection points and positioned on the ground directly above detected soil pipes. The resultant radargrams showed significant changes in reflectance from some cavities and no change from others. Pipe waters were sampled in order to check the radar results. Changes in electrical conductivity of the pipe water could be detected by the GPR, without data post-processing, when background levels were increased by greater than approximately twofold. It was thus possible to rapidly determine hydrological connectivity of soil pipes within dense pipe networks across hillslopes without ground disturbance. It was also possible to remotely measure travel times through pipe systems; the passing of the salt wave below the GPR produced an easily detectable signal on the radargram which required no post-processing. The technique should allow remote sensing of water sources and sinks for soil pipes below the surface. The improved understanding of flowpath connectivity will be important for understanding water delivery, solutional and particulate denudation, and hydrological and geomorphological model development

Sediment and particulate carbon removal by pipe erosion increase over time in blanket peatlands as a consequence of land drainage

Land drainage is common in peatlands. Artificially drained blanket peat catchments have been shown to have a significantly greater soil pipe density than intact catchments. This paper investigates the role of surface land drains in the enhancement of soil piping in blanket peats. The density of piping was found to significantly increase in a linear fashion with the age of the drainage. Thirty-five years after drains were cut, slopes would be expected to have twice the density of soil piping than would an undrained blanket peat catchment. The rate of pipe erosion increases exponentially over time, so that particulate carbon loss from subsurface pipes is greatest where drains are oldest

Geospatial aspects of catchment hydrology

The catchment is a fundamental unit of study in hydrology. It is normally well defined topographically, can be studied as a series of nested units (larger catchments are made of many smaller sub-catchments), and is an open system for measuring inputs and outputs of mass and energy. Catchments are usually delineated by land-surface topography and are made of hillslopes and channels. The proportion of hillslope area to channel density or total channel length may determine how efficiently water can be removed from a catchment since water in channels tends to move much more quickly than water across and through hillslopes. Thus the spatial layout of hillslopes and channels is important. This article describes some basic principles of catchment hydrology and illustrates how determining spatial factors involved is fundamental for understanding how environmental change may impact on runoff production and resulting river flow

Distribution of the Error in Estimated Numbers of Fixed Points of the Discrete Logarithm

Brizolis asked the question: does every prime p have a pair (g,h) such that h is a fixed point for the discrete logarithm with base g? The author and Pieter Moree, building on work of Zhang, Cobeli, and Zaharescu, gave heuristics for estimating the number of such pairs and proved bounds on the error in the estimates. These bounds are not descriptive of the true situation, however, and this paper is a first attempt to collect and analyze some data on the distribution of the actual error in the estimates.Comment: 8 pages, 3 figures, to appear in Communications in Computer Algebr

Restoration of seventeenth century water gardens at Bramham Park

Bramham Park, near Wetherby, is the only large-scale formal garden in the UK to survive virtually unchanged from the late seventeenth and early eighteenth centuries. During the Victorian period most such gardens were altered as styles and trends changed through time