Three-dimensional random Voronoi tessellations: From cubic crystal lattices to Poisson point processes

We perturb the SC, BCC, and FCC crystal structures with a spatial Gaussian noise whose adimensional strength is controlled by the parameter a, and analyze the topological and metrical properties of the resulting Voronoi Tessellations (VT). The topological properties of the VT of the SC and FCC crystals are unstable with respect to the introduction of noise, because the corresponding polyhedra are geometrically degenerate, whereas the tessellation of the BCC crystal is topologically stable even against noise of small but finite intensity. For weak noise, the mean area of the perturbed BCC and FCC crystals VT increases quadratically with a. In the case of perturbed SCC crystals, there is an optimal amount of noise that minimizes the mean area of the cells. Already for a moderate noise (a>0.5), the properties of the three perturbed VT are indistinguishable, and for intense noise (a>2), results converge to the Poisson-VT limit. Notably, 2-parameter gamma distributions are an excellent model for the empirical of of all considered properties. The VT of the perturbed BCC and FCC structures are local maxima for the isoperimetric quotient, which measures the degre of sphericity of the cells, among space filling VT. In the BCC case, this suggests a weaker form of the recentluy disproved Kelvin conjecture. Due to the fluctuations of the shape of the cells, anomalous scalings with exponents >3/2 is observed between the area and the volumes of the cells, and, except for the FCC case, also for a->0. In the Poisson-VT limit, the exponent is about 1.67. As the number of faces is positively correlated with the sphericity of the cells, the anomalous scaling is heavily reduced when we perform powerlaw fits separately on cells with a specific number of faces.

Planning for the reuse of redundant defence estate: disposal processes, policy frameworks and development impacts

This paper reviews recent research and other literature concerning the planning and development of redundant defence estate. It concentrates on UK sources but includes reference to material from Europe and the North America were it is relevant for comparative purposes. It introduces the topic by providing a brief review of the recent restructuring of the UK defence estate and then proceeds to examine the various planning policy issues generated by this process; the policy frameworks used to guide it; comparable approaches to surplus land disposal and the appraisal of impacts; and ending the main body of the review with an analyse of the economic, social and environmental impacts of military base closure and redevelopment. It concludes that there is a significant body of work focusing on the reuse and redevelopment of redundant defence estate in the UK and abroad, but that much of this work is based on limited research or on personal experience. One particular weakness of the current literature is that it does not fully reflect the institutional difficulties posed by the disposal process and the day-to-day pressures which MOD personnel have to deal with. In doing this, it also under-emphasises the embedded cultures of individuals and professional groups who are required to operationalise the policies, procedures and practices for planning and redeveloping redundant defence estate.

On the Convergence of Two-Stage Iterative Processes for Solving Linear Equations

This paper considers two-stage iterative processes for solving the linear system $Af = b$. The outer iteration is defined by $Mf^{k + 1} = Nf^k + b$, where $M$ is a nonsingular matrix such that $M - N = A$. At each stage $f^{k + 1} $ is computed approximately using an inner iteration process to solve $Mv = Nf^k + b$ for $v$. At the $k$th outer iteration, $p_k $ inner iterations are performed. It is shown that this procedure converges if $p_k \geqq P$ for some $P$ provided that the inner iteration is convergent and that the outer process would converge if $f^{k + 1} $ were determined exactly at every step. Convergence is also proved under more specialized conditions, and for the procedure where $p_k = p$ for all $k$, an estimate for $p$ is obtained which optimizes the convergence rate. Examples are given for systems arising from the numerical solution of elliptic partial differential equations and numerical results are presented.

Hydrochemical processes in lowland rivers: insights from in situ, high-resolution monitoring

This paper introduces new insights into the hydrochemical functioning of lowland river systems using field-based spectrophotometric and electrode technologies. The streamwater concentrations of nitrogen species and phosphorus fractions were measured at hourly intervals on a continuous basis at two contrasting sites on tributaries of the River Thames – one draining a rural catchment, the River Enborne, and one draining a more urban system, The Cut. The measurements complement those from an existing network of multi-parameter water quality sondes maintained across the Thames catchment and weekly monitoring based on grab samples. The results of the sub-daily monitoring show that streamwater phosphorus concentrations display highly complex dynamics under storm conditions dependent on the antecedent catchment wetness, and that diurnal phosphorus and nitrogen cycles occur under low flow conditions. The diurnal patterns highlight the dominance of sewage inputs in controlling the streamwater phosphorus and nitrogen concentrations at low flows, even at a distance of 7 km from the nearest sewage treatment works in the rural River Enborne. The time of sample collection is important when judging water quality against ecological thresholds or standards. An exhaustion of the supply of phosphorus from diffuse and multiple septic tank sources during storm events was evident and load estimation was not improved by sub-daily monitoring beyond that achieved by daily sampling because of the eventual reduction in the phosphorus mass entering the stream during events. The results highlight the utility of sub-daily water quality measurements and the discussion considers the practicalities and challenges of in situ, sub-daily monitoring.

Smoothness of scale functions for spectrally negative Lévy processes

Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often require excursion theory rather than Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic.

Small time two-sided LIL behavior for Lévy processes at zero

We wish to characterize when a Lévy process X t crosses boundaries b(t), in a two-sided sense, for small times t, where b(t) satisfies very mild conditions. An integral test is furnished for computing the value of sup t→0|X t |/b(t) = c. In some cases, we also specify a function b(t) in terms of the Lévy triplet, such that sup t→0 |X t |/b(t) = 1.

Extended factorizations of exponential functionals of Lévy processes

Pardo, Patie, and Savov derived, under mild conditions, a Wiener-Hopf type factorization for the exponential functional of proper Lévy processes. In this paper, we extend this factorization by relaxing a finite moment assumption as well as by considering the exponential functional for killed Lévy processes. As a by-product, we derive some interesting fine distributional properties enjoyed by a large class of this random variable, such as the absolute continuity of its distribution and the smoothness, boundedness or complete monotonicity of its density. This type of results is then used to derive similar properties for the law of maxima and first passage time of some stable Lévy processes. Thus, for example, we show that for any stable process with $\rho\in(0,\frac{1}{\alpha}-1]$, where $\rho\in[0,1]$ is the positivity parameter and $\alpha$ is the stable index, then the first passage time has a bounded and non-increasing density on $\mathbb{R}_+$. We also generate many instances of integral or power series representations for the law of the exponential functional of Lévy processes with one or two-sided jumps. The proof of our main results requires different devices from the one developed by Pardo, Patie, Savov. It relies in particular on a generalization of a transform recently introduced by Chazal et al together with some extensions to killed Lévy process of Wiener-Hopf techniques. The factorizations developed here also allow for further applications which we only indicate here also allow for further applications which we only indicate here.

Regulation of surface water quality in a Cretaceous chalk catchment, UK: an assessment of the relative importance of instream and wetland processes

To investigate the relative importance of instream nutrient spiralling and wetland transformation processes on surface water quality, total nitrogen (TN) and total phosphorus (TP) concentrations in a 200 m reach of the River Lambourn in the south-east of England were monitored over a 2-year period. In addition, the soil pore water nutrient dynamics in a riparian ecosystem adjacent to the river were investigated. Analysis of variance indicated that TN, TP and suspended sediment concentrations recorded upstream of the wetland were statistically significantly higher (P<0.05) than those downstream of the site. Such results suggest that the wetland was performing a nutrient retention function. Indeed, analysis of soil pore waters within the site show that up to 85% of TN and 70% of TP was removed from water flowing through the wetland during baseflow conditions, thus supporting the theory that the wetland played an important role in the regulation of surface water quality at the site. However, the small variations observed (0.034 mg TN l-1 and 0.031 mg P l-1) are consistent with the theory of nutrient spiralling suggesting that both instream and wetland retention processes have a causal effect on surface water quality.

The sample autocorrelation function and the detection of long-memory processes

The detection of long-range dependence in time series analysis is an important task to which this paper contributes by showing that whilst the theoretical definition of a long-memory (or long-range dependent) process is based on the autocorrelation function, it is not possible for long memory to be identified using the sum of the sample autocorrelations, as usually defined. The reason for this is that the sample sum is a predetermined constant for any stationary time series; a result that is independent of the sample size. Diagnostic or estimation procedures, such as those in the frequency domain, that embed this sum are equally open to this criticism. We develop this result in the context of long memory, extending it to the implications for the spectral density function and the variance of partial sums of a stationary stochastic process. The results are further extended to higher order sample autocorrelations and the bispectral density. The corresponding result is that the sum of the third order sample (auto) bicorrelations at lags h,k≥1, is also a predetermined constant, different from that in the second order case, for any stationary time series of arbitrary length.

Diabatic processes modifying potential vorticity in a North Atlantic cyclone

Potential vorticity (PV) succinctly describes the evolution of large-scale atmospheric flow because of its material conservation and invertibility properties. However, diabatic processes in extratropical cyclones can modify PV and influence both mesoscale weather and the evolution of the synoptic-scale wave pattern. In this investigation, modification of PV by diabatic processes is diagnosed in a Met Office Unified Model (MetUM) simulation of a North Atlantic cyclone using a set of PV tracers. The structure of diabatic PV within the extratropical cyclone is investigated and linked to the processes responsible for it. On the mesoscale, a tripole of diabatic PV is generated across the tropopause fold extending down to the cold front. The structure results from a dipole in heating across the frontal interface due to condensation in the warm conveyor belt flanking the upper side of the fold and evaporation of precipitation in the dry intrusion and below. On isentropic surfaces intersecting the tropopause, positive diabatic PV is generated on the stratospheric side, while negative diabatic PV is generated on the tropospheric side. The stratospheric diabatic PV is generated primarily by long-wave cooling which peaks at the tropopause itself due to the sharp gradient in humidity there. The tropospheric diabatic PV originates locally from the long-wave radiation and non-locally by advection out of the top of heating associated with the large-scale cloud, convection and boundary layer schemes. In most locations there is no diabatic modification of PV at the tropopause itself but diabatic PV anomalies would influence the tropopause indirectly through the winds they induce and subsequent advection. The consequences of this diabatic PV dipole for the evolution of synoptic-scale wave patterns are discussed.