From symmetry breaking to Poisson Point Process in 2D Voronoi Tessellations: the generic nature of hexagons

We bridge the properties of the regular triangular, square, and hexagonal honeycomb Voronoi tessellations of the plane to the Poisson-Voronoi case, thus analyzing in a common framework symmetry breaking processes and the approach to uniform random distributions of tessellation-generating points. We resort to ensemble simulations of tessellations generated by points whose regular positions are perturbed through a Gaussian noise, whose variance is given by the parameter α2 times the square of the inverse of the average density of points. We analyze the number of sides, the area, and the perimeter of the Voronoi cells. For all valuesα >0, hexagons constitute the most common class of cells, and 2-parameter gamma distributions provide an efficient description of the statistical properties of the analyzed geometrical characteristics. The introduction of noise destroys the triangular and square tessellations, which are structurally unstable, as their topological properties are discontinuous in α = 0. On the contrary, the honeycomb hexagonal tessellation is topologically stable and, experimentally, all Voronoi cells are hexagonal for small but finite noise withα <0.12. For all tessellations and for small values of α, we observe a linear dependence on α of the ensemble mean of the standard deviation of the area and perimeter of the cells. Already for a moderate amount of Gaussian noise (α >0.5), memory of the specific initial unperturbed state is lost, because the statistical properties of the three perturbed regular tessellations are indistinguishable. When α >2, results converge to those of Poisson-Voronoi tessellations. The geometrical properties of n-sided cells change with α until the Poisson- Voronoi limit is reached for α > 2; in this limit the Desch law for perimeters is shown to be not valid and a square root dependence on n is established. This law allows for an easy link to the Lewis law for areas and agrees with exact asymptotic results. Finally, for α >1, the ensemble mean of the cells area and perimeter restricted to the hexagonal cells agree remarkably well with the full ensemble mean; this reinforces the idea that hexagons, beyond their ubiquitous numerical prominence, can be interpreted as typical polygons in 2D Voronoi tessellations.

Symmetry-break in Voronoi tessellations

We analyse in a common framework the properties of the Voronoi tessellations resulting from regular 2D and 3D crystals and those of tessellations generated by Poisson distributions of points, thus joining on symmetry breaking processes and the approach to uniform random distributions of seeds. We perturb crystalline structures in 2D and 3D with a spatial Gaussian noise whose adimensional strength is α and analyse the statistical properties of the cells of the resulting Voronoi tessellations using an ensemble approach. In 2D we consider triangular, square and hexagonal regular lattices, resulting into hexagonal, square and triangular tessellations, respectively. In 3D we consider the simple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC) crystals, whose corresponding Voronoi cells are the cube, the truncated octahedron, and the rhombic dodecahedron, respectively. In 2D, for all values α>0, hexagons constitute the most common class of cells. Noise destroys the triangular and square tessellations, which are structurally unstable, as their topological properties are discontinuous in α=0. On the contrary, the honeycomb hexagonal tessellation is topologically stable and, experimentally, all Voronoi cells are hexagonal for small but finite noise with α<0.12. Basically, the same happens in the 3D case, where only the tessellation of the BCC crystal is topologically stable even against noise of small but finite intensity. In both 2D and 3D cases, already for a moderate amount of Gaussian noise (α>0.5), memory of the specific initial unperturbed state is lost, because the statistical properties of the three perturbed regular tessellations are indistinguishable. When α>2, results converge to those of Poisson-Voronoi tessellations. In 2D, while the isoperimetric ratio increases with noise for the perturbed hexagonal tessellation, for the perturbed triangular and square tessellations it is optimised for specific value of noise intensity. The same applies in 3D, where noise degrades the isoperimetric ratio for perturbed FCC and BCC lattices, whereas the opposite holds for perturbed SCC lattices. This allows for formulating a weaker form of the Kelvin conjecture. By analysing jointly the statistical properties of the area and of the volume of the cells, we discover that also the cells shape heavily fluctuates when noise is introduced in the system. In 2D, the geometrical properties of n-sided cells change with α until the Poisson-Voronoi limit is reached for α>2; in this limit the Desch law for perimeters is shown to be not valid and a square root dependence on n is established, which agrees with exact asymptotic results. Anomalous scaling relations are observed between the perimeter and the area in the 2D and between the areas and the volumes of the cells in 3D: except for the hexagonal (2D) and FCC structure (3D), this applies also for infinitesimal noise. In the Poisson-Voronoi limit, the anomalous exponent is about 0.17 in both the 2D and 3D case. A positive anomaly in the scaling indicates that large cells preferentially feature large isoperimetric quotients. As the number of faces is strongly correlated with the sphericity (cells with more faces are bulkier), in 3D it is shown that the anomalous scaling is heavily reduced when we perform power law fits separately on cells with a specific number of faces.

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.

Location and collocation advantages in international innovation

Purpose – This paper examines the role of location-specific (L) advantages in the spatial distribution of multinational enterprise (MNE) R&D activity. The meaning of L advantages is revisited. In addition to L advantages that are industry-specific, the paper emphasises that there is an important category of L advantages, referred to as collocation advantages. Design/methodology/approach – Using the OLI framework, this paper highlights that the innovation activities of MNEs are about interaction of these variables, and the essential process of internalising L advantages to enhance and create firm-specific advantages. Findings – Collocation advantages derive from spatial proximity to specific unaffiliated firms, which may be suppliers, competitors, or customers. It is also argued that L advantages are not always public goods, because they may not be available to all firms at a similar or marginal cost. These costs are associated with access and internalisation of L advantages, and – especially in the case of R&D – are attendant with the complexities of embeddedness. Originality/value – The centralisation/decentralisation, spatial separation/collocation debates in R&D location have been mistakenly viewed as a paradox facing firms, instead of as a trade-off that firms must make.

Sampling series for determination of probability density of CDMA multiple access interference

Consideration is given to a standard CDMA system and determination of the density function of the interference with and without Gaussian noise using sampling theory concepts. The formula derived provides fast and accurate results and is a simple, useful alternative to other methods

Random-walk-based characterisation of multiple access interference in a DS/SSMA system

The problem of calculating the probability of error in a DS/SSMA system has been extensively studied for more than two decades. When random sequences are employed some conditioning must be done before the application of the central limit theorem is attempted, leading to a Gaussian distribution. The authors seek to characterise the multiple access interference as a random-walk with a random number of steps, for random and deterministic sequences. Using results from random-walk theory, they model the interference as a K-distributed random variable and use it to calculate the probability of error in the form of a series, for a DS/SSMA system with a coherent correlation receiver and BPSK modulation under Gaussian noise. The asymptotic properties of the proposed distribution agree with other analyses. This is, to the best of the authors' knowledge, the first attempt to propose a non-Gaussian distribution for the interference. The modelling can be extended to consider multipath fading and general modulation

Pharmacovigilance of herbal medicines

Pharmacovigilance is essential for developing reliable information on the safety of herbal medicines as used in Europe and the US. The existing systems were developed for synthetic medicines and require some modification to address the specific differences of medicinal herbs. Traditional medicine from many different cultures is used in Europe and the US which adds to the complexities and difficulties of even basic questions such as herb naming systems and chemical variability. Allied to this also is the perception that a ‘natural’ or herbal product must be safe simply because it is not synthetic which means that the safety element of monitoring for such medicines can be overlooked because of the tag associated with such products. Cooperation between orthodox physicians and traditional practitioners is needed to bring together the full case details. Independent scientific assistance on toxicological investigation, botanical verification can be invaluable for full evaluation of any case report. Systematic pharmacovigilance is essential to build up reliable information on the safety of herbal medicines for the development of appropriate guidelines for safe effective use.

Reproductive technologies: the owned child and commodification

This article analyses some complexities in and around the idea that the child, in several recent discussions on reproductive technologies, is constantly brought in relation to the market and commodity, and yet is not simply equated with commodity. When and how is the child a commodity and not a commodity?

A new floating model level scheme for the assimilation of boundary layer top inversions: the univariate assimilation of temperature.

The assimilation of observations with a forecast is often heavily influenced by the description of the error covariances associated with the forecast. When a temperature inversion is present at the top of the boundary layer (BL), a significant part of the forecast error may be described as a vertical positional error (as opposed to amplitude error normally dealt with in data assimilation). In these cases, failing to account for positional error explicitly is shown t o r esult in an analysis for which the inversion structure is erroneously weakened and degraded. In this article, a new assimilation scheme is proposed to explicitly include the positional error associated with an inversion. This is done through the introduction of an extra control variable to allow position errors in the a priori to be treated simultaneously with the usual amplitude errors. This new scheme, referred to as the ‘floating BL scheme’, is applied to the one-dimensional (vertical) variational assimilation of temperature. The floating BL scheme is tested with a series of idealised experiments a nd with real data from radiosondes. For each idealised experiment, the floating BL scheme gives an analysis which has the inversion structure and position in agreement with the truth, and outperforms the a ssimilation which accounts only for forecast a mplitude error. When the floating BL scheme is used to assimilate a l arge sample of radiosonde data, its ability to give an analysis with an inversion height in better agreement with that observed is confirmed. However, it is found that the use of Gaussian statistics is an inappropriate description o f t he error statistics o f t he extra c ontrol variable. This problem is alleviated by incorporating a non-Gaussian description of the new control variable in the new scheme. Anticipated challenges in implementing the scheme operationally are discussed towards the end of the article.

Modelling and inverting complex-valued Wiener systems

We develop a complex-valued (CV) B-spline neural network approach for efficient identification and inversion of CV Wiener systems. The CV nonlinear static function in the Wiener system is represented using the tensor product of two univariate B-spline neural networks. With the aid of a least squares parameter initialisation, the Gauss-Newton algorithm effectively estimates the model parameters that include the CV linear dynamic model coefficients and B-spline neural network weights. The identification algorithm naturally incorporates the efficient De Boor algorithm with both the B-spline curve and first order derivative recursions. An accurate inverse of the CV Wiener system is then obtained, in which the inverse of the CV nonlinear static function of the Wiener system is calculated efficiently using the Gaussian-Newton algorithm based on the estimated B-spline neural network model, with the aid of the De Boor recursions. The effectiveness of our approach for identification and inversion of CV Wiener systems is demonstrated using the application of digital predistorter design for high power amplifiers with memory

Assimilation of RAPID array observations into an ocean model

Experiments assimilating the RAPID dataset of deep temperature and salinity profiles at 26.5°N on the western and eastern Atlantic boundaries into a 1° global NEMO ocean model have been performed. The meridional overturning circulation (MOC) is then assessed against the transports calculated directly from observations. The best initialization found for this short period was obtained by assimilating the EN3 upper-ocean hydrography database prior to 2004, after which different methods of assimilating 5-day average RAPID profiles at the western boundary were tested. The model MOC is strengthened by ∼ 2 Sv giving closer agreement with the RAPID array transports, when the western boundary profiles are assimilated only below 900 m (the approximate depth of the Florida Straits, which are not well resolved) and when the T,S observations are spread meridionally from 10 to 35°N along the deep western boundary. The use of boundary-focused covariances has the largest impact on the assimilation results, otherwise using more conventional Gaussian covariances has a very local impact on the MOC at 26°N with strong adverse impacts on the MOC stream function at higher and lower latitudes. Even using boundary-focused covariances only enables the MOC to be strengthened for ∼ 2 years, after which the increased transport of warm waters leads to a negative feedback on water formation in the subpolar gyre which then reduces the MOC. This negative feedback can be mitigated if EN3 hydrography data continue to be assimilated along with the RAPID array boundary data. Copyright © 2012 Royal Meteorological Society and Crown in the right of Canada.

Sleepwalking certainties: agency, aesthetics, and incapacity in W.G. Sebald’s 'Austerlitz' and Hermann Broch’s 'The Sleepwalkers'

When we first encounter the narrator of Austerlitz, he is wandering around the unfamiliar town of Antwerp with, he tells us, “unsicheren Schritten” (1; 9). As well as reflecting the unfamiliarity of the locale, these “uncertain steps” evince a proud modesty characteristic of the classic Sebaldian narrator, a wanderer who discreetly relays the stories of the people and places he is privileged to encounter. Although Sebald does not use the phrase, steps of this sort, unpurposed yet unerring, are made with what is commonly known in German as somnambule Sicherheit: the legendary surefootedness of the sleepwalker. The convergence of sleepwalking and certainty in a single phrase poses an interesting challenge to one of the central tenets of the English-language canonization of Sebald, for his writing has been most highly valued for its ability to move the reader through apparent certainties towards a salutary uncertainty. But somnambule Sicherheit also presents the possibility that the current may be reversed, that narrative may move under cover of uncertainty towards certainty. That Sebald criticism has not been more troubled by this possibility is in no small part due to the fact that it tends to deploy the notion of sleepwalking with a minimum of reflection on its theoretical ramifications. To evoke some of the complexities of this matter, I first offer a brief cultural history of sleepwalking, as well as a brief account of the topic of uncertainty in Sebald criticism. Most of my argument, however, involves an extended comparative analysis of sleepwalking in Sebald's Austerlitz and Hermann Broch's 1933 trilogy The Sleepwalkers. Although these writers have not previously been the object of any sustained comparison, sleepwalking in Broch's novels illuminates much that is left implicit on the topic in Sebald's fiction and points toward some difficult questions regarding the role of aesthetics and agency in Sebald's work.

On the long time behavior of free stochastic Schrödinger evolutions

We discuss the time evolution of the wave function which is the solution of a stochastic Schrödinger equation describing the dynamics of a free quantum particle subject to spontaneous localizations in space. We prove global existence and uniqueness of solutions. We observe that there exist three time regimes: the collapse regime, the classical regime and the diffusive regime. Concerning the latter, we assert that the general solution converges almost surely to a diffusing Gaussian wave function having a finite spread both in position as well as in momentum. This paper corrects and completes earlier works on this issue.

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.

'Between slavery and freedom': the expulsion and enslavement of free women of colour in the US South before the Civil War

For free black women in the pre-Civil War American South, the status offered by ‘freedom’ was uncertain and malleable. The conceptualization of bondage and freedom as two diametrically opposed conditions therefore fails to make sense of the complexities of life for these women. Instead, notions of enslavement and freedom are better framed as a spectrum. This article develops this idea by exploring two of the ways in which some black women negotiated their status before the law—namely though petitioning for residency or for enslavement. While these petitions are atypical numerically, and often offer tantalizingly scant evidence, when used in conjunction with evidence from the US census, it becomes clear that these women were highly pragmatic. Prioritizing their spousal and broader familial affective relationships above their legal status, they rejected the often theoretical distinction between slavery and liberation. As such, the petitions can be used to reach broader conclusions about the attitudes of women who have left little written testimony.