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.

Spatial variability of flow statistics within regular building arrays

Turbulence statistics obtained by direct numerical simulations are analysed to investigate spatial heterogeneity within regular arrays of building-like cubical obstacles. Two different array layouts are studied, staggered and square, both at a packing density of λp=0.25 . The flow statistics analysed are mean streamwise velocity ( u− ), shear stress ( u′w′−−−− ), turbulent kinetic energy (k) and dispersive stress fraction ( u˜w˜ ). The spatial flow patterns and spatial distribution of these statistics in the two arrays are found to be very different. Local regions of high spatial variability are identified. The overall spatial variances of the statistics are shown to be generally very significant in comparison with their spatial averages within the arrays. Above the arrays the spatial variances as well as dispersive stresses decay rapidly to zero. The heterogeneity is explored further by separately considering six different flow regimes identified within the arrays, described here as: channelling region, constricted region, intersection region, building wake region, canyon region and front-recirculation region. It is found that the flow in the first three regions is relatively homogeneous, but that spatial variances in the latter three regions are large, especially in the building wake and canyon regions. The implication is that, in general, the flow immediately behind (and, to a lesser extent, in front of) a building is much more heterogeneous than elsewhere, even in the relatively dense arrays considered here. Most of the dispersive stress is concentrated in these regions. Considering the experimental difficulties of obtaining enough point measurements to form a representative spatial average, the error incurred by degrading the sampling resolution is investigated. It is found that a good estimate for both area and line averages can be obtained using a relatively small number of strategically located sampling points.

Hierarchical composite regular parallel architecture

The design space of emerging heterogenous multi-core architectures with re-configurability element makes it feasible to design mixed fine-grained and coarse-grained parallel architectures. This paper presents a hierarchical composite array design which extends the curret design space of regular array design by combining a sequence of transformations. This technique is applied to derive a new design of a pipelined parallel regular array with different dataflow between phases of computation.

Efficient FPGA-based regular expression pattern matching

An approach to the automatic generation of efficient Field Programmable Gate Arrays (FPGAs) circuits for the Regular Expression-based (RegEx) Pattern Matching problems is presented. Using a novel design strategy, as proposed, circuits that are highly area-and-time-efficient can be automatically generated for arbitrary sets of regular expressions. This makes the technique suitable for applications that must handle very large sets of patterns at high speed, such as in the network security and intrusion detection application domains. We have combined several existing techniques to optimise our solution for such domains and proposed the way the whole process of dynamic generation of FPGAs for RegEX pattern matching could be automated efficiently.

Compositional technique for synthesising multi-phase regular arrays

We describe a high-level design method to synthesize multi-phase regular arrays. The method is based on deriving component designs using classical regular (or systolic) array synthesis techniques and composing these separately evolved component design into a unified global design. Similarity transformations ar e applied to component designs in the composition stage in order to align data ow between the phases of the computations. Three transformations are considered: rotation, re ection and translation. The technique is aimed at the design of hardware components for high-throughput embedded systems applications and we demonstrate this by deriving a multi-phase regular array for the 2-D DCT algorithm which is widely used in many vide ocommunications applications.

A simple method of calculating eigenvalues and resonances in domains with infinite regular ends

A new approach is presented for the solution of spectral problems on infinite domains with regular ends, which avoids the need to solve boundary-value problems for many trial values of the spectral parameter. We present numerical results both for eigenvalues and for resonances, comparing with results reported by Aslanyan, Parnovski and Vassiliev.

Processing of regular and irregular past tense morphology in highly proficient second language learners of English: a self-paced reading study

Dual-system models suggest that English past tense morphology involves two processing routes: rule application for regular verbs and memory retrieval for irregular verbs (Pinker, 1999). In second language (L2) processing research, Ullman (2001a) suggested that both verb types are retrieved from memory, but more recently Clahsen and Felser (2006) and Ullman (2004) argued that past tense rule application can be automatised with experience by L2 learners. To address this controversy, we tested highly proficient Greek-English learners with naturalistic or classroom L2 exposure compared to native English speakers in a self-paced reading task involving past tense forms embedded in plausible sentences. Our results suggest that, irrespective to the type of exposure, proficient L2 learners of extended L2 exposure apply rule-based processing.

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.

Glimpsing regular lattice arrangements of primary rat hippocampal astrocytes on ultra-thin nodes of Parylene-C

In our seminal work, we reported how the biomaterial Parylene-C has the unique ability to coerce neurons and glial cells to migrate to and then grow in straight lines along serum coated rectangular parylene-C structures mounted on an oxidised silicon substrate. In this brief communication, we report how astrocyte cell bodies, from the dissociated postnatal rat hippocampus, can now to be successfully localised on an ultra-thin 13nm layer of parylene-C mounted on oxidised silicon (Figure 1). What is extremely interesting about this finding is that the astrocyte processes extended mainly in horizontal and vertical directions from the cell body thus creating a regular lattice network of individual cells. In addition, they comfortably extended a 50μm gap (equivalent to ~ 10 cell body diameters) to connect to adjacent astrocytes on neighbouring Parylene-C structures. This was found to occur repeatedly on circular geometries of 20μm diameter. In comparison to our previous work [1], we have decreased the thickness of the parylene-C structures by a factor of 10, to allow such technology to be able to be utilised for passive electrode design that requires extremely thin structures such as these. Thus, being able to culture astrocytes in regular lattice networks will pave the way for precise monitoring and stimulation of such ensembles via multi-electrode arrays, allowing a closer insight into their dynamic behaviour and their network properties.

On the difficulties of present theoretical models to predict the oxidation state of atomic Au adsorbed on regular sites of CeO2(111)

The electronic structure and oxidation state of atomic Au adsorbed on a perfect CeO2(111) surface have been investigated in detail by means of periodic density functional theory-based calculations, using the LDA+U and GGA+U potentials for a broad range of U values, complemented with calculations employing the HSE06 hybrid functional. In addition, the effects of the lattice parameter a0 and of the starting point for the geometry optimization have also been analyzed. From the present results we suggest that the oxidation state of single Au atoms on CeO2(111) predicted by LDA+U, GGA+U, and HSE06 density functional calculations is not conclusive and that the final picture strongly depends on the method chosen and on the construction of the surface model. In some cases we have been able to locate two well-defined states which are close in energy but with very different electronic structure and local geometries, one with Au fully oxidized and one with neutral Au. The energy difference between the two states is typically within the limits of the accuracy of the present exchange-correlation potentials, and therefore, a clear lowest-energy state cannot be identified. These results suggest the possibility of a dynamic distribution of Au0 and Au+ atomic species at the regular sites of the CeO2(111) surface.

Lesions impairing regular versus irregular past tense production

We investigated selective impairments in the production of regular and irregular past tense by examining language performance and lesion sites in a sample of twelve stroke patients. A disadvantage in regular past tense production was observed in six patients when phonological complexity was greater for regular than irregular verbs, and in three patients when phonological complexity was closely matched across regularity. These deficits were not consistently related to grammatical difficulties or phonological errors but were consistently related to lesion site. All six patients with a regular past tense disadvantage had damage to the left ventral pars opercularis (in the inferior frontal cortex), an area associated with articulatory sequencing in prior functional imaging studies. In addition, those that maintained a disadvantage for regular verbs when phonological complexity was controlled had damage to the left ventral supramarginal gyrus (in the inferior parietal lobe), an area associated with phonological short-term memory. When these frontal and parietal regions were spared in patients who had damage to subcortical (n = 2) or posterior temporo-parietal regions (n = 3), past tense production was relatively unimpaired for both regular and irregular forms. The remaining (12th) patient was impaired in producing regular past tense but was significantly less accurate when producing irregular past tense. This patient had frontal, parietal, subcortical and posterior temporo-parietal damage, but was distinguished from the other patients by damage to the left anterior temporal cortex, an area associated with semantic processing. We consider how our lesion site and behavioral observations have implications for theoretical accounts of past tense production.