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.

Investigations into the formation and characterization of phospholipid microemulsions. IV. Pseudo-ternary phase diagrams of systems containing water-lecithin-alcohol and oil; The influence of oil

Phase studies have been performed for quaternary systems composed of egg lecithin, cosurfactant, water and oil. The lecithin used was the commercially available egg lecithin Ovothin 200 (which comprises ≥ 92% phosphatidylcholine). The cosurfactants employed were propanol and butanol, and these were used at lecithin/cosurfactant mixing ratios (Km) of 1:1 and 1.94:1 (weight basis). Six polar oils were investigated, including the alkanoic acids, octanoic and oleic, their corresponding ethyl esters and the medium and long chain triglycerides, Miglyol 812 and soybean oil. All oils, irrespective of the alcohol and the Km used, gave rise to systems that produced a stable isotropic region along the surfactant/oil axis (designated as a reverse microemulsion system). In addition, the systems incorporating propanol at both Km and butanol at a Km of 1.94: 1, generally gave rise to a liquid crystalline region and, in some cases, a second isotropic non-birefingent area (designated as a normal microemulsion system). The phase behaviour observed was largely dependent upon the alcohol and Km used and the size and the polarity of the oil present.

Plague of diagrams

Group exhibition curated by Plastique Fantastique. Organised by David Burrows and Dean Kenning in collaboration with Ami Clarke, Andrew Conio, John Cussans and David Osbaldeston. Contributors: David Burrows, Rachel Cattle & Jenna Collins, Neil Chapman & Gillian Wylde, Ami Clarke, Richard Cochrane, Andrew Conio, John Cussans, David Burrows, Benedict Drew, English Heretic, Nikolaus Gansterer, Joey Holder, Dean Kenning, Christoph Lueder, Stine Llungdalh, Adelheid Mers, Mike Nelson, Paul O'Kane, David Osbaldeston, Plastique Fantastique, Patricia Reed, John Russell, Erica Scourti, Andy Sharp, Kamini Vellodi, Martin Westwood and Carey Young.

Diagnostic tool to analyse colour-magnitude diagrams of poorly populated stellar concentrations

The dynamical processes that lead to open cluster disruption cause its mass to decrease. To investigate such processes from the observational point of view, it is important to identify open cluster remnants (OCRs), which are intrinsically poorly populated. Due to their nature, distinguishing them from field star fluctuations is still an unresolved issue. In this work, we developed a statistical diagnostic tool to distinguish poorly populated star concentrations from background field fluctuations. We use 2MASS photometry to explore one of the conditions required for a stellar group to be a physical group: to produce distinct sequences in a colour-magnitude diagram (CMD). We use automated tools to (i) derive the limiting radius; (ii) decontaminate the field and assign membership probabilities; (iii) fit isochrones; and (iv) compare object and field CMDs, considering the isochrone solution, in order to verify the similarity. If the object cannot be statistically considered as a field fluctuation, we derive its probable age, distance modulus, reddening and uncertainties in a self-consistent way. As a test, we apply the tool to open clusters and comparison fields. Finally, we study the OCR candidates DoDz 6, NGC 272, ESO 435 SC48 and ESO 325 SC15. The tool is optimized to treat these low-statistic objects and to separate the best OCR candidates for studies on kinematics and chemical composition. The study of the possible OCRs will certainly provide a deep understanding of OCR properties and constraints for theoretical models, including insights into the evolution of open clusters and dissolution rates.

Alternative diagnostic diagrams and the `forgotten` population of weak line galaxies in the SDSS

A numerous population of weak line galaxies (WLGs) is often left out of statistical studies on emission-line galaxies (ELGs) due to the absence of an adequate classification scheme, since classical diagnostic diagrams, such as [O iii]/H beta versus [N ii]/H alpha (the BPT diagram), require the measurement of at least four emission lines. This paper aims to remedy this situation by transposing the usual divisory lines between star-forming (SF) galaxies and active galactic nuclei (AGN) hosts and between Seyferts and LINERs to diagrams that are more economical in terms of line quality requirements. By doing this, we rescue from the classification limbo a substantial number of sources and modify the global census of ELGs. More specifically, (1) we use the Sloan Digital Sky Survey Data Release 7 to constitute a suitable sample of 280 000 ELGs, one-third of which are WLGs. (2) Galaxies with strong emission lines are classified using the widely applied criteria of Kewley et al., Kauffmann et al. and Stasinska et al. to distinguish SF galaxies and AGN hosts and Kewley et al. to distinguish Seyferts from LINERs. (3) We transpose these classification schemes to alternative diagrams keeping [N ii]/H alpha as a horizontal axis, but replacing H beta by a stronger line (H alpha or [O ii]), or substituting the ionization-level sensitive [O iii]/H beta ratio with the equivalent width of H alpha (W(H alpha)). Optimized equations for the transposed divisory lines are provided. (4) We show that nothing significant is lost in the translation, but that the new diagrams allow one to classify up to 50 per cent more ELGs. (5) Introducing WLGs in the census of galaxies in the local Universe increases the proportion of metal-rich SF galaxies and especially LINERs. In the course of this analysis, we were led to make the following points. (i) The Kewley et al. BPT line for galaxy classification is generally ill-used. (ii) Replacing [O iii]/H beta by W(H alpha) in the classification introduces a change in the philosophy of the distinction between LINERs and Seyferts, but not in its results. Because the W(H alpha) versus [N ii]/H alpha diagram can be applied to the largest sample of ELGs without loss of discriminating power between Seyferts and LINERs, we recommend its use in further studies. (iii) The dichotomy between Seyferts and LINERs is washed out by WLGs in the BPT plane, but it subsists in other diagnostic diagrams. This suggests that the right wing in the BPT diagram is indeed populated by at least two classes, tentatively identified with bona fide AGN and `retired` galaxies that have stopped forming stars and are ionized by their old stellar populations.

THE VORONOI TESSELLATION CLUSTER FINDER IN 2+1 DIMENSIONS

We present a detailed description of the Voronoi Tessellation (VT) cluster finder algorithm in 2+1 dimensions, which improves on past implementations of this technique. The need for cluster finder algorithms able to produce reliable cluster catalogs up to redshift 1 or beyond and down to 10(13.5) solar masses is paramount especially in light of upcoming surveys aiming at cosmological constraints from galaxy cluster number counts. We build the VT in photometric redshift shells and use the two-point correlation function of the galaxies in the field to both determine the density threshold for detection of cluster candidates and to establish their significance. This allows us to detect clusters in a self-consistent way without any assumptions about their astrophysical properties. We apply the VT to mock catalogs which extend to redshift 1.4 reproducing the ACDM cosmology and the clustering properties observed in the Sloan Digital Sky Survey data. An objective estimate of the cluster selection function in terms of the completeness and purity as a function of mass and redshift is as important as having a reliable cluster finder. We measure these quantities by matching the VT cluster catalog with the mock truth table. We show that the VT can produce a cluster catalog with completeness and purity > 80% for the redshift range up to similar to 1 and mass range down to similar to 10(13.5) solar masses.

Plane curve diagrams and geometrical applications

We look at plane curve diagrams (f,alpha), which are given by a plane curve multigerm alpha : (R, S) -> R(2) and a function on it f : (R, S) -> R. We obtain a classification of all such diagrams, where alpha has e-codimension <= 2 and f has finite order. Then we define an equivalence between plane curves which we call Ah(alpha)-equivalence and which is determined by the class of the diagram (h(alpha), alpha). Here, h alpha denotes the height function of alpha with respect to its normal vector. This is an equivalence which not only takes into account the topology of the singularity of alpha, but also its flat geometry. Finally, we apply our results in order to obtain a classification of all the plane projections of a generic space curve gamma embedded in R(3).

Phase diagrams of a CTAB/organic solvent/buffer system applied to extraction of enzymes by reverse micelles

A partial pseudo-ternary phase diagram has been studied for the cethyltrimethylammonium bromide/isooctane:hexanol:butanol/potassium phosphate buffer system, where the two-phase diagram consisting of the reverse micelle phase (L-2) in equilibrium with the solvent is indicated. Based on these diagrams two-phase systems of reverse micelles were prepared with different compositions of the compounds and used for extraction and recovery of two enzymes, and the percentage of enzyme recovery yield monitored. The enzymes glucose-6-phosphate dehydrogenase (G6PD) and xylose redutase (XR) obtained from Candida guilliermondii yeast were used in the extraction procedures. The recovery yield data indicate that micelles having different composition give selective extraction of enzymes. The method can thus be used to optimize enzyme extraction processes. (c) 2007 Elsevier B.V. All rights reserved.

Phase Diagrams of the Aqueous Two-Phase Systems of Poly(ethylene glycol)/Sodium Polyacrylate/Salts

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

CP and T trajectory diagrams for a unified graphical representation of neutrino oscillations

Recently the CP trajectory diagram was introduced to demonstrate the difference between the intrinsic CP violating effects to those induced by matter for neutrino oscillation. In this Letter we introduce the T trajectory diagram. In these diagrams the probability for a given oscillation process is plotted versus the probability for the CP- or the T-conjugate processes, which forms an ellipse as the CP- or T-violating phase is varied. Since the CP- and the T-conjugate processes are related by CPT symmetry, even in the presence of matter, these two trajectory diagrams are closely related with each other and form a unified description of neutrino oscillations in matter. (C) 2002 Published by Elsevier B.V. B.V.

Divergent diagrams of folds and simultaneous conjugacy of involutions

In this work we show that the smooth classification of divergent diagrams of folds (f(1),..., f(s)) : (R-n, 0) -> (R-n x(...)xR(n), 0) can be reduced to the classification of the s-tuples (p(1)., W) of associated involutions. We apply the result to obtain normal forms when s <= n and {p(1),...,p(s)} is a transversal set of linear involutions. A complete description is given when s = 2 and n >= 2. We also present a brief discussion on applications of our results to the study of discontinuous vector fields and discrete reversible dynamical systems.