Geometric diagnostics of complex patterns: Spiral defect chaos

Motivated by the observation of spiral patterns in a wide range of physical, chemical, and biological systems, we present an automated approach that aims at characterizing quantitatively spiral-like elements in complex stripelike patterns. The approach provides the location of the spiral tip and the size of the spiral arms in terms of their arc length and their winding number. In addition, it yields the number of pattern components (Betti number of order 1), as well as their size and certain aspects of their shape. We apply the method to spiral defect chaos in thermally driven Rayleigh- Bénard convection and find that the arc length of spirals decreases monotonically with decreasing Prandtl number of the fluid and increasing heating. By contrast, the winding number of the spirals is nonmonotonic in the heating. The distribution function for the number of spirals is significantly narrower than a Poisson distribution. The distribution function for the winding number shows approximately an exponential decay. It depends only weakly on the heating, but strongly on the Prandtl number. Large spirals arise only for larger Prandtl numbers. In this regime the joint distribution for the spiral length and the winding number exhibits a three-peak structure, indicating the dominance of Archimedean spirals of opposite sign and relatively straight sections. For small Prandtl numbers the distribution function reveals a large number of small compact pattern components.

Dissociation of HIV-1 from follicular dendritic cells during HAART: Mathematical analysis

Follicular dendritic cells (FDC) provide a reservoir for HIV type 1 (HIV-1) that may reignite infection if highly active antiretroviral therapy (HAART) is withdrawn before virus on FDC is cleared. To estimate the treatment time required to eliminate HIV-1 on FDC, we develop deterministic and stochastic models for the reversible binding of HIV-1 to FDC via ligand–receptor interactions and examine the consequences of reducing the virus available for binding to FDC. Analysis of these models shows that the rate at which HIV-1 dissociates from FDC during HAART is biphasic, with an initial period of rapid decay followed by a period of slower exponential decay. The speed of the slower second stage of dissociation and the treatment time required to eradicate the FDC reservoir of HIV-1 are insensitive to the number of virions bound and their degree of attachment to FDC before treatment. In contrast, the expected time required for dissociation of an individual virion from FDC varies sensitively with the number of ligands attached to the virion that are available to interact with receptors on FDC. Although most virions may dissociate from FDC on the time scale of days to weeks, virions coupled to a higher-than-average number of ligands may persist on FDC for years. This result suggests that HAART may not be able to clear all HIV-1 trapped on FDC and that, even if clearance is possible, years of treatment will be required.

Fluorescence correlation spectroscopy reveals fast optical excitation-driven intramolecular dynamics of yellow fluorescent proteins

Fast excitation-driven fluctuations in the fluorescence emission of yellow-shifted green fluorescent protein mutants T203Y and T203F, with S65G/S72A, are discovered in the 10−6–10−3-s time range, by using fluorescence correlation spectroscopy at 10−8 M. This intensity-dependent flickering is conspicuous at high pH, with rate constants independent of pH and viscosity with a minor temperature effect. The mean flicker rate increases linearly with excitation intensity for at least three decades, but the mean dark fraction of the molecules undergoing these dynamics is independent of illumination intensity over ≈6 × 102 to 5 × 106 W/cm2. These results suggest that optical excitation establishes an equilibration between two molecular states of different spectroscopic properties that are coupled only via the excited state as a gateway. This reversible excitation-driven transition has a quantum efficiency of ≈10−3. Dynamics of external protonation, reversibly quenching the fluorescence, are also observed at low pH in the 10- to 100-μs time range. The independence of these two bright–dark flicker processes implies the existence of at least two separate dark states of these green fluorescent protein mutants. Time-resolved fluorescence measurements reveal a single exponential decay of the excited state population with 3.8-ns lifetime, after 500-nm excitation, that is pH independent. Our fluorescence correlation spectroscopy results are discussed in terms of recent theoretical studies that invoke isomerization of the chromophore as a nonradiative channel of the excited state relaxation.

Polarity and permeation profiles in lipid membranes

The isotropic 14N-hyperfine coupling constant, a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{o}^{N}}}\end{equation*}\end{document}, of nitroxide spin labels is dependent on the local environmental polarity. The dependence of a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{o}^{N}}}\end{equation*}\end{document} in fluid phospholipid bilayer membranes on the C-atom position, n, of the nitroxide in the sn-2 chain of a spin-labeled diacyl glycerophospholipid therefore determines the transmembrane polarity profile. The polarity variation in phospholipid membranes, with and without equimolar cholesterol, is characterized by a sigmoidal, trough-like profile of the form {1 + exp [(n − no)/λ]}−1, where n = no is the point of maximum gradient, or polarity midpoint, beyond which the free energy of permeation decreases linearly with n, on a characteristic length-scale, λ. Integration over this profile yields a corresponding expression for the permeability barrier to polar solutes. For fluid membranes without cholesterol, no ≈ 8 and λ ≈ 0.5–1 CH2 units, and the permeability barrier introduces an additional diffusive resistance that is equivalent to increasing the effective membrane thickness by 35–80%, depending on the lipid. For membranes containing equimolar cholesterol, no ≈ 9–10, and the total change in polarity is greater than for membranes without cholesterol, increasing the permeability barrier by a factor of 2, whereas the decay length remains similar. The permeation of oxygen into fluid lipid membranes (determined by spin-label relaxation enhancements) displays a profile similar to that of the transmembrane polarity but of opposite sense. For fluid membranes without cholesterol no ≈ 8 and λ ≈ 1 CH2 units, also for oxygen. The permeation profile for polar paramagnetic ion complexes is closer to a single exponential decay, i.e., no lies outside the acyl-chain region of the membrane. These results are relevant not only to the permeation of water and polar solutes into membranes and their permeabilities, but also to depth determinations of site-specifically spin-labeled protein residues by using paramagnetic relaxation agents.

Recurrence interval statistics of cellular automaton seismicity models

The recurrence interval statistics for regional seismicity follows a universal distribution function, independent of the tectonic setting or average rate of activity (Corral, 2004). The universal function is a modified gamma distribution with power-law scaling of recurrence intervals shorter than the average rate of activity and exponential decay for larger intervals. We employ the method of Corral (2004) to examine the recurrence statistics of a range of cellular automaton earthquake models. The majority of models has an exponential distribution of recurrence intervals, the same as that of a Poisson process. One model, the Olami-Feder-Christensen automaton, has recurrence statistics consistent with regional seismicity for a certain range of the conservation parameter of that model. For conservation parameters in this range, the event size statistics are also consistent with regional seismicity. Models whose dynamics are dominated by characteristic earthquakes do not appear to display universality of recurrence statistics.

Computational mechanics of molecular systems:quantifying high-dimensional dynamics by distribution of Poincaré recurrence times

A framework that connects computational mechanics and molecular dynamics has been developed and described. As the key parts of the framework, the problem of symbolising molecular trajectory and the associated interrelation between microscopic phase space variables and macroscopic observables of the molecular system are considered. Following Shalizi and Moore, it is shown that causal states, the constituent parts of the main construct of computational mechanics, the e-machine, define areas of the phase space that are optimal in the sense of transferring information from the micro-variables to the macro-observables. We have demonstrated that, based on the decay of their Poincare´ return times, these areas can be divided into two classes that characterise the separation of the phase space into resonant and chaotic areas. The first class is characterised by predominantly short time returns, typical to quasi-periodic or periodic trajectories. This class includes a countable number of areas corresponding to resonances. The second class includes trajectories with chaotic behaviour characterised by the exponential decay of return times in accordance with the Poincare´ theorem.

Data methods in optometry. Part 10: non-linear regression analysis

1. The techniques associated with regression, whether linear or non-linear, are some of the most useful statistical procedures that can be applied in clinical studies in optometry. 2. In some cases, there may be no scientific model of the relationship between X and Y that can be specified in advance and the objective may be to provide a ‘curve of best fit’ for predictive purposes. In such cases, the fitting of a general polynomial type curve may be the best approach. 3. An investigator may have a specific model in mind that relates Y to X and the data may provide a test of this hypothesis. Some of these curves can be reduced to a linear regression by transformation, e.g., the exponential and negative exponential decay curves. 4. In some circumstances, e.g., the asymptotic curve or logistic growth law, a more complex process of curve fitting involving non-linear estimation will be required.

Accurate WKB Approximation for a 1D Problem with Low Regularity

2000 Mathematics Subject Classification: 34L40, 65L10, 65Z05, 81Q20.

Molecular alignment relaxation in polymer optical fibers for sensing applications

A systematic study of annealing behavior of drawn PMMA fibers was performed. Annealing dynamics were investigated under different environmental conditions by fiber longitudinal shrinkage monitoring. The shrinkage process was found to follow a stretched exponential decay function revealing the heterogeneous nature of the underlying molecular dynamics. The complex dependence of the fiber shrinkage on initial degree of molecular alignment in the fiber, annealing time and temperature was investigated and interpreted. Moreover, humidity was shown to have a profound effect on the annealing process, which was not recognized previously. Annealing was also shown to have considerable effect on the fiber mechanical properties associated with the relaxation of molecular alignment in the fiber. The consequences of fiber annealing for the climatic stability of certain polymer optical fiber-based sensors are discussed, emphasizing the importance of fiber controlled pre-annealing with respect to the foreseeable operating conditions.

Representing spray zone with cross flow as a well-mixed compartment in a high shear granulator

The spray zone is an important region to control nucleation of granules in a high shear granulator. In this study, a spray zone with cross flow is quantified as a well-mixed compartment in a high shear granulator. Granulation kinetics is quantitatively derived at both particle-scale and spray zone-scale. Two spatial decay rates, DGSDR (droplet-granule spatial decay rate) ζDG and DPSDR (droplet-primary particle spatial decay rate) ζDP, which are functions of volume fraction and diameter of particulate species within the powder bed, are defined to simplify the deduction. It is concluded that in cross flow, explicit analytical results show that the droplet concentration is subject to exponential decay with depth which produces a numerically infinite depth of spray zone in a real penetration process. In a well-mixed spray zone, the depth of the spray zone is 4/(ζDG + ζDP) and π2/3(ζDG + ζDP) in cuboid and cylinder shape, respectively. The first-order droplet-based collision rates of, nucleation rate B0 and rewetting rate RW0 are uncorrelated with the flow pattern and shape of the spray zone. The second-order droplet-based collision rate, nucleated granule-granule collision rate RGG, is correlated with the mixing pattern. Finally, a real formulation case of a high shear granulation process is used to estimate the size of the spray zone. The results show that the spray zone is a thin layer at the powder bed surface. We present, for the first time, the spray zone as a well-mixed compartment. The granulation kinetics of a well-mixed spray zone could be integrated into a Population Balance Model (PBM), particularly to aid development of a distributed model for product quality prediction.

Periphyton light transmission relationships in Florida Bay and the Florida Keys, USA

Light transmission was measured through intact, submerged periphyton communities on artificial seagrass leaves. The periphyton communities were representative of the communities on Thalassia testudinum in subtropical seagrass meadows. The periphyton communities sampled were adhered carbonate sediment, coralline algae, and mixed algal assemblages. Crustose or film-forming periphyton assemblages were best prepared for light transmission measurements using artificial leaves fouled on both sides, while measurements through three-dimensional filamentous algae required the periphyton to be removed from one side. For one-sided samples, light transmission could be measured as the difference between fouled and reference artificial leaf samples. For two-sided samples, the percent periphyton light transmission to the leaf surface was calculated as the square root of the fraction of incident light. Linear, exponential, and hyperbolic equations were evaluated as descriptors of the periphyton dry weight versus light transmission relationship. Hyperbolic and exponential decay models were superior to linear models and exhibited the best fits for the observed relationships. Differences between the coefficients of determination (r2) of hyperbolic and exponential decay models were statistically insignificant. Constraining these models for 100% light transmission at zero periphyton load did not result in any statistically significant loss in the explanatory capability of the models. In most all cases, increasing model complexity using three-parameter models rather than two-parameter models did not significantly increase the amount of variation explained. Constrained two-parameter hyperbolic or exponential decay models were judged best for describing the periphyton dry weight versus light transmission relationship. On T. testudinum in Florida Bay and the Florida Keys, significant differences were not observed in the light transmission characteristics of the varying periphyton communities at different study sites. Using pooled data from the study sites, the hyperbolic decay coefficient for periphyton light transmission was estimated to be 4.36 mg dry wt. cm−2. For exponential models, the exponential decay coefficient was estimated to be 0.16 cm2 mg dry wt.−1.

Quantitative and Qualitative Aspects of Dissolved Organic Carbon Leached from Senescent Plants in an Oligotrophic Wetland

We conducted a series of experiments whereby dissolved organic matter (DOM) was leached from various wetland and estuarine plants, namely sawgrass (Cladium jamaicense), spikerush (Eleocharis cellulosa), red mangrove (Rhizophora mangle), cattail (Typha domingensis), periphyton (dry and wet mat), and a seagrass (turtle grass; Thalassia testudinum). All are abundant in the Florida Coastal Everglades (FCE) except for cattail, but this species has a potential to proliferate in this environment. Senescent plant samples were immersed into ultrapure water with and without addition of 0.1% NaN3 (w/ and w/o NaN3, respectively) for 36 days. We replaced the water every 3 days. The amount of dissolved organic carbon (DOC), sugars, and phenols in the leachates were analyzed. The contribution of plant leachates to the ultrafiltered high molecular weight fraction of DOM (>1 kDa; UDOM) in natural waters in the FCE was also investigated. UDOM in plant leachates was obtained by tangential flow ultrafiltration and its carbon and phenolic compound compositions were analyzed using solid state 13C cross-polarization magic angle spinning nuclear magnetic resonance (13C CPMAS NMR) spectroscopy and thermochemolysis in the presence of tetramethylammonium hydroxide (TMAH thermochemolysis), respectively. The maximum yield of DOC leached from plants over the 36-day incubations ranged from 13.0 to 55.2 g C kg−1 dry weight. This amount was lower in w/o NaN3 treatments (more DOC was consumed by microbes than produced) except for periphyton. During the first 2 weeks of the 5 week incubation period, 60–85% of the total amount of DOC was leached, and exponential decay models fit the leaching rates except for periphyton w/o NaN3. Leached DOC (w/ NaN3) contained different concentrations of sugars and phenols depending on the plant types (1.09–7.22 and 0.38–12.4 g C kg−1 dry weight, respectively), and those biomolecules comprised 8–34% and 4–28% of the total DOC, respectively. This result shows that polyphenols that readily leach from senescent plants can be an important source of chromophoric DOM (CDOM) in wetland environments. The O-alkyl C was found to be the major C form (55±9%) of UDOM in plant leachates as determined by 13C CPMAS NMR. The relative abundance of alkyl C and carbonyl C was consistently lower in plant-leached UDOM than that in natural water UDOM in the FCE, which suggests that these constituents increase in relative abundance during diagenetic processing. TMAH thermochemolysis analysis revealed that the phenolic composition was different among the UDOM leached from different plants, and was expected to serve as a source indicator of UDOM in natural water. Polyphenols are, however, very reactive and photosensitive in aquatic environments, and thus may loose their plant-specific molecular characteristics shortly. Our study suggests that variations in vegetative cover across a wetland landscape will affect the quantity and quality of DOM leached into the water, and such differences in DOM characteristics may affect other biogeochemical processes.

Ablation of Steel by Microsecond Pulse Trains

Laser micromachining is an important material processing technique used in industry and medicine to produce parts with high precision. Control of the material removal process is imperative to obtain the desired part with minimal thermal damage to the surrounding material. Longer pulsed lasers, with pulse durations of milli- and microseconds, are used primarily for laser through-cutting and welding. In this work, a two-pulse sequence using microsecond pulse durations is demonstrated to achieve consistent material removal during percussion drilling when the delay between the pulses is properly defined. The light-matter interaction moves from a regime of surface morphology changes to melt and vapour ejection. Inline coherent imaging (ICI), a broadband, spatially-coherent imaging technique, is used to monitor the ablation process. The pulse parameter space is explored and the key regimes are determined. Material removal is observed when the pulse delay is on the order of the pulse duration. ICI is also used to directly observe the ablation process. Melt dynamics are characterized by monitoring surface changes during and after laser processing at several positions in and around the interaction region. Ablation is enhanced when the melt has time to flow back into the hole before the interaction with the second pulse begins. A phenomenological model is developed to understand the relationship between material removal and pulse delay. Based on melt refilling the interaction region, described by logistic growth, and heat loss, described by exponential decay, the model is fit to several datasets. The fit parameters reflect the pulse energies and durations used in the ablation experiments. For pulse durations of 50 us with pulse energies of 7.32 mJ +/- 0.09 mJ, the logisitic growth component of the model reaches half maximum after 8.3 us +/- 1.1 us and the exponential decays with a rate of 64 us +/- 15 us. The phenomenological model offers an interpretation of the material removal process.

La metrica di Agmon ed il decadimento esponenziale delle soluzioni di equazioni alle derivate parziali

The aim of this master thesis is to study the exponential decay of solutions of elliptic partial equations. This work is based on the results obtained by Agmon. To this purpose, first, we define the Agmon metric, that plays an important role in the study of exponential decay, because it is related to the rate of decay. Under some assumptions on the growth of the function and on the positivity of the quadratic form associated to the operator, a first result of exponential decay is presented. This result is then applied to show the exponential decay of eigenfunctions with eigenvalues whose real part lies below the bottom of the essential spectrum. Finally, three examples are given: the harmonic oscillator, the hydrogen atom and a Schrödinger operator with purely discrete spectrum.

Unravelling the nanostructure of strawberry fruit pectins by atomic force microscopy

Atomic force microscopy (AFM) allows the analysis of individual polymers at nanostructural level with a minimal sample preparation. This technique has been used to analyse the pectin disassembly process during the ripening and postharvest storage of several fleshy fruits. In general, pectins analysed by AFM are usually visualized as isolated chains, unbranched or with a low number of branchs and, occasionally, as large aggregates. However, the exact nature of these structures is unknown. It has been suggested that pectin aggregates represent a mixture of rhamnonogalacturonan I and homogalacturonan, while isolated chains and their branches are mainly composed by polygalacturonic acid. In order to gain insight into the nature of these structures, sodium carbonate soluble pectins from ripe strawberry (Fragaria x ananassa, Duch.) fruits were subjected to enzymatic digestion with endo-Polygalacturonase M2 from Aspergillus aculeatus, and the samples visualized by AFM at different time intervals. Pectins isolated from control, non-transformed plants, and two transgenic genotypes with low level of expression of ripening-induced pectinase genes encoding a polygalacturonase (APG) or a pectate lyase (APEL) were also included in this study. Before digestion, isolated pectin chains from control were shorter than those from transgenic fruits, showing number-average (LN) contour length values of 73.2 nm vs. 95.9 nm and 91.4 nm in APG and APEL, respectively. The percentage of branched polymers was significantly higher in APG polyuronides than in the remaining genotypes, 33% in APG vs. 6% in control and APEL. As a result of the endo-PG treatment, a gradual decrease in the main backbone length of isolated chains was observed in the three samples. The minimum LN value was reached after 8 h of digestion, being similar in the three genotypes, 22 nm. By contrast, the branches were not visible after 1.5-2 h of digestion. LN values were plotted against digestion time and the data fitted to a first-order exponential decay curve, obtaining R2 values higher than 0.9. The half digestion time calculated with these equations were similar for control and APG pectins, 1.7 h, but significantly higher in APEL, 2.5 h, indicating that these polymer chains were more resistant to endo-PG digestion. Regarding the pectin aggregates, their volumes were estimated and used to calculate LN molecular weights. Before digestion, control and APEL samples showed complexes of similar molecular weights, 1722 kDa, and slightly higher than those observed in APG samples. After endo-PG digestion, size of complexes diminished significantly, reaching similar values in the three pectin samples, around 650 kDa. These results suggest that isolated polymer chains visualized by AFM are formed by a HG domain linked to a shorter polymer resistant to endo-PG digestion, maybe xylogalacturonan or RG-I. The silencing of the pectate lyase gene slightly modified the structure and/or chemical composition of polymer chains making these polyuronides more resistant to enzymatic degradation. Similarly, polygalacturonic acid is one of the main component of the aggregates.