Multiplicative Operator Splittings in Nonlinear Diffusion: From Spatial Splitting to Multiple Timesteps

Operator splitting is a powerful concept used in many diversed fields of applied mathematics for the design of effective numerical schemes. Following the success of the additive operator splitting (AOS) in performing an efficient nonlinear diffusion filtering on digital images, we analyze the possibility of using multiplicative operator splittings to process images from different perspectives.We start by examining the potential of using fractional step methods to design a multiplicative operator splitting as an alternative to AOS schemes. By means of a Strang splitting, we attempt to use numerical schemes that are known to be more accurate in linear diffusion processes and apply them on images. Initially we implement the Crank-Nicolson and DuFort-Frankel schemes to diffuse noisy signals in one dimension and devise a simple extrapolation that enables the Crank-Nicolson to be used with high accuracy on these signals. We then combine the Crank-Nicolson in 1D with various multiplicative operator splittings to process images. Based on these ideas we obtain some interesting results. However, from the practical standpoint, due to the computational expenses associated with these schemes and the questionable benefits in applying them to perform nonlinear diffusion filtering when using long timesteps, we conclude that AOS schemes are simple and efficient compared to these alternatives.We then examine the potential utility of using multiple timestep methods combined with AOS schemes, as means to expedite the diffusion process. These methods were developed for molecular dynamics applications and are used efficiently in biomolecular simulations. The idea is to split the forces exerted on atoms into different classes according to their behavior in time, and assign longer timesteps to nonlocal, slowly-varying forces such as the Coulomb and van der Waals interactions, whereas the local forces like bond and angle are treated with smaller timesteps. Multiple timestep integrators can be derived from the Trotter factorization, a decomposition that bears a strong resemblance to a Strang splitting. Both formulations decompose the time propagator into trilateral products to construct multiplicative operator splittings which are second order in time, with the possibility of extending the factorization to higher order expansions. While a Strang splitting is a decomposition across spatial dimensions, where each dimension is subsequently treated with a fractional step, the multiple timestep method is a decomposition across scales. Thus, multiple timestep methods are a realization of the multiplicative operator splitting idea. For certain nonlinear diffusion coefficients with favorable properties, we show that a simple multiple timestep method can improve the diffusion process.     

Studies of substrate specificities of lipases from different sources

Although lipases are widely applied in a wide variety of reactions, available information on the factors that are responsible for the substrate specificities of lipases from different sources is scarce. In this paper, nine lipase‐producing microorganism strains were isolated from oil‐containing samples. The properties of these enzymes, including pH optima, temperature optima, thermal stability, and pH stability, vary significantly with the different sources from which these lipases were isolated. The substrate specificities of the nine lipases, including fatty acid and positional specificities, were also studied. The fatty acid specificities of the nine lipases were observably different toward 15 substrates with different carbon chain lengths, different saturation degrees and different side chains. The lipases from S₃ Penicillium citrinum (PCL), MJ₁ Aspergillus niger (ANL), MJ₂ Aspergillus oryzae (AOL), YM Bacillus coughing (BCL), S₉ Geotrichum candidum (GCL), and S₁₁ Candida lypolytica (CLL) showed the strongest specificities to short‐chain esters, and the other lipases showed strong selectivity for medium‐ or long‐chain and branched esters. The positional specificities were examined by analyzing the hydrolytic products of triolein catalyzed by the nine lipases, using TLC. The lipases can be mainly classified into two groups by their specificities for the ester bond at position 2 of triglycerides; one group can selectively hydrolyze the ester bond at position 2 of the triglycerides, the other group cannot. All these nine lipases were divided into four groups by hierarchical clustering analysis on the basis of the results of fatty acid and positional specificities.     

Determination of the total concentration and speciation of Al(III) in tea infusions

The aluminium species in different tea infusions were investigated, by determining their stability constants and concentration. This was done for some particular samples using a simple experimental method based on the sorption of aluminium on the strongly sorbing resin Chelex 100, by a batch procedure. From the thermodynamic information obtained it is possible to calculate the concentration of the different species, and in particular that of the free metal ion, which is very important for evaluating the adsorption of aluminium on biological membranes. It was found that aluminium in the tea infusions here considered is present at high total concentration, approximately 0.1 mM, but mainly linked to strong complexes, for instance with side reaction coefficient higher than 10(5.11) at pH 3.95 in one case (tea 1). This could be the reason for the low toxicity of aluminium in tea. These strong complexes were not dissociated even in the presence of Chelex 100. In this case only a limiting value of the reaction coefficient could be evaluated. The presence of the very strong complexes was found in all the tea sample here considered. In two of the considered samples (one black and one green tea) a part of Al(III) was linked to less strong complexes, for example with a reaction coefficient 10(4.14) (tea 2, pH 4.20). The presence in the considered tea infusions of other substances able to complex aluminium was also detected, by the well known ligand titration procedure, at concentration ranging from 0.65 to 3.37 mM in three tea infusions, and at somewhat higher concentration in the case of the ready drink, which was also considered for comparison.     

Parametric and nonparametric Bayesian model specification: A case study involving models for count data

In this paper we present the results of a simulation study to explore the ability of Bayesian parametric and nonparametric models to provide an adequate fit to count data of the type that would routinely be analyzed parametrically either through fixed-effects or random-effects Poisson models. The context of the study is a randomized controlled trial with two groups (treatment and control). Our nonparametric approach uses several modeling formulations based on Dirichlet process priors. We find that the nonparametric models are able to flexibly adapt to the data, to offer rich posterior inference, and to provide, in a variety of settings, more accurate predictive inference than parametric models.     

Assessment of factors considered to be important in handling tasks by expert handlers

Twenty-eight experienced handlers, considered the best in two transport companies, were interviewed using a questionnaire consisting of partially closed-ended questions. They had to assess 20 videotape handling sequences showing two different handling methods to determine, for 10 factors and overall, which one was the best. They were asked about the inportance of these factors in their choices of the best method. The aims of the study were to verify whether there was a consensus among experienced handlers as to what was a good handling method and to identify the factors taken into consideration by them. Results show the lack of consensus on the best methods. Handlers favor different strategies depending on the worjk context and their own priorities.     

Influence of Two Salicylate Components on the Particle Size of an Oil-in-Water Emulsion with Nonionic Surfactants

This paper reports a study undertaken using techniques of static and dynamic light scattering to investigate the influence of sodium salicylate and methyl salicylate on droplet size of oil-in-water emulsions. The rates of changes were measured by determining the size and distribution of the oil droplet in the material. All emulsions showed a bimodal size distribution; the mean diameters and polydispersity were calculated from intensity. These data were analyzed with nonlinear regressions and bootstrap methodology. An amount of methyl salicylate component induced a decrease of mean diameter and standard deviation. On the contrary, sodium salicylate entailed the growth of all droplet populations and coalescence for the highest concentration.     

Stratified institutions and elementary homomorphisms

For conventional logic institutions, when one extends the sentences to contain open sentences, their satisfaction is then parameterized. For instance, in the first-order logic, the satisfaction is parameterized by the valuation of unbound variables, while in modal logics it is further by possible worlds. This paper proposes a uniform treatment of such parameterization of the satisfaction relation within the abstract setting of logics as institutions, by defining the new notion of stratified institutions. In this new framework, the notion of elementary model homomorphisms is defined independently of an internal stratification or elementary diagrams. At this level of abstraction, a general Tarski style study of connectives is developed. This is an abstract unified approach to the usual Boolean connectives, to quantifiers, and to modal connectives. A general theorem subsuming Tarski's elementary chain theorem is then proved for stratified institutions with this new notion of connectives.