Hierarchical Dobinski-type relations via substitution and the moment problem

We consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson ([a,a*]=1) monomials of the form exp(x (a*)^r a), r=1,2,..., under the composition of their exponential generating functions (egf). They turn out to be of Sheffer-type. We demonstrate that two key properties of these sequences remain preserved under substitutional composition: (a)the property of being the solution of the Stieltjes moment problem; and (b) the representation of these sequences through infinite series (Dobinski-type relations). We present a number of examples of such composition satisfying properties (a) and (b). We obtain new Dobinski-type formulas and solve the associated moment problem for several hierarchically defined combinatorial families of sequences.Comment: 14 pages, 31 reference

Insight into the evolution of the proton concentration during autohydrolysis and dilute-acid hydrolysis of hemicellulose

Background: During pretreatment, hemicellulose is removed from biomass via proton-catalyzed hydrolysis to produce soluble poly- and mono-saccharides. Many kinetic models have been proposed but the dependence of rate on proton concentration is not well-defined; autohydrolysis and dilute-acid hydrolysis models apply very different treatments despite having similar chemistries. In this work, evolution of proton concentration is examined during both autohydrolysis and dilute-acid hydrolysis of hemicellulose from green bamboo. An approximate mathematical model, or “toy model”, to describe proton concentration based upon conservation of mass and charge during deacetylation and ash neutralization coupled with a number of competing equilibria, was derived. The model was qualitatively compared to experiments where pH was measured as a function of time, temperature, and initial acid level. Proton evolution was also examined at room temperature to decouple the effect of ash neutralization from deacetylation. Results The toy model predicts the existence of a steady-state proton concentration dictated by equilibrium constants, initial acetyl groups, and initial added acid. At room temperature, it was found that pH remains essentially constant both at low initial pH and autohydrolysis conditions. Acid is likely in excess of the neutralization potential of the ash, in the former case, and the kinetics of neutralization become exceedingly small in the latter case due to the low proton concentration. Finally, when the hydrolysis reaction proceeded at elevated temperatures, one case of non-monotonic behavior in which the pH initially increased, and then decreased at longer times, was found. This is likely due to the difference in rates between neutralization and deacetylation. Conclusions The model and experimental work demonstrate that the evolution of proton concentration during hydrolysis follows complex behavior that depends upon the acetyl group and ash content of biomass, initial acid levels and temperature. In the limit of excess added acid, pH varies very weakly with time. Below this limit, complex schemes are found primarily related to the selectivity of deacetylation in comparison to neutralization. These findings indicate that a more rigorous approach to models of hemicellulose hydrolysis is needed. Improved models will lead to more efficient acid utilization and facilitate process scale-up.Applied Science, Faculty ofForestry, Faculty ofNon UBCChemical and Biological Engineering, Department ofWood Science, Department ofReviewedFacult