Approximate dynamic programming for stochastic N-stage optimization with application to optimal consumption under uncertainty

Stochastic optimization problems with an objective function that is additive over a finite number of stages are addressed. Although Dynamic Programming allows one to formally solve such problems, closed-form solutions can be derived only in particular cases. The search for suboptimal solutions via two approaches is addressed: approximation of the value functions and approximation of the optimal decision policies. The approximations take on the form of linear combinations of basis functions containing adjustable parameters to be optimized together with the coefficients of the combinations. Two kinds of basis functions are considered: Gaussians with varying centers and widths and sigmoids with varying weights and biases. The accuracies of such suboptimal solutions are investigated via estimates of the error propagation through the stages. Upper bounds are derived on the differences between the optimal value of the objective functional and its suboptimal values corresponding to the use at each stage of approximate value functions and approximate policies. Conditions under which the number of basis functions required for a desired approximation accuracy does not grow “too fast” with respect to the dimensions of the state and random vectors are provided. As an example of application, a multidimensional problem of optimal consumption under uncertainty is investigated, where consumers aim at maximizing a social utility function. Numerical simulations are provided, emphasizing computational pros and cons of the two approaches (i.e., value-function approximation and optimal-policy approximation) using the above-mentioned two kinds of basis functions. To investigate the dependencies of the performances on dimensionality, the numerical analysis is performed for various numbers of consumers. In the simulations, discretization techniques exploiting low-discrepancy sequences are used. Both theoretical and numerical results give insights into the possibility of coping with the curse of dimensionality in stochastic optimization problems whose decision strategies depend on large numbers of variables.     

A novel enantiopure tricyclic β-lactam via stereoselective intramolecular nitrilimine cycloaddition

Starting from the Staudinger [2+2] cycloaddition between the 1-azadiene 1 and acetoxyacetylketene, generated in situ from acetoxyacetyl chloride, we developed a seven-step synthesis of the novel azeto[3′,4′:2,3]pyrano[4,5-c]pyrazole skeleton 9 in both racemic and enantiopure forms. Silver carbonate treatment of the functionalised hydrazonoyl chloride 7 promoted in situ generation of the corresponding nitrilimine 8, which underwent a clean, stereoselective cycloaddition to the target tricyclic β-lactam 9. The key step of the synthetic pathway leading to enantiopure 9 was the production of the enantiopure azetidinone (3R,4S)-3 by enzymatic resolution with Amano Lypase PS of the racemic precursor (3S1,4R1)-2.     

Preparative Isoelectric Focusing

Electrophoretic mobility has been used for years as a basis for separation and characterization of proteins. The resolving power of conventional electrophoresis is limited, and the characterization by mobility is not unique because of the dependence of the protein mobilities on the pH and ionic strength of the buffers.     

The First Synthesis of a Ribo-Hexos-5-Ulose: the L-Enantiomer

ABSTRACT

The title compound, previously unreported in either enantioform, and its 2,6-di-O-benzyl derivative have been synthesized through a stereocontrolled epimerization at C-2 of 6-O-protected methyl 3,4-O-isopropylidene-5-C-methoxy-β-D-galactopyranosides. The epimerization, performed through a high yielding sequence of oxidation-reduction owing to the cooperative role of the equatorial C-1 aglycon and the steric hindrance of the isopropylidene group, turned out to be completely diastereoselective. Whereas the unprotected L-ribo-hexos-5-ulose exists, as proved by NMR in D₂O, in five main tautomeric forms in a ratio of about 4:2:2:1:1, only two anomeric 1,4-furanosic forms are present at equilibrium in its 2,6-di-O-benzyl derivative, in ratios ranging from 10:1 to 7:3, depending on the prevalence of D₂O or CD₃CN in the solvent mixture.     

Preparation of L-Lyxo-Hexos-5-Ulose Through C-3 Epimerization of Bis-Glycopyranosides of L-Arabino-Hexos-5-Ulose1

Abstract

The unreported title compound and its 2,6-di-O-benzyl derivative have been prepared from methyl β-D-galactopyranoside through a sequence involving the bisglycoside methyl 2,6-di-O-benzyl-5-O-methoxv-β-D-galactopyranoside 8, the precursor of L-orabino-hexos-5-ulose, that was converted to the L-lyxo series by inversion at C-3. The inversion was achieved in acceptable yields by selective triflation, followed by displacement with benzoate, and by an oxidation/reduction sequence. Whereas 2,5-di-O-benzyl-L-lyxo-hexos-5-ulose exists entirely as a mixture of the two anomeric 1,4-furanosic forms, the unprotected hexos-5-ulose involves at equilibrium in CD₃CN/D₂O at least eight tautomers, one of which is predominant.     

Long time behavior of heat kernels of operators with unbounded drift terms

Given a second-order elliptic operator on Rd, with bounded diffusion coefficients and unbounded drift, which is the generator of a strongly continuous semigroup on L²(Rd) represented by an integral, we study the time behavior of the integral kernel and prove estimates on its decay at infinity. If the diffusion coefficients are symmetric, a local lower estimate is also proved.     

Fibre-optic pH sensor for sea-water monitoring using a single dye

A sensor for the continuous monitoring of pH in sea water, having the configuration of a probe, is described. The characteristics of this probe, which uses phenol red adsorbed on Amberlite resin as a chemichromatic element on top of an optical fibre, are reported and the effects of interfering parameters (variable salinity and temperature of sea water) are assessed. The possibility of enhancing the working range is discussed, and a procedure for rendering the range suitable for marine monitoring is presented.