MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

[¹⁸F]fluorination of biorelevant arylboronic acid pinacol ester scaffolds synthesized by convergence techniques

Aim: The development of small molecules through convergent multicomponent reactions (MCR) has been boosted during the last decade due to the ability to synthesize, virtually without any side-products, numerous small drug-like molecules with several degrees of structural diversity.(1) The association of positron emission tomography (PET) labeling techniques in line with the “one-pot” development of biologically active compounds has the potential to become relevant not only for the evaluation and characterization of those MCR products through molecular imaging, but also to increase the library of radiotracers available. Therefore, since the [18F]fluorination of arylboronic acid pinacol ester derivatives tolerates electron-poor and electro-rich arenes and various functional groups,(2) the main goal of this research work was to achieve the 18F-radiolabeling of several different molecules synthesized through MCR. Materials and Methods: [18F]Fluorination of boronic acid pinacol esters was first extensively optimized using a benzaldehyde derivative in relation to the ideal amount of Cu(II) catalyst and precursor to be used, as well as the reaction solvent. Radiochemical conversion (RCC) yields were assessed by TLC-SG. The optimized radiolabeling conditions were subsequently applied to several structurally different MCR scaffolds comprising biologically relevant pharmacophores (e.g. β-lactam, morpholine, tetrazole, oxazole) that were synthesized to specifically contain a boronic acid pinacol ester group. Results: Radiolabeling with fluorine-18 was achieved with volumes (800 μl) and activities (≤ 2 GBq) compatible with most radiochemistry techniques and modules. In summary, an increase in the quantities of precursor or Cu(II) catalyst lead to higher conversion yields. An optimal amount of precursor (0.06 mmol) and Cu(OTf)2(py)4 (0.04 mmol) was defined for further reactions, with DMA being a preferential solvent over DMF. RCC yields from 15% to 76%, depending on the scaffold, were reproducibly achieved. Interestingly, it was noticed that the structure of the scaffolds, beyond the arylboronic acid, exerts some influence in the final RCC, with electron-withdrawing groups in the para position apparently enhancing the radiolabeling yield. Conclusion: The developed method with high RCC and reproducibility has the potential to be applied in line with MCR and also has a possibility to be incorporated in a later stage of this convergent “one-pot” synthesis strategy. Further studies are currently ongoing to apply this radiolabeling concept to fluorine-containing approved drugs whose boronic acid pinacol ester precursors can be synthesized through MCR (e.g. atorvastatin)