From Geometry to Numerics: interdisciplinary aspects in mathematical and numerical relativity

This article reviews some aspects in the current relationship between mathematical and numerical General Relativity. Focus is placed on the description of isolated systems, with a particular emphasis on recent developments in the study of black holes. Ideas concerning asymptotic flatness, the initial value problem, the constraint equations, evolution formalisms, geometric inequalities and quasi-local black hole horizons are discussed on the light of the interaction between numerical and mathematical relativists.Comment: Topical review commissioned by Classical and Quantum Gravity. Discussion inspired by the workshop "From Geometry to Numerics" (Paris, 20-24 November, 2006), part of the "General Relativity Trimester" at the Institut Henri Poincare (Fall 2006). Comments and references added. Typos corrected. Submitted to Classical and Quantum Gravit

Numerical Study of the High-Contrast Stokes Equation and Its Robust Preconditioning

We numerically study the Stokes equation with high-contrast viscosity coefficients. The high-contrast viscosity values create complications in the convergence of the underlying solver methods. To address this complication, we construct a preconditioner that is robust with respect to contrast size and mesh size simultaneously based on the preconditioner proposed by Aksoylu et al. (Comput. Vis. Sci. 11:319-331, 2008). We examine the performance of our preconditioner against multigrid and provide a comparative study reflecting the effect of the underlying discretization and the aspect ratio of the mesh by utilizing the preconditioned inexact Uzawa and Minres solvers. Our preconditioner turns out to be most effective when used as a preconditioner to the inexact p-Uzawa solver and we observe contrast size and mesh size robustness simultaneously. As the contrast size grows asymptotically, we numerically demonstrate that the inexact p-Uzawa solver converges to the exact one. We also observe that our preconditioner is contrast size and mesh size robust under p-Minres when the Schur complement solve is accurate enough. In this case, the multigrid preconditioner loses both contrast size and mesh size robustness. © Springer Science+Business Media New York 2013.293978 DMS 1016190 National Science Foundation: 101619

Incorporation of opuntia spp. into food systems

Opuntia spp. has widespread species which are well-adapted to arid lands and climates over the world. This cactus plant was originated in America and then spread to other regions such as Europe, Africa, and Mediterranean countries. It is mainly cultivated for its seed, edible fruit (prickly pear), and cladodes. The nutritional and health benefits of the Opuntia genus are provided by various compounds such as phenolic compounds, pigments, polysaccharides, mucilage, dietary fibre, vitamins (B1, B2, A, and C), and minerals including magnesium, iron, calcium, potassium, and phosphorus. Owing to several health benefits, including prevention of diabetes, cancer, cardiovascular disease, and inhibition of inflammation, the number of studies focusing on developing novel foods and bioactive compounds by using different parts of this plant has increased recently. Several attempts have been made to integrate this plant into other foods, including bread, cake, pasta, gluten-free products, extrudates, cereal bars, juices, and meat products to improve their nutritional quality. In particular, Opuntia has gained importance as an excellent food source as desertification areas have increased and water resources have decreased globally. This chapter will discuss the recent studies dealing with the use of Opuntia spp. for edible purposes and the development of appropriate processing techniques to incorporate various parts of this valuable plant into other food matrices. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021

Optimisation of microwave effect on bioactives contents and colour attributes of aqueous green tea extracts by central composite design

The aim of the present study was to investigate the effects of microwave-assisted extraction parameters on total phenolic content (TPC), total flavonoid content (TFC), condensed tannins content (CTC), DPPH-scavenging activity, and colour attributes of aqueous green tea extracts and to optimise the microwaving conditions using response surface methodology. Microwave power (120–360–600 W) and irradiation time (1–3–5 min) were selected as independent variables. Statistical analysis revealed that all responses were significantly affected by extraction parameters. The optimum microwave-assisted extraction conditions were 350.65 W microwave power and 5 min irradiation time to maximise TPC, TFC, CTC, DPPH-scavenging activity, and L* values and to minimise a* and b* values of aqueous green tea extracts. The predicted TPC was 116.58 mg gallic acid equivalents/g (GAE/g), TFC was 49.33 mg catechin equivalents/g (CE/g), CTC was 9.89 mg catechin equivalents/g (CE/g), DPPH-scavenging activity (IC50) was 294.46 µg/mL, L* value was 12.31, a* value was 2.61 and b* value was 7.02 under optimum microwave-assisted extraction conditions. This study revealed that microwave extraction conditions should be controlled to obtain aqueous green tea extracts with both high bioactivity and acceptable colour quality. © 2020, Springer Science+Business Media, LLC, part of Springer Nature