Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains

Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solutions are under-explored. This paper aims at developing accurate spectral methods using rational basis (or modified mapped Gegenbauer functions) for such models in unbounded domains. The main building block of the spectral algorithms is the explicit representations for the Fourier transform and fractional Laplacian of the rational basis, derived from some useful integral identites related to modified Bessel functions. With these at our disposal, we can construct rational spectral-Galerkin and direct collocation schemes by pre-computing the associated fractional differentiation matrices. We obtain optimal error estimates of rational spectral approximation in the fractional Sobolev spaces, and analyze the optimal convergence of the proposed Galerkin scheme. We also provide ample numerical results to show that the rational method outperforms the Hermite function approach

Metaphorical Language Change Is Self-Organized Criticality

One way to resolve the actuation problem of metaphorical language change is to provide a statistical profile of metaphorical constructions and generative rules with antecedent conditions. Based on arguments from the view of language as complex systems and the dynamic view of metaphor, this paper argues that metaphorical language change qualifies as a self-organized criticality state and the linguistic expressions of a metaphor can be profiled as a fractal with spatio-temporal correlations. Synchronously, these metaphorical expressions self-organize into a self-similar, scale-invariant fractal that follows a power-law distribution; temporally, long range inter-dependence constrains the self-organization process by the way of transformation rules that are intrinsic of a language system. This argument is verified in the paper with statistical analyses of twelve randomly selected Chinese verb metaphors in a large-scale diachronic corpus

Pharmacological activities and pharmacokinetic study of hyperoside: A short review

Hyperoside (quercetin-3-O-D-galactoside) is a flavonol glycoside which has been  isolated from different plants. It has different pharmacological actions such as  anti-inflammatory, anti-depressant, neuroprotective, cardio-protective, anti-diabetic, anti-cancer, anti-fungal, radio-protective, gastro-protective, and antioxidant activities. Studies on its pharmacokinetic (PK) properties revealed that it is a stable compound with no significant gender variation in its activities. Other significant details on its pharmacological properties and information for future investigations on its components are provided.Keywords: Hyperoside; Anti-inflammatory, Antidepressant, Neuroprotective, Antidiabetic, Anticancer, Antioxidant, Cytochrome P45